Flocks, Generalised Quadrangles and Hyperovals

Ivano Pinneri

This thesis is presented for the degree of

Doctor of Philosophy of The University of Western Australia

Department of Mathematics June 1996

Acknowledgments

Special thanks goes to Tim Penttila, my supervisor and one time drinking partner (and I have the medical certificates to prove it) and a pretty smart cookie to boot who has made “linking” my eternal quest of mathematics.

To my dearest friend Nick. Life on the PhD trail would have been pretty lonely and down right boring without you. May all your toast be crackin’ and all your dwarves red.

A very special thanks to Gordon — he of the many questions and great programming flair. For being my defacto supervisor and much much more I thank you with a nice bottle of SB in my hand!

A big thanks to the secretaries for allowing me to disrupt their routine when I needed someone to talk to, keeping me sane while I wrote up and allowing me to have the room to myself after Nick went to Brisbane,^{1} amongst other things!

Thanks to the Adelaide gang: Matt Brown and Cath Quinn for showing me that you can be human and still be in the PhD game; and especially Christine O’Keefe for her help and occasional confidence boosting.

Big thanks to Bill Cherowitzo, Franck De Clerck, Marialuisa de Resmini, Stan Payne and Jef Thas for making me feel like I’m actually doing something of importance and for just being nice!

Thanks go to my flatmate par excellence Cathy, the Message Moose and Kenneth. Oh! and Mum. ii

1The best place in the world, I hear.

Preface

In 1959 structures called generalised quadrangles were introduced by J. Tits [132] and much later Stan Payne [92] showed how certain generalised quadrangles of order (q^{2}, q) were related to a set of 2 × 2 matrices. P. Dembowski [36] popularised the term flocks in relation to a partition of an inversive plane, minus two points, into a set of disjoint circles. In turn the idea of partitioning quadrics of P G(3, q) into disjoint conics occured. The concept of oval, or at least the study of these configurations in themselves, may have started with the paper [8] of Baer. The work expanded from the study of Bose [16] and a number of Finnish astronomers, most notably the work of Qvist [114]. Under the guide of Beniamino Segre and the numerous number of Italians who followed him, the area broadened in scope. Segre’s theorem classifying ovals for q odd stimulated the area even further with most of the initial effort to proof a similar theorem for q even. Needless to say we are still working on that problem, (amongst others).

The relationship between generalised quadrangles and hyperovals originated (in the present form) in Stan Payne’s paper [93] and the relationship between flocks of quadratic cones and generalised quadrangles by the work of Jef Thas [127], Bill Kantor [67] and Stan Payne [93]. Underlying these objects are the matrices forming a q–clan, a term invented by Payne. Together Thas, Kantor and Payne have made q–clans the central idea behind these structures to study.

This thesis deals more fully with the relationships between generalised quadrangles, flocks and hyperovals. The links have been investigated and added to, in particular with the introduction of the concept of a herd of ovals; concluding in the construction of the Subiaco family of ovals, flocks of quadratic cone, etc. for all q even.

The use of computers has been of extreme importance in this work. Most notably the construction of the first irregular hyperovals in P G(2, 64) by Penttila and myself, and the introduction of the technique of “prime at a time”. Special mention should go to Gordon Royle who pushed most of the computing techniques described here (and in the many papers co-authored by Royle) to present computing^{2} power; and allowed me to extensively quote the results from these papers.

2It should be affordable computing power, both in time and money resources.

The thesis itself is in straightforward chapters. Unfortunately, due to the nature of this work a large amount of forward and backward referencing is made. It is virtually impossible to write this work without doing so.

Chapter I and II initially started as a Masters thesis, with a view of trying to write a definitive work on hyperovals in projective planes (Desarguesian and non-Desarguesian). I then transferred to a PhD and the original aim of the masters was discarded. But most of the original work on hyperovals has been included and an abridged version of the original notes on group theory as well. In these chapters most results are explained in detail, while the last two deal with the subject in a less thorough manner.

The main influences on this work have been: the paper [93] by Stan Payne, the book [103] by Jef Thas and Payne and the series of lectures Thas gave in 1992 at the University of WA. Also in need of special mention is Payne’s work on the Subiaco constructions, most notably his feedback with proofs and counterexamples to Penttila’s and my conjectures on the hyperovals in P G(2, q), q = 4^{e}, e odd. Payne’s Subiaco Notebook [90] were a source of great help in understanding the creature Bill Cherowitzo, Penttila, Royle and myself brought onto the world.

(The reader should be aware that Section 11 on page 159 gives a historical perspective on the ideas, and influences, of the Subiaco constructions.)

When I started this area Dembowski [36] and the first two volumes of projective geometries by Hirschfeld [55][56] were about the only decent books to find on projective geometry. Today the amount of material on the area has grown substantially. Of particular importance is the text called “The Handbook” [24]. The chapters by Thas (Chapter 8 and Chapter 9) and F. De Clerck and H. van Maldeghem (Chapter 10) in the Handbook are a main reference for this work.

The reader should be warned that for Chapters I, II and IV, groups are acting on the left, but in Chapter III they are acting on the right! The reason for this change in handiness is given on page 94.

Nearly all of the work in this thesis is collaborative. In particular, special mention must go to Tim Penttila, my supervisor. ALL results have his influence and most of them contain his original ideas. I have given credit to original sources for theorems that have not been published elsewhere.

In summary, this work investigates the connection between hyperovals, ovals, flocks of quadratic cones and generalised quadrangles and their subquadrangles.

Summary

This thesis is divided into five chapters:

Chapter 1: Introduction Definitions of projective plane and spaces, generalised quadrangles, duality and other relevant material of an introductory nature are included (and in fact can be skipped for the more able reader). A small collection of group theoretical results are given and the important “Samosa Lemma” is proved.

Chapter 2: Hyperovals in Desarguesian planes This chapter is an attempt at a comprehensive look at hyperovals in Desarguesian planes. Firstly, many results are stated for ovals and collineation groups of ovals and hyperovals in arbitrary projective planes. Results on when hyperovals can be stabilised by elations and generalised homologies are given. Secondly, a list and some properties of the known hyperovals are given. From then on most of the chapter is devoted to constructing the first two irregular hyperovals in P G(2, 64). The rest of the chapter is devoted to the very recent construction of hyperovals in other Desarguesian projective planes. The last section deals with related areas of hyperovals.

Chapter 3: Generalised Quadrangles We first look at the parameters of a generalised quadrangle, classical GQs, regularity and other properties of GQs. Ovoids, spreads, subquadrangles and the concept of rosettes is introduced. Four-gonal families and q-clans are introduced with some pointers to the new ideas of tensor q-clan geometries. A specific look is given at rosettes and subquadrangles of the GQ of order (q, q^{2}) with T_{2}(O), O a pointed conic of P G(2, 8), as a subquadrangle. The concept of herd is introduced and q– clan geometries and elation generalised quadrangles are described. The groups of the Subiaco GQs are investigated and the proof that there are two inequivalent hyperovals for q = 4^{e}, e > 1, odd is given.

Chapter 4: Flocks and Chapter 5: Open questions Flocks of quadrics are discussed, with the rest of the introductory sections being devoted to flocks of cones, especially quadratic cones and a brief mention of translation oval cones. Most of the section is taken up with proofs of the Subiaco herds. A brief historical view of the Subiaco constructions forms the last part of the thesis with some open questions about hyperovals, flocks, generalised quadrangles and their links. vi

Contents

Acknowledgments i Preface iii Summary v Chapter I. Introduction 1 1. Projective and Affine spaces 1 2. Duality 2 3. Projective Spaces and Collineation Groups 2 4. Group theory results 4 5. The Exponential Convention 9 6. The Trace Equation 9 7. Spreads and Reguli 10 8. Bruck-Bose Construction 10 9. Generalised Quadrangles 11 10. End Notes 14 Chapter II. Hyperovals in Desarguesian Planes 15 1. Arcs in Arbitrary Projective Planes 15 2. Conics 18 3. Collineation Groups 23 3.1. Fixed Configurations 24 3.2. Collineations of P G(2, q) 32 3.3. Fixed configurations for P GL(3, q) 37 3.4. Subgroups of P GL(3, q) 38 3.5. Polarities 40 4. Hyperovals in P G(2, q) 41 5. Glynn’s Inequality 43 5.1. Lagrange Interpolation 43 5.2. Glynn’s inequality 44 6. Known Hyperovals 46 6.1. Regular hyperovals 46 6.2. Translation hyperovals 47 6.3. Lunelli-Sce hyperoval 47 6.4. Segre-Bartocci hyperoval 49 6.5. Glynn hyperovals 50 6.6. Payne hyperoval 50 6.7. Cherowitzo hyperoval 51 6.8. O’Keefe-Penttila hyperoval 53 6.9. Irregular hyperovals in P G(2, 64) 54 7. Fano Configuration 54 8. Irregular Hyperovals in P G(2, 64) 55 8.1. Hyperovals, H, in P G(2, 64) with |P Γ L(3, 64)_{H}| divisible by 13. 55 8.2. Hyperovals, H, in P G(2, 64) with |P Γ L(3, 64)_{H}| divisible by 7. 56 8.3. Hyperovals, H, in P G(2, 64) with |P Γ L(3, 64)_{H}| divisible by 5. 57 9. Searching Algorithms for Hyperovals 62 9.1. More on Hyperovals in P G(2, 64) 65 10. A Quick Look in P G(2, q), q > 64 68 11. Applications of Hyperovals 69 11.1. Generalised Quadrangles 69 11.2. Codes 69 11.3. Ovoids 71 11.4. Inversive Planes 74 11.5. Partial Geometries and Strongly Regular Graphs 75 11.6. B-ovals 76 11.7. Howell Designs 76 12. End Notes 77

Chapter III. Generalised Quadrangles 81 1. Parameters and Regularity 81 2. Some examples of generalised quadrangles 83 2.1. Sylvester’s Syntheme-Duad geometry 83 2.2. T_{2}^{∗}(H) 84 2.3. T_{2}(O) 84 2.4. P (𝒮 , x) 86 2.5. T_{3}(Ω) 86 2.6. Other GQs from arcs 87 3. Classical Generalised Quadrangles 87 3.1. Trace and span in W (q) 88 3.2. Regularity of W (q) 88 4. Ovoids and Spreads 89 5. Isomorphisms of some GQs 91 6. Subquadrangles 91 6.1. Rosette of ovoids 93 6.2. Automorphisms 94 7. 4-gonal families and EGQs 94 8. Constructing GQs of order (q^{2}, q) 97 8.1. Incidence of 𝒮 (G, ℱ) of order (q^{2}, q) 98 9. q–clan geometry 99 9.1. Herds and q–clans 100 10. Elation Generalised Quadrangles from q–clans 102 10.1. Ovoids from q–clans 103 11. Subquadrangles of EGQs 104 12. A view of the Subiaco q–clan 105 13. Fundamental Theorem of q–clan Geometry 105 14. Subiaco q–clans 108 14.1. Collineations of the Subiaco q–clan 108 14.2. Subiaco q–clans, for q = 4^{e}, e odd 110 14.3. Ovals for q = 4^{e}, e odd 111 14.4. Ovals for q = 4^{e}, e odd, 5|(e/2) 113 14.5. Remarks on the Subiaco hyperovals 119 15. Tensor q–clan geometries 119 15.1. Cyclic q–clans 121 16. Subquadrangles in GQs of order (s, s^{2}) 122 16.1. Flocks and subquadrangles 123 17. Generalised quadrangles of order (8, 64) with a subquadrangle of order 8 124 17.1. Preliminaries 125

17.1.1. Lines of the form A(t)g 126 17.1.2. Lines of the form A(∞)g 126

17.2. Fisher-Thas-Walker-Kantor-Betten q–clan 127 17.2.1. Lines of type A(t)g 127 17.2.2. Lines of type A(∞)g 128

17.3. Constructing ovoids from spreads 129 17.3.1. Ovoids from lines of type A(t)g 130 17.3.2. Ovoids from lines of type A(∞)g 131

17.4. Ovoids from the polarity 131 17.5. Constructing GQ(q, q^{2}) from T_{2}(O): FTWKB case 132 17.6. Subtended ovoids from T_{3}(Ω) 132

17.6.1. Planar ovoids 134 17.7. Constructing GQ(q, q^{2}) from T_{2}(O): FTWKB and T_{2}(Ω) cases 134 17.8. Ovoids of T_{2}(O), O pointed conic, q = 8 134 18. End Note 136

Chapter IV. Flocks 141 1. Introduction 141 2. Translation Planes from Flocks of Quadrics 141 3. Classification of Flocks of Ovoids 143 4. Classification of Flocks of a Hyperbolic Quadric 143 5. Flocks of Cones 144 6. Flocks of QuadraticCones 145 7. Derivation of flocks of quadratic cones in P G(3, q), q odd 146 8. Known q–clans for q Even 147 9. The Subiaco q–clans 149 9.1. The case q = 2^{e} where e is odd. 149 9.2. The case q = 4^{e}, where e is odd 151 9.3. The case q = 16^{e} where e is odd 153 9.4. Existence of Subiaco q–clans, in general 154 10. Concluding Remarks 158 10.1. The Subiaco q–clans 158 10.2. The Subiaco flocks 159 10.3. The Subiaco planes 159 10.4. The Subiaco hyperovals 159 11. History of the Subiaco Constructions 159 12. End Notes 161 Chapter V. Open Problems 163 Bibliography 165 Index 171

CHAPTER I Introduction

1. Projective and Affine spaces

Throughout this work we will be dealing with incidence structures and some of the configurations within them. An incidence structure, ℐ, is a triple of objects, (𝒫, ℬ, I), where 𝒫 is a set of points, ℬ is a set of blocks, with 𝒫 ∩ ℬ = ∅ and 𝒫, ℬ are non-empty, and I is the relation I⊆ 𝒫 × ℬ called the incidence relation. We define the dual of an incidence structure ℐ = (𝒫, ℬ, I), denoted ℐ̂, with ℐ̂= (ℬ, 𝒫, Î) where Î = {(B, P) | (P, B) ∈I}.

We define a projective plane as an incidence structure ℐ = (𝒫, ℒ , I), where ℒ is a set of lines, such that ℐ satisfies these axioms:

PP1: Any two distinct points are incident with a unique line. PP2: Any two distinct lines are incident with a unique point. PP3: There exists at least four points, no three collinear.

Theorem 1 ([46]). If ℐ is a projective plane then so is ℐ̂.

A finite projective plane is a projective plane with 𝒫 and ℒ being finite.

Theorem 2 ([46]). Let q ≥ 2 be an integer. In a projective plane, π, any one of the following implies the rest.

(i) One line contains exactly q + 1 points.

(ii) One point is on exactly q + 1 lines.

(iii) Every line contains exactly q + 1 points.

(iv) Every point is on exactly q + 1 lines.

(v) There are q^{2} + q + 1 points in π.

(vi) There are q^{2} + q + 1 lines in π.

If any of the conditions of Theorem 2 are satisfied then the projective plane, π, has order q.

Theorem 3 ([46]). A finite projective plane has order q, for some q.

Similar treatment can be given to incidence structures called affine planes. This is an incidence structure ℐ = (𝒫, ℬ, I) such that;

A1: Any two distinct points are incident with a unique line.

A2: Given a point P and a non-incident line l then there exists a unique line m

incident with P but disjoint from l. A3: There exists three points not all incident with the same line.

Two lines, l and m are parallel if they are equal or disjoint. Parallelism is an equivalence relation.

A finite affine plane is one where both 𝒫 and ℒ are finite sets.

It is well known [36] that one can get a projective plane from an affine plane and an affine plane from a projective plane.

2. Duality

We take a closer look at duality. Recall that the dual of an incidence structure ℐ = (𝒫, ℬ, I) is ℐ̂= (ℬ, 𝒫, Î) where Î = {(B, P) | (P, B) ∈I}.

A finite projective plane, π, can be defined as a bipartite graph of girth 6 and diameter 3 (and a thickness condition), where one set of vertices are labelled points and the other set of vertices are labelled lines.

Given a theorem about finite projective planes if we swap the words “points” and “lines” (and reword incidence accordingly), then this theorem is now concerned with a bipartite graph where the labels of “points” and “lines” have been interchanged. Hence it is still a theorem about projective planes (with the same order).

We will use this fact throughout this work and say dually, to mean swap the words “lines” and “points” from a previous result to obtain the current result.

3. Projective Spaces and Collineation Groups

Let V be a finite dimensional vector space over a field 𝔽. The projective space, P G(V), associated with V is the space whose elements are proper subspaces of V and incidence is symmetrised inclusion. We have points being the one dimensional subspaces of V, lines being the two dimensional subspaces of V, planes being the three dimensional subspaces of V,…, and hyperplanes being the dim_{𝔽} V − 1 subspaces of V. The projective dimension of a subspace W of V is dim_{𝔽} W − 1. The projective dimension of P G(V) is dim_{𝔽} V − 1. Elements of P G(V) are called subspaces of the projective space. An isomorphism between projective spaces is a bijection preserving projective dimension and symmetrised inclusion.

Theorem 4. Let V, W be vector spaces. If V ≅ W then P G(V) ≅ P G(W) as projective spaces.

We write P G(d, 𝔽) for P G(𝔽^{d+1}) and P G(d, q) for P G(GF (q)^{d+1}).

The points of P G(d, q) are one dimensional subspaces ⟨(a_{1}, …, a_{d+1})⟩ of GF (q)^{d+1}. We use homogeneous coordinates for points, ⟨(a_{1}, …, a_{d+1})⟩, abbreviated to

(a_{1}, …, a_{d+1}) where (a_{1}, …, a_{d+1}) ≠ 0.

Hence

(λa_{1}, …, λa_{d+1}) = (a_{1}, …, a_{d+1}) for λ ∈ GF (q), λ ≠ 0.

The set of all invertible linear maps V → V is denoted GL(V). The set of all semilinear maps V → V is denoted by Γ L(V). We see that GL(V) ◁ Γ L(V) and

Aut 𝔽 ≅ \frac{Γ L(V)}{GL(V)}

where 𝔽 is the field V is defined over. Let Z = {λI | λ ∈ 𝔽 \ {0}} then Z is normal in Γ L(V) and GL(V). An automorphism of P G(V) is an isomorphism P G(V) → P G(V). We define Aut P G(V) to be the set of all automorphisms, and is a group under composition.

By Theorem 4, p. 3, we have any A ∈ GL(d + 1, q) gives an automorphism

S ↦ AS for all subspaces S of P G(d, q),

of P G(d, q). Automorphisms of this form are called homographies. They form a group, P GL(d + 1, q),

P GL(d + 1, q) ≅ \frac{GL(d + 1, q)}{Z}.

For an automorphism, α, of GF (q), α acts on GF (q)^{d+1} by
α(a_{1}, …, a_{d+1}) = (α(a_{1}), …, α(a_{d+1})),

so α acts on subspaces S of GF (q)^{d+1} by

α(S) = {α(v) | v ∈ S}

and

S ↦ α(S) for all subspaces S of P G(d, q)

is an automorphism of P G(d, q), an automorphic collineation. The automorphic collineations and homographies form a group

S ↦ Aα(S) for all subspaces S of P G(d, q)

called P Γ L(d + 1, q), the collineations of P G(d, q),

P Γ L(d + 1, q) ≅ \frac{Γ L(d + 1, q)}{Z}.

Let SL(d, q) be the subgroup of elements of GL(d, q) with determinant 1. Then we can define a subgroup, P SL(d, q), of P GL(d, q) as:

P SL(d, q) ≅ \frac{SL(d, q)Z}{Z} ≅ \frac{SL(d, q)}{SL(d, q) ∩ Z} .

For reference we list the orders of the above groups:

|GL(d, q)| = q^{d(d−1)/2} ∏_{i=1}^{d}(q^{i} − 1)

|P GL(d, q)| = q^{d(d−1)/2} ∏_{i=2}^{d}(q^{i} − 1) = \frac{|GL(d, q)|}{q − 1}
|P SL(d, q)| = \frac{q^{d(d−1)/2} ∏_{i=2}^{d}(q^{i} − 1)}{(d, q − 1)} = \frac{|P GL(d, q)|}{(d, q − 1)}
|P Γ L(d, q)| = |P GL(d, q)|| Aut GF (q)|

Theorem 5 (Fundamental Theorem of Projective Geometry). For d > 1 we have

Aut P G(d, q) = P Γ L(d + 1, q).

An example of a projective space is the Desarguesian projective plane P G(2, q). This incidence structure satisfies axioms PP1, PP2, and PP3 and so is a projective plane. These projective planes satisfy the Theorem of Desargues (see [59]). We label points homogeneously as for projective spaces, and we label lines in P G(2, q) homogeneously, so that the line

ax + by + cz = 0 for points (x, y, z)

is denoted [a, b, c] such that for any λ ∈ GF (q)^{∗} we have [a, b, c] = [λa, λb, λc].

We can construct an affine plane on a two dimensional vector space. Points are vectors of the space, lines are cosets of one-dimensional subspaces and incidence is inclusion. When the vector space is GF (q)^{2} we call this affine space AG(2, q).

4. Group theory results

Lemma 6. Let f be a polynomial defined over GF (q). If x − a|f(x) then x − a| f(a).

Proof. By the remainder theorem we have

f(x) = g(x)(x − a) + f(a)

hence if x − a divides f(x) then it must divide f(a).

Theorem 7 ([136]). Let H and K be subgroups of G with |G : H| = m and |G : K| = n and (m, n) = 1. Then |K : H ∩ K| = m and |H : H ∩ K| = n.

Corollary 8. Let G act transitively on X, let H = G_{x} for some x ∈ X and |X| = n. If (|G : H|, n) = 1 then H is transitive on X.

Let G be a group and H ≤ G. The normaliser of H in G is denoted N_{G}(H), and the centraliser of H in G is denoted C_{G}(H).

Theorem 9. Let G act on X and H ≤ G. The normaliser of H in G permutes orbits of H on a set X of size m for all m ∈ ℤ^{+}. Particularly, N_{G}(H) permutes fixed points.

Theorem 10. Let H ◁ G. Then

(i) C_{G}(H) ◁ N_{G}(H),

(ii) N_{G}(H) ≤ N_{G}(C_{G}(H))), and

(iii) C_{G}(H) ≤ N_{G}(C_{G}(H))).

Theorem 11. The number of conjugates of S under H is the index in H of the normaliser of S in H.

Theorem 12. Let G be a group acting transitively on the set X. Then G is primitive on X if and only if G_{x} is a maximal subgroup of G, for all x ∈ X.

Theorem 13. G acts on X and let α ∈ X. If N is regular on X then G = NG_{α}, and we say G_{α} is a supplement for N in G.

If N was normal, then G = N ⋊ G_{α} and G_{α} is a complement for N in G.

The number of subsets of G conjugate to S is |G : N_{G}(S)|.

The number of elements in the conjugacy class that contains a is |G : C_{G}(a)|.

Theorem 14 (Cauchy’s Theorem). Let G be a finite group and p be a prime divisor of |G|. Then G contains an element of order p.

A subgroup H of a group G is called a p-subgroup of G if H is a p-group. Let G be a finite group of order p^{r}m where r ≥ 1 and p does not divide m. A p-subgroup H of G is called a Sylow p-subgroup of G if |H| = p^{r}, that is, if |G : H| is coprime to p.

Theorem 15 (Sylow’s Theorem). Let G be a finite group of order p^{r}m where r ≥ 1 and p does not divide m. Then

(i) Sylow p-subgroups of G exist; (ii) any two Sylow p-subgroups of G are conjugate in G; (iii) if H is any p-subgroup of G then H is contained in some Sylow p-subgroup of

G; (iv) the number of Sylow p-subgroups of G divides m and is congruent to 1 modulo

p.

The set of all Sylow p-subgroups of G is denoted by Syl_{p}(G).

Corollary 16. Let P ∈ Syl_{p}(G). Then P ◁ G if and only if P is the unique Sylow p-subgroup of G.

Corollary 17 (Frattini Argument). Let H ◁ G and P ∈ Syl_{p}(H). Then

G = HN_{G}(P).

Theorem 18. Let p be a prime and p||G|. Let P ∈ Syl_{p}(G). If P is cyclic then there exists a unique conjugacy class of subgroups of G of order p^{i} for all i such that p^{i} ||G|.

The below theorem is one of the most widely used in this work. Unfortunately, most references are obscure, either they are in German [61], or hard to find [64]. The proof of the theorem requires some knowledge of representation theory (see [33]) and primitive divisors. The so called Samosa Lemma is used to calculate N_{Γ L(d,q)}(P) for any p-subgroup of GL(d, q) where p is a q-primitive prime divisor of q^{e} − 1 for e > d/2. We concentrate on the case |P | = p, the others follow as a corollary.

A q-primitive prime divisor, or q-ppd, of q^{d} − 1 is a prime divisor of q^{d} − 1 which does not divide q^{e} − 1 for 1 ≤ e < d. If p is a q-ppd of q^{d} − 1 then (p, q) = 1 so q ∈ U(ℤ_{p}) has an order. Let |q|_{p} be the order of q as an element of U(ℤ_{p}). Then |q|_{p} = d (obviously), hence p|q^{i} − 1 if and only if d|i.

A q-primitive divisor, or q-pd of q^{d} − 1 is a divisor which is coprime to q^{e} − 1 for 1 ≤ e < d (so q-ppd if and only if q-pd and prime). Let Φ_{d}^{∗}(q) be the largest q-primitive divisor of q^{d} − 1, the q-primitive part of q^{d} − 1. Let

Φ_{n}(x) = \frac{x^{n} − 1}{(x^{n} − 1, ∏_{i=1}^{n−1}(x^{i} − 1))}

be the nth cyclotomic polynomial (see [50][51]). Then we have Φ_{d}^{∗}(q) | Φ_{d}(q), so if l is a prime and l |Φ_{d}(q) but l | Φ̸_{d}^{∗}(q), then l |d.

Theorem 19 ([138]). We have

Φ_{d}^{∗}(q) = \frac{1}{f^{α}} Φ_{d}(q),

where f = (d, Φ_{d}(q)) and f^{α} | Φ_{d}(q) but f^{α+1} | Φ̸_{d}(q). In particular, Φ_{d}^{∗}(q) ≠ 1 for d ≥ 2 unless (d, q) equals (6, 2) or (2, 2^{a} − 1) for some a.

Suppose a prime p | |GL(d, q)| with p | q̸ then there exists an e ≤ d such that p | q^{e} − 1. If we choose e to be minimal such that p | q^{e} − 1 implies that p is a q-ppd of q^{e} − 1. So primes not the characteristic of q can be classified by e.

Suppose (d/2) < e. Then |q|_{p} = e with p | q^{i} − 1 if and only if e | i, in fact, since (d/2) < e, we have that p|q^{i} − 1 for some i, 1 ≤ i ≤ d, if and only if e = i. Hence

(p, \frac{|GL(d, q)|}{q^{e} − 1}) = 1, if |q|_{p} = e > \frac{d}{2}. (1)

Let V = GF (q)^{d}. Let U be a subspace of dimension e, with W a complement of U in V such that V = U ⊕ W. So U ≅ GF (q)^{e} and identify U with GF (q^{e}). Multiplication in GF (q^{e}) is GF (q)-linear, thus there exists a C_{q^{e}−1} ≤ GL(U) via Singer’s Theorem (page 27). We can extend this linearity to the whole space (namely C_{q^{e}−1} × I_{W}) so there exists a C_{q^{e}−1} ≤ GL(V).

Let P ∈ Syl_{p}(C_{q^{e}−1}). By Eq. (1) we have P ∈ Syl_{p}(GL(d, q)). All p-subgroups of GL(d, q) are cyclic; any two of the same order are conjugate: Since if P_{1}, P_{2} are p-subgroups of GL(d, q) of equal order implies that there exists P_{1}^{∗} and P_{2}^{∗} from Syl_{p}(GL(d, q)) with P ≤ P_{1}^{∗} and P_{2} ≤ P_{2}^{∗} by Sylow’s Theorem which, also by Sylow’s Theorem, implies there exists a g ∈ GL(d, q) such that gP_{1}^{∗}g^{−1} = P_{2}^{∗} so gP_{1}g^{−1} ≤ P_{2}^{∗} of order |P_{2}|. So P_{2}^{∗} is cyclic implies that there is a unique subgroup of each order dividing |P_{2}^{∗}|. Hence gP_{1}g^{−1} = P_{2}.

A representation of a group G is a homomorphism G → GL(V) for a vector space V. Two representations T_{1} : G → GL(V), T_{2} : G → GL(W) are equivalent if there exists an invertible linear map φ: V → W such that

φT_{1}(g)φ^{−1} = T_{2}(g) for all g ∈ G

that is,

φT_{1}(g) = T_{2}(g)φ for all g ∈ G.

An intertwiner (or G-endomorphism) between T_{1} and T_{2} is a linear map φ: V → W such that

φT_{1}(g) = T_{2}(g)φ for all g ∈ G.

(We note that the zero map is always an intertwiner.)

A subspace is G-invariant if for all g ∈ G we have T (g)W = W.

A representation T: G → GL(V) is reducible if there is a non-zero, proper subspace W of V such that W is G-invariant. Otherwise we say T is irreducible.

Let T: G → GL(V) be a representation, then

End_{G}(V) = {intertwiners between T and T }

is a division ring under addition and composition by Schur’s Lemma [72]. Now

U(End_{G}(V)) = {φ ∈ GL(V) | φT (g) = T (g)φ for all g ∈ G}
= C_{GL(V)}(T (G))

Theorem 20 (Samosa Lemma). Let q and d be integers strictly greater than 1. Let p be a q-ppd of q^{e} − 1 with e > (d/2). Let P be a p-subgroup of GL(d, q). Then

N_{GL(d,q)}(P) ≅ (GF (q^{e})^{∗} × GL(d − e, q)) ⋊ Gal (\frac{GF (q^{e})}{GF (q)}) .
N_{Γ L(d,q)}(P) ≅ (GF (q^{e})^{∗} × GL(d − e, q)) ⋊ Aut GF (q^{e}).
C_{GL(d,q)}(P) ≅ GF (q^{e})^{∗} × GL(d − e, q).

Proof. We have P acting on V = U ⊕ W with dim U = e as gw = w for all g ∈ P and for all w ∈ W. Now GF (q^{e})^{∗} acts regularly on U − {0} implies that P acts semiregularly on U − {0} which implies that all orbits of P on U − {0} have length |P |. (Noting that p divides q^{i} − 1 if and only if e divides i for i > 0.) Let {0} ≠ U′ ≤ U be P -invariant. Then |U′| = 1 + pn for some n. Let e′ = dim U′ ≠ 0. Then q^{e′} − 1 = pn which implies p divides q^{e′} − 1 which implies e divides e′. But e′ ≤ e so e′ = e. Hence P is irreducible on U (strictly speaking T (P) is).

Now N_{Γ L(V)}(P) permutes fixed vectors of P, so N_{Γ L(V)}(P) fixes W. Also N_{Γ L(V)}(P) permutes the fixed subspaces of P and preserves irreducibility, so N_{Γ L(V)}(P) fixes U (there are no other irreducible subspaces of dimensions not equal to one).

We now look at the action of N_{Γ L(V)}(P) on U.

What is the kernel of this action? The kernel acts trivially on U so implying that its companion automorphism is 1, hence kernel is a subgroup of GL(V). Fix W and we have the kernel a subgroup of I_{U} × GL(W). Now

I_{U} × GL(W) ≤ C_{GL(V)}(P) ∩ kernel.

Hence the kernel of the action is equal to I_{U} × GL(W).

Now we look at the action induced on U. Without loss of generality let V = U and d = e. By Schur’s Lemma [72], End_{P} (U) is a division ring and so by Wedderburn’s Theorem [72], End_{P} (U) is a field. Hence End_{P} (U) ≅ GF (q^{e}). (Since End_{P} (U)^{∗} is cyclic and so abelian, it acts faithfully on V implying that it acts semiregularly on U − {0}. So we have | End_{P} (U)^{∗}| dividing |U − {0}| = q^{e} − 1. Now P is cyclic and so abelian, making P a subgroup of C_{GL(V)}(P), giving that p divides | End_{P} (U)^{∗}|. But p is primitive, so given {λI | λ ∈ GF (q)} ≤ End_{P} (U) we have End_{P} (U) ≅ GF (q^{e′}) for some e′ with e′ ≥ 1. Hence p divides q^{e′} − 1 so e divides e′ so e = e′.)

This gives C_{GL(U)}(P) = GF (q^{e})^{∗} ≅ C_{q^{e}−1}. Now C_{GL(U)}(P) is normal in N_{GL(U)}(P) and

\frac{N_{GL(U)}(C_{q^{e}−1})}{C_{GL(U)}(C_{q^{e}−1})} ≤ Aut C_{q^{e}−1} ≅ U(ℤ_{q^{e}−1}).

Let n ∈ N_{GL(U)}(C_{GL(U)}(P)), n1 = 1 under the identification of U with GF (q^{e}) so n: λ ↦ λ^{j} for some j. Now n is additive so j = q_{0}^{k} for some k where q_{0} is the characteristic of GF (q). Also since n preserves GF (q)-scalars we have that j = q^{l} for some l.

Hence

N_{GL(U)}(P) ≤ GF (q^{e})^{∗} ⋊ Gal (\frac{GF (q^{e})}{GF (q)}) ,

and

N_{Γ L(U)}(P) ≤ GF (q^{e})^{∗} ⋊ Aut GF (q^{e}).

The theorem follows.

5. The Exponential Convention

We shall need to compute within the field GF (q), q even, with functions f: GF (q) → GF (q). It will prove to be very convenient to use exponential notation for functions f of the form f: x ↦ x^{n}, for some integer n, that is, for monomial functions.

The natural map from polynomials to polynomial functions is a ring homomorphism from GF (q)[x] onto the ring of all functions GF (q) → GF (q) (with pointwise operations) with kernel the principal ideal generated by x^{q} − x. It follows that, for integers m and n the polynomials x^{m} and x^{n} map to the same function if and only if m = n = 0 or m ≡ n (mod q − 1). For this reason we set ℤ_{q−1} = {1, 2, …, q − 1}.

For n ∈ ℤ_{q−1} we denote the function x ↦ x^{n} from GF (q) to GF (q) by n. With this convention for m, n ∈ ℤ_{q−1}, m + n is the pointwise product of the functions x ↦ x^{m} and x ↦ x^{n}, while mn is their composition.

If u is a unit of ℤ_{q−1} we denote its inverse in ℤ_{q−1} by 1/u (since ℤ_{q−1} is commutative). Now u is a unit in ℤ_{q−1} if and only if the function x ↦ x^{u} is invertible and in this case 1/u denotes the inverse function. Also the function q/2 is equal to the function 1/2, and q − 2 is equal to −1.

6. The Trace Equation

Let q = 2^{e}. The map trace_{GF (q)→GF (q_{0})} : GF (q) → GF (q_{0}) defined to be

trace_{GF (q)→GF (q_{0})}(x) = ∑_{α∈Gal( \frac{GF (q)}{GF (q_{0}} )} x^{α} = ∑_{i=0}^{e′−1} x^{q_{0}^{i}},

where q_{0} = 2^{e′}, x ∈ GF (q), is the trace map from GF (q) to GF (q_{0}).

Let GF (q) = GF (q_{0}^{h}) with h odd. Then

trace_{GF (q_{0})→GF (2)}(x) = trace_{GF (q)→GF (2)}(x)

for all x ∈ GF (q_{0}). If h was even then we have trace_{GF (q)→GF (2)}(x) = 0 for all x ∈ GF (q_{0}).

We define the map trace = trace_{GF (q)→GF (2)} : GF (q) → GF (2) to be

trace(x) = ∑_{i=0}^{e−1} x^{2^{i}}

for x ∈ GF (q), and simply call this the trace map (sometimes referred to as the absolute trace map).

Let K be the kernel of the trace map.

Theorem 21 ([108]). (i) The trace map is additive.

(ii) The trace map is invariant under automorphisms of GF (q).

(iii) We have K being a subgroup of (GF (q), + ) of index 2 and trace has image

GF (2). (iv) We have K being automorphism invariant.

(v) There are q−1 subgroups of (GF (q), +) of index 2, namely, aK for a ∈ GF (q)^{∗} and if aK = bK then a = b.

(vi) The only subgroup of (GF (q), + ) of index 2 which is automorphism invariant

is K. (vii) For α ∈ Aut GF (q), the set

{x^{α} + x | x ∈ GF (q)} ⊆ K

with equality if and only if α generates Aut GF (q). (viii) For a ≠ 0 we have ax^{2} + bx + c being irreducible over GF (q) if and only if

trace (\frac{ac}{b^{2}}) = 1 and b ≠ 0.

7. Spreads and Reguli

Let Π be a projective space of odd dimension 2t − 1. Let S be a collection of (t − 1)- dimensional projective subspaces of Π. Then S is a spread of Π if each point of Π is contained in one and only one member of S.

If Π = P G(2t − 1, q), then a spread of Π is a collection of q^{t} − 1 distinct (t − 1)- dimensional subspaces having no points in common.

Let R be a set of q + 1 pairwise skew lines of P G(3, q). A line l is a transversal of R if it meets all elements of R. A transversal meets every element of R in a unique point, and conversely.

Let Π be a projective space of dimension greater or equal to three. A regulus is a set of transversals to three pairwise skew lines.

Given a regulus R, and l_{1}, l_{2}, l_{3} ∈ R, distinct lines, the transversals to l_{1}, l_{2}, l_{3} are transversal to all the lines of R. This regulus is called the opposite regulus to R.

Hence any triple of pairwise skew lines lies on a unique regulus.

8. Bruck-Bose Construction

In this section we mention a way of constructing affine planes (and, hence, projective planes) from projective spaces.

Theorem 22 (Bruck-Bose Construction [20][2]). Let S be a t−1-spread of P G(2t− 1, q). Embed P G(2t − 1, q) as a hyperplane, H, of P G(2t, q). Let

(i) Let P equal the points of P G(2t, q) not on H.

(ii) Let L be the subspaces of P G(2t, q) of dimension t which meet H in an element

of S. (iii) Let I be the natural incidence of P G(2t, q).

Then ℐ = (𝒫, ℒ , I) is an affine plane of order q^{t}. Note ℐ may be non-Desarguesian.

9. Generalised Quadrangles

We say two points, x and y, are collinear or adjacent, denoted x ∼ y, if there exists a line they are both incident to. If no such line exists then we say x and y are noncollinear or non-adjacent, denoted x ≁ y. Two lines, L and M, are concurrent or intersect, denoted L ∼ M, if there exists a point incident with both. If no such point exists then we say that the lines L and M are non-concurrent or disjoint, denoted by L ≁ M.

A firm generalised quadrangle, abbreviated to firm GQ, is an incidence structure (see page 1), ℐ = (𝒫, ℬ, I) , with the following axioms:

GQ1: Two points are incident with at most one line;

GQ2: If a point x is not incident with a line L, then there exists a unique point, y, and a unique line, M, such that x is incident with M which is incident with y which in turn is incident with L.

ptGQ: There exists no point that is collinear with all points. Axiom GQ2 is sometimes referred to as “there are no triangles” axiom.

From GQ1, GQ2 and ptGQ we can get the dual axioms: GQ1*: Two lines are concurrent with at most one point; GQ2*: If a point x is not incident with a line L, then there exists a unique point, y, and a unique line, M, such that x is incident with M which is incident with y which in turn is incident with L. That is GQ2.

ptGQ*: There exists no line that is concurrent with all other lines.

Theorem 23. We can determine the axioms GQ1*, GQ2* and ptGQ* from the axioms GQ1, GQ2 and ptGQ.

Proof. As GQ1* and GQ2* are straightforward we will prove ptGQ*.

Suppose that there exists a line L that is concurrent with all lines of the firm generalised quadrangle. Let y be a point of the firm generalised quadrangle not incident to L (this exists else we would contradict ptGQ). By GQ2 there exists a line M and a point x such that x I L I y I M. By ptGQ there is a point z that is not collinear with x and y. By GQ2 there exists a line N such that z I N I x I M. But as L is concurrent with all lines of the firm generalised quadrangle then y is collinear with two points on L contradicting GQ2. So ptGQ* is true.

Hence the dual of a firm generalised quadrangle is a firm generalised quadrangle. A firm generalised quadrangle has at least two points incident with every line

and two lines incident with every point (from now on we will rephrase comments like the one above to “... at least two points incident with every line and dually,” taking advantage of the fact that the dual of a firm GQ is a firm GQ). Suppose not: Then there is a line L with only one point on it, p say. By axiom GQ2 every point not on L must be collinear with p (as this is the only point on L), and so p is collinear with all points, contradicting axiom ptGQ.

Two examples of firm generalised quadrangles are complete bipartite graphs and their duals, grids. A grid of type (s, s′), s, s′ > 0, is a set of s + s′ + 2 lines and (s + 1)(s′ + 1) points such that every point is on exactly two lines and that there are s + 1 lines with s′ + 1 points on each line and s′ + 1 lines with s + 1 points on each. The dual of a grid is a complete bipartite graph of type (t, t′), t, t′ > 0, denoted K_{t+1,t′+1}, with every line having two points and a partitioning of points into two sets, one of size t + 1 and one of t′ + 1; additionally, every point in the set of size t + 1 is adjacent to every point of the set of size t′ + 1 and conversely.

A generalised quadrangle is a firm generalised quadrangle with two extra axioms:

thickGQ: There are at least three points on every line, and thickGQ*: There are at least three lines on every point. These are referred to as the thickness axioms. Thickness is self-dual(!), so again,

we have that the dual of a generalised quadrangle is a generalised quadrangle. A firm generalised quadrangle need not be a generalised quadrangle.

Theorem 24 (Thickness implies order [13]). In a generalised quadrangle the number of points on a line is a constant and the number of lines on a point is a constant (not necessarily the same constant).

Proof.

(1) Let L and M be disjoint lines of the generalised quadrangle (existence via ptGQ axiom). For every point p on L there is a unique point, p′, on M, and conversely. So there is a one-to-one correspondence of points between disjoint lines. Hence disjoint lines have the same cardinality.

(2) Suppose L and M meet at a point y (by axiom GQ1 there are no more points of intersection). Let x be a point on neither L or M and not collinear with y. By GQ2 there is one line on x meeting L and another line meeting M. By thickness there exists another line, N, which is disjoint from L and M (else we will have triangles). So we use part (1) to get that L has the same cardinality as N and similarly for M.

Dually every point has the same number of lines on it.

We say that the constants of the above theorem are the parameters of the generalised quadrangle. A finite generalised quadrangle is a generalised quadrangle where the parameters are finite. We say a finite generalised quadrangle has order (s, t) where the number of points on a line is s + 1 and the number of lines on a point is t + 1 (note: one line having s + 1 points implies all lines have s + 1 points, and dually). If s = t then we say the generalised quadrangle has order s. The dual of a generalised quadrangle of order (s, t) is a generalised quadrangle of order (t, s).

Note. Payne and Thas [103] define a generalised quadrangle as an incidence structure satisfying GQ1, GQ2 with the existence of the parameters s and t (as above) for s, t ≥ 1.

For a grid of type (s, s) we have a firm generalised quadrangle with a constant number of points on every line (equal to s + 1) and a constant number of lines on every point (equal to two), hence this is a Payne and Thas generalised quadrangle of order (s, 1), but it is not a generalised quadrangle. For a complete bipartite graph of type (t, t) we have a firm generalised quadrangle with a constant number of lines on a point (equal to t + 1) and a constant number of points on a line (equal to two). Again this is not a generalised quadrangle, but it is a Payne and Thas generalised quadrangle of order (1, t).

Theorem 25 (A poor man’s classification of firm GQs). A firm generalised quadrangle is either

(i) a grid, or (ii) a complete bipartite graph, or (iii) a generalised quadrangle.

Proof. Let L and M be disjoint lines of the generalised quadrangle. By GQ2 there is a one-to-one correspondence between the points of L and M. Let s+1 be the number of points on disjoint lines. Let the line N meet L. This line can not meet any other line meeting L, else this will contradict GQ2. Let y be the point of intersection of L and N. Any other line meeting L, other than at point y, is disjoint from N, so there is a one-to-one correspondence between the points of N and the other line by GQ2. We let s′ + 1 be the number of points on N.

If we have thickness then s = s′ (as there is a line disjoint from L and N, and so by GQ2, we have that there is one-to-one correspondence with the lines L and N), so we have a generalised quadrangle.

If no such line exists, then all lines meet L or N. From a consequence of axiom ptGQ we have at least two lines on any point. So a point not on L or N must have one line meeting L and the other meeting N, (with GQ2 forbidding both lines from the point meeting only L or N). Hence we have a grid.

Dually, we arrive at a constant number of lines through a point (and a generalised quadrangle) or we have a complete bipartite graph.

Corollary 26. A firm generalised quadrangle with only two points on any line is a complete bipartite graph, or if dually, then the firm generalised quadrangle is a grid.

10. End Notes

Notes on Section 1, p. 1. Incidence structures that satisfy PP1, A3 and every line has at least two points are called linear spaces. A large body of work has been written about such spaces, of particular interest is the work of F. Buekenhout (see [12] for an overview of finite linear spaces).

The results in this Section are well known.

Notes on Section 2, p. 2. The results in this Section are well known.

Notes on Section 3, p. 2. The definition of collineation on page 3 is equivalent to that on page 23 due to the Fundamental Theorem of Projective Geometry (Theorem 5, p. 4).

The results in this Section are well known.

Notes on Section 4, p. 4. Theorem 20, p. 7, is a specific case of the more general theorem that says for a given irreducible W we have

End(W^{n}) = M_{n}(End(W)).

The results in this Section are well known, though Theorem 20 is hard to find.

Notes on Section 5, p. 9. The results in this Section are well known.

Notes on Section 6, p. 9. The results in this Section are well known.

Notes on Section 7, p. 10. The results in this Section are well known.

Notes on Section 8, p. 10. The results in this Section are well known.

Notes on Section 9, p. 11. The results in this Section are well known. The author’s treatment is different from the standard text [103] as the author prefers synthetic definitions without assuming order.

CHAPTER II Hyperovals in Desarguesian Planes

1. Arcs in Arbitrary Projective Planes

Let K be a set of k points, a k-set, in a projective plane of order q. For each i with 0 ≤ i ≤ q + 1, let

t_{i} = |{ lines l | |l ∩ K| = i}|.

A k-set K has class [m_{1}, m_{2}, …, m_{s}] if 0 ≤ m_{1} < m_{2} < ⋯ < m_{s} ≤ q + 1 and {i | t_{i} ≠ 0} ⊆ {m_{1}, m_{2}, …, m_{s}}; it has type (m_{1}, m_{2}, …, m_{r}) if 0 ≤ m_{1} < m_{2} < ⋯ < m_{r} ≤ q + 1 and {i | t_{i} ≠ 0} = {m_{1}, m_{2}, …, m_{r}}.

Theorem 27 ([118] [55]). Let K be a set of k points of a projective plane π. Then

(1)  ∑_{i=0}^{q+1} t_{i} = q^{2} + q + 1.

(2)  ∑_{i=1}^{q+1} it_{i} = k(q + 1).

(3)  ∑_{i=2}^{q+1} i(i − 1)t_{i} = k(k − 1).

Proof.

(1) Count the number of lines in two ways.

(2) Count |{(P, l) | P ∈ K, P I l}| in two ways: For each of the k points P of K, there are q + 1 lines on P; for each of the t_{i} lines l with |l ∩ K| = i, there are i points of K on l.

(3) Count {(P, Q, l) | P, Q ∈ K, P ≠ Q, P I l, Q I l}| in two ways; for each of the k points P of K, there are k − 1 points Q of K with P ≠ Q, and each pair (P, Q) has a unique line on it; for each of the t_{i} lines l with |l ∩ K| = i, there are i points P of K on l and i − 1 points Q of K on l with Q ≠ P.

An arc in a finite projective plane is a set of points where no three are collinear. A k-arc is an arc containing k points. A quadrangle is another way of referring to a 4-arc and triangle for a 3-arc.

Secant, tangent, and external lines are lines that meet an arc in exactly 2, 1 or 0 points, respectively.

Hence a k-arc is of class [0, 1, 2].

Theorem 28 ([16]). A k-arc in a projective plane of order q has k ≤ q+2. When equality holds there are no tangent lines to the arc.

Proof. Let P be a point of the k-arc. Each line on P is incident with at most one point from the arc (other than P) as no three are collinear. We have q + 1 lines on P, so that there are at most q + 1 points on the arc not including P. Hence k ≤ (q + 1) + 1 = q + 2.

Suppose we have a (q + 2)-arc, H. Let P be a point on H. Let l be a line on P. Each line on P meets H − {P } in exactly one point. Therefore l meets H − {P } in exactly one point. Hence l meets H in exactly two points. Thus every line that meets H is secant to H.

We call a (q + 2)-arc a hyperoval. The proof above shows that any hyperoval has type (0, 2).

Theorem 29 ([16]). If there exists a hyperoval in a projective plane of order q then q is even.

Proof. Let H be a hyperoval. Choose a point P not on H. This point exists as q^{2} + q + 1 > q + 2. By Theorem 28, each line on P meets H in zero or two points. Hence the lines on P partition H into sets of size two. Hence 2 divides q + 2, which implies 2 divides q, which implies that q is even.

Corollary 30 ([16]). In a projective plane of order q, q odd, a k-arc has k ≤ q + 1.

Irrespective of whether q is odd or even, a (q + 1)-arc in a projective plane of order q is called an oval.

Lemma 31 ([114]). In a projective plane of order q every point of an oval has a unique tangent on it.

Proof. Let P be a point of an oval O. Then

number of tangents on P = (q + 1) − number of secants on P
= (q + 1) − |O − {P }|
= (q + 1) − q
= 1.

\underline{1. Arcs 17}

Hence the number of tangents of an oval in a projective plane of order q is q + 1.

Theorem 32 ([114]). In a projective plane of order q, with q odd, every point not on the oval is on zero or two tangents of the oval.

Proof. Let O be an oval in a projective plane of odd order q. The number of points on O is even. Thus every point not on O lies on an even number of tangent lines. Now let l be a tangent line to O. Every point on l not in O lies on at least one other tangent line to O. Since there are q tangent lines to O other than l and q points on l not in O each point on l not in O lies on exactly one tangent line other than l.

So every point not in O either lies on no tangent lines to O or exactly two tangent lines to O.

We call all points not on the oval O that have two tangents to O external points to O, and those with no tangents to O the internal points to O. In fact, we can see that all points in our projective plane of order q, q odd, can be characterised as external points, internal points, or the q + 1 points of O.

Let P be a point of the oval O. Then there is a unique tangent line, l, to P of O. The line l has q points different from P, with these points being incident with another tangent line to O not equal to l. Hence there are \frac{1}{2}q(q + 1) external points in a projective plane of order q, q odd. Since there are q + 1 points on O, this gives \frac{1}{2}q(q − 1) internal points.

Remark. Theorem 32, says that for q odd, the tangent lines to an oval form an oval of the dual plane.

Theorem 33 ([114]). In a projective plane of order q, q even, an oval, O, has q + 1 tangents all concurrent in a point.

Proof. By Lemma 31, we have q + 1 tangents. Since q + 1 is odd, each point lies on an odd number of tangents to O. Let l be a secant line to O. Each point of l lies on at least one tangent to O, but there are only q + 1 tangents so each lie on exactly one tangent. Hence any point incident with two (or more) tangents is incident with no secant lines, and so is on q + 1 tangents.

Equivalently any intersection of two tangents does not lie on a secant of O. So all the lines on a point of intersection of two tangents are tangents.

Corollary 34 ([114]). In a projective plane of order q, with q even, every oval can be uniquely completed to form a hyperoval.

Proof. All the tangents of the oval O meet in a point N and this point lies on no secant line to O. Hence O ∪ {N} is a hyperoval. This is unique by Lemma 31.

In a plane of even order, we call the point of intersection of all the tangents from an oval the nucleus of the oval.

Theorem 35 ([114]). Let K be a k-arc contained in a hyperoval with k > q/2+1. If K ∪ {P } is a (k + 1)-arc, then P is on the hyperoval containing K.

Proof. Let K ∪ {P } be an arc with P not on the hyperoval containing K, which we shall call H. Then all the lines from P must meet H in two or no points by Theorem 28, p. 16. But as K has more than q/2 + 1 points on it, there must be a line from P, intersecting H, that meets K in two points, contradicting the fact that K ∪ {P } is an arc. Hence P is on H.

2. Conics

We investigate a special example of an oval. The reader should be familar with the area of sesquilinear forms and Witt’s Theorem. See [124] for an introduction.

The radical of a sesquilinear form f: V × V → GF (q) on a vector space V is

Rad f = V^{⊥} = {w ∈ V | f(v, w) = 0 for all v ∈ V }.

A form f is nondegenerate if Rad f = {0}, otherwise f is degenerate.

A quadratic form on a vector space V is a function Q: V → GF (q) with Q(cv) = c^{2}Q(v) for all v ∈ V and c ∈ GF (q) and we define f_{Q} the symmetric bilinear form associated with Q as

f_{Q}(v, w) = Q(v + w) − Q(v) − Q(w) for all v, w ∈ GF (q).

The singular radical of a quadratic form Q is

{v ∈ Rad f_{Q} | Q(v) = 0}.

A quadratic form is nondegenerate (or nonsingular) if the singular radical is {0}, degenerate (or singular) otherwise.

For q odd, Q is nondegenerate if and only if f_{Q} is nondegenerate. For q even, there are nondegenerate quadratic forms Q with a degenerate form f_{Q}.

A quadratic form Q on V defines a quadric,

Q = {⟨x⟩ | Q(x) = 0}

in P G(n, q) where dim_{GF (q)} V = n + 1. The one-dimensional subspaces ⟨x⟩ where Q(x) = 0 are called totally singular points of Q, since they are points of P G(n, q). When a nondegenerate quadratic form Q has f_{Q} degenerate, then the radical of f_{Q} is called the nucleus of Q. Quadrics in P G(2, q) are called conics.

Example. Let Q be the quadratic form on GF (q)^{3} defined by

Q(x, y, z) = ax^{2} + by^{2} + cz^{2} + dxy + exz + fyz.

\underline{2. Conics 19}

Then Q defines a conic in P G(2, q). The corresponding form is

f_{Q}(x, x′) = Q(x + x′, y + y′, z + z′) − Q(x, y, z) − Q(x′, y′, z′)

= (x y z) (\begin{matrix}2a & d & e \\ d & 2b & f \\ e & f & 2c \end{matrix}) (\begin{matrix}x′ \\ y′ \\ z′ \end{matrix}) ,

with f_{Q} being non-degenerate if and only if

|\begin{matrix}2a & d & e \\ d & 2b & f \\ e & f & 2c \end{matrix}| ≠ 0.

Now suppose that Q is a degenerate conic. We define f̄_{Q} over GF (q)^{3}/ Rad f_{Q} so f̄_{Q} is nondegenerate.

Let q be even. Then dim(V/ Rad f_{Q}) is even so dim Rad f_{Q} is odd. If dim Rad f_{Q} = 3 then f_{Q} = 0 so Q(x, y, z) = ax^{2} + by^{2} + cz^{2} = (\sqrt{a}x + \sqrt{b}y + \sqrt{c}z)^{2}, a repeated line, or the whole plane. If dim Rad f_{Q} = 1 then Rad f_{Q} is the singular radical of Q since Q is degenerate. Then Q̄ defined on GF (q)^{3} over the singular radical of Q has dimension two. If it is of plus type we have two distinct concurrent lines, if it is of minus type then it is a point (see [124]).

Let q be odd. If dim Rad f_{Q} = 3 then Q = 0 and Q is the whole plane. If dim Rad f_{Q} = 2 then Q̄ is a one dimensional non-degenerate form with no singular points, hence Q is a repeated line. When dim Rad f_{Q} = 1 we have the same situation as for q even.

Hence we can say that a degenerate conic is one of the following:

(i) The whole plane, (ii) a point, (iii) two concurrent distinct lines, or (iv) a repeated line.

Theorem 36 (see [55] Lemma 7.2.3). A nondegenerate conic is an oval of P G(2, q).

Proof. A nondegenerate conic is the set of totally singular points with respect to a nondegenerate quadratic form Q. Given a line l (that is, a two-dimensional subspace of GF (q)^{3}) the restriction Q|_{l} of Q to l is a (possibly degenerate) quadratic form. Since Q has Witt index 1, that is, the common dimension of the maximal totally singular subspaces is 1, then Q|_{l} ≠ 0. If Q|_{l} is nondegenerate, it has either zero or two singular points on l, since Q|_{l} is isometric to O^{−}(2, q) or O^{+}(2, q), respectively [124]. If degenerate, the radical is one-dimensional, and is the only totally singular point. Hence every line l meets the conic in 0, 1, or 2 points. The form Q is similar to xz − y^{2} which has q + 1 totally singular points, those being {(1, t, t^{2}) | t ∈ GF (q)} ∪ {(0, 0, 1)}.

Theorem 37. Let q = 2^{e}. Let i satisfy (i, e) = 1 and α = 2^{i} be an automorphism of GF (q). Then (in the notation of Section 6, p. 46) we have

D(α) − {(0, 1, 0)} = {(1, t, t^{α}) | t ∈ GF (q)} ∪ {(0, 0, 1)},

being an oval.

Proof. The lines on the point (0, 0, 1) are of the form [0, 1, 0] and [1, b, 0] where b ∈ GF (q). These only meet the oval in at most one other point. What is left to show is that no three points of {(1, t, t^{α}) | t ∈ GF (q)} are collinear.

Suppose the points (1, x, x^{α}), (1, y, y^{α}), and (1, z, z^{α}) are collinear for distinct x, y, z ∈ GF (q). Then

\frac{x^{α} + y^{α}}{x + y} = \frac{x^{α} + z^{α}}{x + z} = \frac{y^{α} + z^{α}}{y + z} ,

which implies that (x + y)^{α−1} = (x + z)^{α−1} = (y + z)^{α−1} as α ∈ Aut GF (q). Since (α − 1, q − 1) = (2^{i} − 1, 2^{e} − 1) = 2^{(i,e)} − 1 = 1 we have α − 1 being one-to-one. Hence x + y = x + z = y + z. Contradiction. Hence D(α) − {(0, 1, 0)} is an oval.

Lemma 38 (Lemma of Tangents [55]). Assume Q is a q-arc of P G(2, q), q odd. Let t_{1}, …, t_{r} be tangents of Q passing through the point P. Put Q_{i} = t_{i} ∩ Q and denote by s_{i} the second tangent of Q through Q_{i}. Then there is a conic C of P G(2, q) containing all the Q_{i} and such that s_{i} is a tangent of C for all i.

In 1949 Järnefelt and Kustaanheimo [62] conjectured that for q odd every oval is a conic. Marshall Hall Jr. wrote when reviewing the article [45] “The reviewer finds this conjecture implausible.” This moved Segre into proving the conjecture and he announced the proof in 1954 in Italian and 1955 in English [116].

Theorem 39 (Segre’s Theorem). Let O be an oval in P G(2, q), q odd. Then O is a conic.

Proof. This theorem is true if q = 3, so assume q ≥ 5.

Pick P ∈ O. Then all the secants of O through P are tangents of the q-arc O − {P }. Hence there is a conic C with C ⊇ O − {P }, such that all the tangents of O which have their points of tangency in C ⊇ O − {P } are also tangents of C by Lemma 38, p. 20.

Pick Q ∈ O − {P }. The same argument yields a conic C′ with C′ ⊇ C − {Q}, such that the tangents of C′ through points of C − {Q} are also tangents of C. Hence C and C′ have at least q − 1 ≥ 4 points and at least as many tangents in common.

This is more than enough to make C = C′.

A hyperoval in P G(2, q), q even, is regular if it is the union of a conic and the nucleus of the conic. A pointed conic is an oval equal to C ∪ {N} − {P } where C is a conic with nucleus N and P is a point of the conic.

Lemma 40. A regular hyperoval in P G(2, q) contains a unique conic if q ≥ 8.

Proof. By definition there exist a conic, C, such that H = C ∪ {N}, where N is the nucleus of the conic. Suppose C′ ⊆ H. Then |C ∩ C′| ≥ q ≥ 5, and since there exists a unique conic on a 5-arc ([55] corollary to theorem 7.2.1) we have C = C′.

The quadrangle {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1)} is referred to as the fundamental quadrangle.

Lemma 41 ([55], theorem 8.4.2, corollary 6). Let H be a regular hyperoval in P G(2, 2) then P Γ L(3, 2)_{H} ≅ S_{4}. In particular, all hyperovals in P G(2, 2) are regular.

Proof. In P G(2, 2) H is a 4-arc. As P Γ L(2, 2) = P GL(2, 2) is regular on ordered quadrangles we have P Γ L(3, 2)_{H} ≅ S_{4}, so all hyperovals are equivalent in P G(2, 2). The four hyperovals, C ∪ {N}, for a conic C and nucleus N, on the fundamental quadrangle are:

(i) C defined as xy + xz + yz = 0 with nucleus N = (1, 1, 1).

(ii) C defined as x^{2} + yz = 0 with nucleus N = (1, 0, 0).

(iii) C defined as y^{2} + xz = 0 with nucleus N = (0, 1, 0).

(iv) C defined as z^{2} + xy = 0 with nucleus N = (0, 0, 1).

Lemma 42 ([55], theorem 8.4.2, corollary 6). Let H be a regular hyperoval in P G(2, 4), then P Γ L(3, 4)_{H} ≅ S_{6}. In particular, all hyperovals in P G(2, 4) are regular.

Proof. Let q = 4. Let Q be a 4-arc in P G(2, 4). There are two points P_{1} and P_{2} of P G(2, 4) such that Q ∪ {P_{i}} is a 5-arc for i = 1, 2. In fact Q ∪ {P_{1}, P_{2}} is the unique hyperoval on Q. Hence there exists a unique hyperoval on Q. Since P GL(3, 4) is regular on 4-arcs then there exists a unique g ∈ P GL(3, 4) such that gQ_{1} = Q_{2} with Q_{1} and Q_{2} being 4-arcs contained in a hyperoval H. Since there is a unique hyperoval on each 4-arc we have gH = H. Hence P GL(3, 4)_{H} is sharply 4-transitive on H. Hence P GL(3, 4)_{H} ≅ A_{6}, so G ≅ S_{6}.

Let ω be a primitive element of GF (4) such that ω^{2} = ω + 1. Let

Q = {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)}

with {P_{1}, P_{2}} = {(1, ω, ω^{2}), (1, ω^{2}, ω)}, so

Q ∪ {P_{1}, P_{2}} = {(1, t, t^{2}) | t ∈ GF (q)} ∪ {(0, 1, 0), (0, 0, 1)}

so Q ∪ {P_{1}, P_{2}} is equal to a conic, C, plus its nucleus N. Then we have the conics and nuclei being:

(i) C xy + ωxz + ω^{2}yz = 0 with nucleus N = (1, ω^{2}, ω).

(ii) C : ω^{2}xy + ωxz + yz = 0 with nucleus N = (1, ω, ω^{2}).

(iii) C : x^{2} + yz = 0 with nucleus N = (1, 0, 0).

(iv) C : y^{2} + xz = 0 with nucleus N = (0, 1, 0).

(v) C : xy + xz + yz = 0 with nucleus N = (1, 1, 1).

(vi) C : z^{2} + xy = 0 with nucleus N = (0, 0, 1).

Hence Lemma 40, p. 20, needs q ≥ 8.

Theorem 43 ([55], theorem 8.4.2, corollary 6). The stabiliser of a regular hyperoval, H in P GL(3, q) is isomorphic to

\{(\begin{matrix}a^{2} & 0 & b^{2} \\ ac & ad + bc & bd \\ c^{2} & 0 & d^{2} \end{matrix}) | a, b, c, d ∈ GF (q), ad + bc ≠ 0\} .

since this is the stabiliser of D(2) (in the notation of Section 4, p. 41).

Proof. If A is the set of all field automorphisms of GF (q) (acting on P G(2, q) as in Section 3, p. 2) then

P Γ L(3, q)_{H} = ⟨P GL(3, q)_{H}, A⟩.

Also P GL(3, q)_{H} is generated by these matrices:

(\begin{matrix}1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t^{2} \end{matrix}) , (\begin{matrix}1 & 0 & 0 \\ t & 1 & 0 \\ t^{2} & 0 & 1 \end{matrix}) , (\begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}) , t ∈ GF (q).

The first two types of matrices generate a maximal subgroup in P GL(3, q)_{H}. The proof follows:

Let

H = ⟨\{(\begin{matrix}1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t^{2} \end{matrix}) | t ∈ GF (q)\} , \{(\begin{matrix}1 & 0 & 0 \\ t & 1 & 0 \\ t^{2} & 0 & 1 \end{matrix}) | t ∈ GF (q)\}⟩ ,

so H ≤ P GL(3, q)_{H,(0,0,1)}. Since |P GL(3, q)_{H}| = (q + 1)q(q − 1) and |P GL(3, q)_{H} : P GL(3, q)_{H,(0,0,1)}| = q + 1 with P GL(3, q)_{H} transitive on points of the conic, we conclude that |P GL(3, q)_{H,(0,0,1)}| = q(q−1), so H = P GL(3, q)_{H,(0,0,1)}. Since P GL(3, q)_{H} is 2-transitive on points of the conic, then P GL(3, q)_{H} is primitive on points of the conic and so H = P GL(3, q)_{H,(0,0,1)} is a maximal subgroup of P GL(3, q)_{H} by Theorem 12, p. 5.

Theorem 44 ([55], lemma 7.2.3, corollary 7). The action of P Γ O(3, q) on the set of totally singular points of the O(3, q) polar space is equivalent to the action of P Γ L(2, q) on the points of P G(1, q).

Proof. The bijection from P G(1, q) to O(3, q) with form xz = y^{2}

f : \begin{cases}(1, t) ↔ (1, t, t^{2}) \\ (0, 1) ↔ (0, 0, 1) \end{cases}

plus the isomorphism from P GL(2, q) to P GO(3, q) (with field automorphisms acting coordinate wise)

ψ : (\begin{matrix}a & b \\ c & d \end{matrix}) Z_{2} ↔ (\begin{matrix}a^{2} & 2ab & b^{2} \\ ac & bc + ad & bd \\ c^{2} & 2cd & d^{2} \end{matrix}) Z_{3}

(where Z_{d} is the centre of GL(d, q)) form an equivalence since ψ(g)f(x) = f(gx) for all x ∈ P G(1, q) and all g ∈ P Γ L(2, q).

Let

D = (\begin{matrix}a^{2} & 2ab & b^{2} \\ ac & bc + ad & bd \\ c^{2} & 2cd & d^{2} \end{matrix}) .

Since D(2) ⊇ quadrangle (for q > 2) then

f (\begin{matrix}a & b \\ c & d \end{matrix}) f^{−1} and D

agree on a quadrangle, and so

f (\begin{matrix}a & b \\ c & d \end{matrix}) f^{−1} = D|_{D(2)},

with D unique.

Note. Let O be a conic in P G(2, 4) and H = O ∪ {N}, where N is the nucleus of O. So there are 15 secant lines and hence 6 external lines to H in P G(2, 4); these external lines make a dual hyperoval. Since P Γ L(3, 4)_{O} ≅ S_{5} ≅ P GL(2, 4) and by Witt’s Theorem [124], P Γ L(3, 4)_{O} is transitive on external lines to H.

Theorem 45. Let H be a regular hyperoval in P G(2, q). Let G = P Γ L(3, q)_{H}. Then for q ≥ 8 we have G ≅ P Γ L(2, q).

Proof. Let q ≥ 8. By Lemma 40, p. 20, we have a unique conic C, such that, H = C ∪{N} where N is the nucleus of C. Hence G = P Γ L(3, q)_{C∪{N}} = P Γ L(3, q)_{C}. The result follows from Theorem 44, p. 22.

3. Collineation Groups

A collineation of a projective plane, π, is an automorphism of π. We quite happily use both collineation and automorphism interchangeably.

Let P and Q be points of a projective plane π. Let g be a collineation of π. Let l be the line, P Q, joining P and Q. Then gl is the unique line joining gP and gQ, that is,

g(P Q) = (gP)(gQ).

Dually, if l and m are lines of π with l ≠ m, then

g(l ∩ m) = (gl) ∩ (gm).

We will define lines by the points on them, so P ∈ l is equivalent to saying P I l.

3.1. Fixed Configurations

Let G be a group of collineations of a projective plane π with P a point and l a line of π.

(i) If gP = P for all g ∈ G then P is fixed under G.

(ii) If gl = l for all g ∈ G, that is, {P | P ∈ l} = {gP | P ∈ l}, then l is fixed

under G. (iii) l is fixed!pointwise by G if gP = P for all P ∈ l, for all g ∈ G. (iv) P is fixed!linewise by G if gl = l for all l incident with P, for all g ∈ G.

The fixed configuration for a group G of collineations is the configuration of fixed lines and fixed points under G.

We call a subset of points 𝒫′ and a subset of lines ℒ′ of a projective plane a closed configuration if

(i) Given P, Q ∈ 𝒫′ with P ≠ Q then P Q ∈ ℒ′, and

(ii) given l, m ∈ ℒ′ with l ≠ m then l ∩ m ∈ 𝒫′.

Theorem 46 ([134]). Let G be a group of collineations of a projective plane π. Then for all g ∈ G

(i) If distinct points P and Q are fixed then so is the line P Q.

(ii) If distinct lines l and m are fixed so is the point l ∩ m.

(iii) If the distinct lines l and m are fixed pointwise with l ≠ m, then g = 1.

(iv) If the distinct points P and Q are fixed linewise with P ≠ Q, then g = 1.

Proof. (i) g(P Q) = gP gQ = P Q as P and Q are fixed.

(ii) Dually.

(iii) Let O = l ∩ m. Let P be any point not on l or m. Then take two lines r and s through P distinct from P O (these exist as there are at least three lines on any point).

Let R = l ∩ r, S = l ∩ s, C = m ∩ s and U = m ∩ r. Since gR = R and gU = U we have gr = r. Similarly, gs = s. Hence P is fixed by (ii). Since P is any arbitrary point not initially fixed by g, it follows that g = 1.

(iv) Dually.

A fixed configuration of a group of collineations G is always a closed configuration. In fact the incidence structure of a fixed configuration satisfies axioms PP1 and PP2.

A subplane, π′, of a projective plane, π, is a closed configuration that contains a quadrangle. A subplane π′ is a Baer subplane if every point of π is on a line of π′ and every line of π is on a point of π′. If a collineation has a subplane as a fixed configuration then we say the collineation is planar, those planar collineations whose fixed subplane is Baer are called Baer collineations.

Corollary 47. Let g be a collineation of a projective plane such that

(i) l is fixed pointwise by g, and (ii) gP = P and gQ = Q with P ∉ l and Q ∉ l,

then g = 1.

Proof. Take any point R not on l and P Q. Let U = RP ∩ l and V = RQ ∩ l. Since U, V, P and Q are fixed, the lines UP and V Q are fixed. Hence R = UP ∩ V Q is fixed. Since R was arbitrary, the line UR is fixed pointwise as well as l. Since we have two distinct lines UR and l being fixed pointwise by Theorem 46, we have g = 1.

Let ℐ = (𝒫, ℬ, I) be a projective plane of order q. An incidence matrix A for ℐ is a matrix with rows indexed by points, columns indexed by lines with entry in row P and column l being 1 if P I l and 0 if P I̸ l.

Lemma 48 ([46]). Suppose that ℐ = (𝒫, ℬ, I) is a finite projective plane of order q then

AA^{T} = qI + J,

where J is the all one’s matrix. Also A is invertible.

Proof. For P_{1}, P_{2} ∈ 𝒫, the entry in row P_{1} and column P_{2} of AA^{T} is

|\{l ∈ ℬ | P_{1} I l I P_{2}\}| = \begin{cases}1 & if P_{1} ≠ P_{2}, \\ q + 1 & if P_{1} = P_{2}. \end{cases}

So AA^{T} = qI + J. Now det(qI + J) = (n + 1)^{2}n^{n^{2}+n} ≠ 0 so A is invertible.

Theorem 49 ([46]). In a finite projective plane the permutation σ of points and τ of lines of a collineation has the same cycle structure.

Proof. Let ℐ = (𝒫, ℒ , I) be a point-line incidence structure with a point-line incidence matrix A. Let g ∈ Sym(𝒫 ∪ ℒ)_{𝒫∪ℒ} . Then g ∈ Aut ℐ if and only if we have gP ∈ gl ⟺ P ∈ l. Let g = (σ, τ) where σ = g|_{𝒫} and τ = g|_{ℒ} . So g ∈ Aut ℐ if and only if σP ∈ τ ⟺ P ∈ l.

Let the permutation matrix of σ be S and τ have permutation matrix T. Then P SAT l = P Al for all points P and all lines l, hence SAT = A.

By Theorem 48 A is invertible so

S = AT^{−1}A^{−1} ⟹ S ∼ T^{−1}.

Corollary 50 ([46]). In a finite projective plane a collineation fixes the same number of points and lines.

By Corollary 50, in a group generated by a collineation (that is, a cyclic group of collineations) the number of fixed points equals the number of fixed lines. It is necessary that the group be cyclic since the group AGL(2, q) fixes no points of P G(2, q) but does fix a line.

If g fixes a line l pointwise then we say that g has axis l. If g fixes a point linewise then we say that g has centre P.

If g is not equal to the identity then it has at most one axis and centre by Theorem 46.

Let g ≠ 1 have axis l and centre P, then g is an elation if P I l and a homology if not. By convention, the identity is both an elation and a homology.

Notes. (i) For a collineation g with axis l and centre P, apart from l, only lines through P are fixed. Since, if we suppose there exists a line n not through P fixed by g and a line m through P with l ≠ m, then m ∩ n is fixed, giving a contradiction.

(ii) Given P and l the set of all collineations with centre P and axis l form a group. We have (Aut π)_{(P,l)} acting on {lines on P } ∪ {points on l}. The kernel of the action equals {collineations with centre P and axis l} ◁ (Aut π)_{(P,l)}.

(iii) If gP = P then for any collineation h, we have

(hgh^{−1})(hP) = hP,

so hP is fixed by hgh^{−1}. (iv) Given a collineation g with axis l and centre P and h a collineation, then hgh^{−1}

is a collineation with centre hP and axis hl, since, if P ∈ hl, then h^{−1}P ∈ l. Also

hgh^{−1}(P) = hg(h^{−1}P)
= h(h^{−1}P)
= P,

so hl is fixed pointwise by hgh^{−1}. Similarly, we can prove hP is fixed linewise by hgh^{−1}.

The (cyclic) group generated by an elation or a homology acts semiregularly on points not equal to the centre and not on the axis by Corollary 47.

Theorem 51 ([36]). In a projective plane of order q.

(i) The order of an elation divides q.

(ii) The order of a homology divides q − 1.

Proof. Look at a line l on the centre but not on the axis. Since l is on the centre it is fixed, so has q points not fixed on l if it is an elation, and q − 1 points not fixed on l if it is a homology.

Theorem 52. A non-trivial homology does not stabilise a hyperoval in a projective plane of order q.

Proof. Let g be a homology with axis l and centre C. Let P be a point not on l and distinct from C, then C, P, gP are collinear. So the orbits of P under ⟨g⟩ are collinear (on a line incident with the centre C). The length of this orbit is equal to |g|, since g acts semi-regularly. Since q is even we have q − 1 being odd, so |g| is odd, hence as g is non-trivial |g| ≥ 3 (Theorem 51), hence the orbits under g are not arcs.

This is the reason we concentrate on elations more than homologies.

We say l is a translation line if for all P I l we have for every pair of distinct points Q, R where Q, R, P collinear and Q, R not on l, there exists an elation g with centre P and axis l such that gQ = R.

If a projective plane, π, has a translation line then we say π is a translation plane. The projective plane P G(2, q) is a translation plane where every line is a translation line.

In a projective plane, π, of order q a translation q-arc, K, is a q-arc whose stabiliser in the full automorphism group of π contains a group of elations acting transitively on its q points. A translation hyperoval is a hyperoval that contains a translation q-arc K. Since the group is transitive, the axis of the group of elations is external to K.

A semi-translation hyperoval [112] is a hyperoval with an elation group with common axis in its stabiliser each of whose orbits not on the axis, together with the points of the hyperoval on the common axis forms a translation hyperoval of a Baer subplane (cf. Ostrom’s definition [89] of a semi-translation plane).

Theorem 53 ([36]). Let g and h be elations with common axis l but distinct centres C and B, respectively, with g ≠ h ≠ 1 ≠ g. Then gh = hg.

Proof. Take a point P. If P ∈ l then gh(P) = hg(P). So suppose P ∉ l. Then C, P, and gP are collinear, and B, P, and hP are collinear. Let U = ChP ∩ BgP. Then hP = BP ∩ CU, so ghP = g(BP) ∩ g(CU) = BgP ∩ CU = U. Now gP = BU ∩ CP, so hgP = h(BU) ∩ h(CP) = BU ∩ ChP = U. So gh = hg.

Theorem 54 (Singer’s Theorem [122]). A Desarguesian plane π of order q has a cyclic collineation group of order q^{2} + q + 1 which is regular on the points and also on the lines.

An involution is a collineation of order 2.

Theorem 55. An elation stabilising a hyperoval is involutory.

Proof. The group generated by the elation acts semi-regularly on the non-fixed points, but interchanges the points of secancy on any non-axial secant line on the centre; thus the group has order 2.

Example. The complement of a line is a hyperoval in P G(2, 2). It is stabilised by all elations with centre on the line. Those with axis different from the line have axis secant to the hyperoval, while those with axis equal to the line have axis external to the hyperoval. The stabiliser of the hyperoval is S_{4} (see Theorem 41, p. 21). The elations with axis external to the hyperoval generate a Klein four group C_{2}^{2}. The elations with axis secant to the hyperoval generate the whole stabiliser. Each point not on the hyperoval is the centre of exactly three elations stabilising the hyperoval.

Let Fix(G) denote the configuration of fixed points and lines of the group G and let Fix(g) denote the configuration of fixed points and lines of the cyclic group generated by the element g.

Theorem 56 ([9]). An involutory collineation, g, of a projective plane of order q is

(i) an elation and q is even; or (ii) a homology and q is odd; or (iii) a Baer collineation (that is, Fix(g) is a Baer subplane) and q is square.

Proof. Every non-fixed line contains a fixed point (namely, the intersection of the line and its image under g). Thus, if the collineation is planar, then it is Baer. If it is not planar, then it is a generalised homology or a generalised elation (see page 31), and so it is a homology or an elation, by above.

Theorem 57 ([58]). If π is a projective plane of order q ≡ 2 (mod 4) and if π possess a collineation of order two, then q = 2.

Theorem 58 ([111]). An elation stabilising a hyperoval has centre not on the hyperoval and, if the plane has order greater than two, has axis secant to the hyperoval.

Proof. By the proof of Theorem 55, the centre is not on the hyperoval. By Theorems 55 and 57, the order of the plane is not congruent to 2 modulo 4 if it is bigger than 2. Thus, there are an odd number of secant lines on the fixed point not equal to the centre of the elation, from which it follows that the unique fixed line on this point (namely, the axis) is a secant line.

Theorem 59. A planar collineation fixing a hyperoval either has its fixed subplane disjoint from the hyperoval or they intersect in a hyperoval of the fixed subplane.

Proof. If a fixed line contains a fixed point on the hyperoval, then it is a secant line, so it contains two fixed points. Thus the set of fixed points on the hyperoval is set of points of the fixed subplane of class [0, 2], so is either empty or a hyperoval.

Example. A regular hyperoval in P G(2, 4) has stabiliser isomorphic to S_{6} (see Theorem 42, p. 21). It has three types of involutions in its stabiliser: Baer involutions fixing no points of the hyperoval, Baer involutions fixing four points of the hyperoval and elations fixing two points of the hyperoval. Each point not on the hyperoval is the centre of exactly three elations stabilising the hyperoval.

Theorem 60 ([15]). Two distinct nonidentity elations stabilising a hyperoval in a projective plane of order q with q > 4 have distinct centres.

Proof. If not, then, for each secant line on the common centre that is not the axis of either of the elations, the points of secancy are interchanged by both elations, so their product fixes an arc of size at least q − 2 which is greater than \sqrt{q} + 2, so lies in no proper subplane. Thus their product is the identity, contrary to their being distinct.

Theorem 61 ([15]). Two distinct Baer involutions fixing a hyperoval have distinct fixed planes.

Proof. If not, any point P of the hyperoval not in the common fixed subplane has the line joining it and its image (under either of the involutions) in the subplane. This is a secant line and the image of P under each involution must therefore be the other point of secancy, so P is fixed by the product of the involutions; thus the fixed configuration of the product of the involutions is the whole plane, that is, the product is the identity, that is, the involutions are not distinct.

Theorem 62 ([113]). The product of two involutorial elations, the centre of each of which is not incident with the axis of the other, fixes no point not on the line joining the two centres and not equal to the intersection of the two axes.

Proof. Suppose not and let g be the product of the two involutions. Then, since the intersection P of the axes, the line l joining the centres, and some point Q not on the line are fixed by g, g fixes the line m = P Q, and so the point R of intersection of l and m. Thus g fixes three collinear points on m. It follows that if g is not planar, then m is the unique line containing more than two fixed points of g. The elations invert g, but do not both fix m, so g is planar. Thus the elations fix the fixed subplane π_{0} of g. They induce involutions of π_{0} with all fixed points collinear, so these must be elations. But these induced involutions commute, contrary to the original hypothesis and the axes and centres.

Theorem 63 ([113]). The group H generated by the product of two involutorial elations g and h, the centre of each of which is not incident with the axis of the other, acts semiregularly on the points not on the line joining the two centres and not equal to the intersection of the two axes.

Proof. Let G be the group generated by g and h. Then G is a dihedral group with H as a cyclic subgroup of index 2. Each nonidentity element of H is the product of two distinct elements of G that are not in H. Each of the elements of G that are not in H are involutions that are conjugate either to g or to h, and so are elations.

Moreover, any two such have the property that the centre of each of them is not incident with the axis of the other. Thus Theorem 62 applies, and any nonidentity element of H fixes no point not on the line joining the two centres and not equal to the intersection of the two axes.

Theorem 64 ([113]). The product of two involutorial elations g and h of a projective plane of order q, the centre of each of which is not incident with the axis of the other, has order dividing q^{2} − 1 and not equal to one.

Proof. This follows from Theorem 63.

Theorem 65 ([15]). If two distinct elations stabilise a hyperoval in a plane of order q, then one of the following situations occurs:

(i) the elations have different centres but the same axis, which is a secant line to the hyperoval, and the product of the two elations is an involutory elation with the same axis but a different centre; or

(ii) the axes are distinct and meet at a point of the hyperoval, the centres are

distinct and the line joining the centres is (a) a secant line and the product of the elations has order dividing q − 1; or (b) an external line, and the product of the elations has order dividing q + 1;

or

(iii) the axes are distinct secant lines which meet at a point not on the hyperoval, the centres are distinct and the line joining them is external to the hyperoval, the product of the elations has order 3, and q is congruent to 1 modulo 3; or

(iv) q is 2 or 4.

Proof. Suppose that q > 4, so that we are not in case (iv). Then by Theorem 58, p. 28, the axes are secant lines. If they are distinct, then, by Theorem 60, p. 29, they have different centres and case (i) arises. If they are distinct then their intersection is not the centre of either of the elations, as then, conjugating the first elation (which has centre the intersection of the two axes) by the second elation we obtain an elation with the same centre as the first elation but a different axis. Thus there would be two elations with the same centre fixing the hyperoval contrary to Theorem 60, p. 29. Now Theorems 63 and 64, apply, and the group generated by their product has order dividing q^{2} − 1 and acts semi-regularly on the points of the hyperoval not on the line joining the centres and not the intersection of the axes. If the axes meet at a point of the hyperoval, then either the line joining the centres is secant and the order of the product divides q − 1 or the line joining the centres is external and the order of the product divides q + 1, which are cases (ii)(a) and (ii)(b), respectively. If the axes meet at a point not on the hyperoval, then the line joining the centres cannot be a secant line, for then the order of the product would be a nontrivial divisor of q^{2} − 1 and q, which are coprime. So it is external and the order of the product is a nontrivial divisor of q^{2} − 1 and q + 2, from which it follows that it is 3 and that q is congruent to 1 modulo 3. This is case (iii).

The examples after Theorems 55, p. 27 and Theorem 59, p. 28 show that case (iv) in the above theorem is genuinely exceptional.

Examples. (i) Translation hyperovals and semi-translation hyperovals (see

Section 3.1, p. 27) give examples of (i) of Theorem 65. (ii) Regular hyperovals in Desarguesian plane give examples of (ii)(a) of Theo-

rem 65. (iii) Regular hyperovals and the first class of Subiaco hyperovals (see Section 9,

p. 149), for q = 4^{e}, e odd, give examples of (ii)(b) of Theorem 65. (iv) The regular hyperoval in P G(2, 4) and the Lunelli-Sce hyperoval (see Section 9,

p. 149) in P G(2, 16) give examples of (iii) of Theorem 65.

Theorem 66 ([15][105]). If q is not congruent to 1 modulo 3 and not equal to 2 then there is no hyperoval in a projective plane of order q which is stabilised by three elations whose axes form a triangle.

Proof. In these cases, (i), (iii) and (iv) cannot apply to any two of these elations, so that (ii) holds (referring to the cases in Theorem 65). But now two of the elations generate a dihedral group containing an odd number of elations, all with distinct axes, all meeting at a point on the hyperoval; the third elation cannot have axis meeting all of these axes on the hyperoval.

Theorem 67 ([113]). In a group of a projective plane of order q > 4 containing involutory elations and fixing no point or line all elations are conjugate.

Let g ∈ Aut π, with t + 1 fixed points on l and t + 1 fixed lines on C, with C ∉ l. Then g is a generalised homology (of order t). We say g has pseudo-axis l and pseudo-centre C. If t = 2 then the centre and axis are not unique for a generalised homology. Similarly we can define a generalised elation (of order t) with centre C incident with axis l. Note, the terms centre and axis have different meanings here to those on page 26, but the context will make it clear which definition we will be using.

Theorem 68 ([105], cf Theorem 52, p. 27). Any nonidentity generalised homology of prime order stabilising a hyperoval in a projective plane of order q has at most (q + 2)/2 fixed points. If it has more than three fixed points, then it has order 3, and q is congruent to 1 modulo 3.

Proof. A generalised homology of a projective plane containing a hyperoval has odd order. Thus, if the generalised homology is not the identity, every orbit of points on a fixed line that is not just a fixed point contains a triple of collinear points. If the generalised homology has more than three fixed points then it has unique centre and axis, the centre is not on the hyperoval and the axis is external to the hyperoval. (If the centre were on the hyperoval, then each fixed line on the centre would be secant to the hyperoval, so each fixed point of the axis would be on the hyperoval, so the axis would meet the hyperoval in more than two points. So the centre is not on the hyperoval, making the fixed lines on the centre external, and so the axis external.) There are (q + 2)/2 secant lines on the centre, none of which are fixed. So there are at most q/2 fixed lines on the centre, giving at most (q + 2)/2 fixed points. The group generated by the generalised homology acts semi-regularly on the (q + 2)/2 secant lines through the centre. Thus the generalised homology has order 3, since a generalised homology of prime order has order dividing q − 1.

Example ([112]). In the dual Hall plane of order 16, there is a hyperoval stabilised by a generalised homology of order 3 with 6 fixed points. The full stabiliser is dihedral of order 6 with the involutions being Baer. In the dual Dempwolff plane of order 16, there is a hyperoval stabilised by a generalised homology of order 3 with 9 fixed points, which gives equality in the bound. The full stabiliser is cyclic of order 6 with the involution being Baer. In the semifield plane of order 16 with kernel GF (2), there is a hyperoval stabilised by a generalised homology of order 3 with 6 fixed points. In this case, the full stabiliser has order 3. These are the only hyperovals in the known planes of order 16 stabilised by generalised homologies with more than three fixed points.

Theorem 69 (cf [78]). Let H be a hyperoval in a projective plane π of order q, q even, and let G = (Aut π)_{H}. The ovals obtained by deleting the points P and Q of H are equivalent if and only if P and Q lie in the same orbit of G in H.

Proof. Straightforward.

Theorem 70 (cf [78]). Let O be an oval of projective plane π of order q, q even, with nucleus N, so that H = O ∪ {N}. Let G = (Aut π)_{H} and suppose that the orbit of G on H which contains N has n points. Then the stabiliser J in Aut π of O has order |G|/n.

3.2. Collineations of P G(2, q)

P Γ L(3, q) acts faithfully on P G(2, q). We have P GL(3, q) being regular on ordered 4-arcs in P G(2, q), and P GL(2, q) being sharply 3-transitive on points of P G(1, q). In fact, P GL(n, q) is 2-transitive on points of P G(n − 1, q)

The stabiliser of any three distinct points of P G(1, q) in P GL(2, q) is trivial.

Theorem 71 ([36]). If a homography of P G(2, q) fixes three collinear points then it fixes every point on the line joining them.

Proof. Let l be a line in P G(2, q). Then we have

P GL(3, q)_{l}^{l} ≅ P GL(2, q)^{P G(1,q)} ≅ P GL(2, q)_{P G(1,q)}^{P G(1,q)}.

We have for a line l, P GL(3, q)_{l}^{l} is sharply 3-transitive on the points of l. So if h ∈ P GL(3, q)_{l}^{l} fixes three distinct points of l, then h = 1. So if h ∈ P GL(3, q)_{l} fixes three distinct points of l, then h has an axis. If h ∈ P GL(3, q) fixes three collinear points, then h has axis l.

So we can say that if h is a non-trivial homography its fixed configuration does not contain a quadrangle.

Let A ∈ P GL(3, q). We would like to know what is the image of the line [a, b, c] under A. Now

(x, y, z) I [a, b, c] ⟺ ax + by + cz = 0

⟺ (a b c) (\begin{matrix}x \\ y \\ z \end{matrix}) = 0

Since collineations preserve incidence we have (x, y, z) I [a, b, c] if and only if A(x, y, z) I A[a, b, c]. Let A(x, y, z) = (x′, y′, z′), so (x, y, z) = A^{−1}(x′, y′, z′). So

(a b c) (\begin{matrix}x \\ y \\ z \end{matrix}) = 0 ⟺ (x′, y′, z′) I A[a, b, c]

⟺ (a b c) A^{−1} (\begin{matrix}x′ \\ y′ \\ z′ \end{matrix}) = 0.

Hence A[a, b, c] is defined to be (a b c) A^{−1}. So the action of GL(3, q) on the dual of P G(2, q) is via A^{−T}.

A flag is an incident point-line pair. An antiflag is a nonincident point-line pair. We have P GL(3, q) being transitive on flags of P G(2, q). The centre and axis of

an elation is a flag hence without loss of generality the flag is ((0, 1, 0), [1, 0, 0]) up to conjugation.

Theorem 72.

G = \{(\begin{matrix}1 & 0 & 0 \\ a & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) | a ∈ GF (q)\} ,

is the set of all elations with centre (0, 1, 0) and axis [1, 0, 0]. G is transitive on the q points of [1, 0, 0] not equal to (0, 1, 0).

Proof. We have (x, y, z) I [1, 0, 0] if and only if x = 0, and

(\begin{matrix}1 & 0 & 0 \\ a & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}0 \\ y \\ z \end{matrix}) = (\begin{matrix}0 \\ y \\ z \end{matrix}) ,

so [1, 0, 0] is the axis.

Now (0, 1, 0) I [d, e, f] if and only if e = 0. So we look at the points on the lines [d, 0, f]. Since (x, y, z) I [d, 0, f] if and only if dx + fz = 0 and elements of G take (x, y, z) to (x, ax+y, z), we have (x, y, z) I [d, 0, f] if and only if (x, ax+y, z) I [d, 0, f]. So the lines [d, 0, f] are fixed, that is, (0, 1, 0) is the centre.

Let a, b ∈ GF (q)^{∗}. Then

(\begin{matrix}1 & 0 & 0 \\ 0 & b/a & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 & 0 \\ a & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 & 0 \\ 0 & b/a & 0 \\ 0 & 0 & 1 \end{matrix})^{−1} = (\begin{matrix}1 & 0 & 0 \\ 0 & b/a & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 & 0 \\ a & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 & 0 \\ 0 & a/b & 0 \\ 0 & 0 & 1 \end{matrix})

= (\begin{matrix}1 & 0 & 0 \\ b & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

The above proof shows that all non-identity elations are conjugate.

By the above techniques we can see that the map

(\begin{matrix}1 & 0 & 0 \\ a & 1 & 0 \\ b & 0 & 1 \end{matrix})

has centre (0, a, b) and axis [1, 0, 0]. Hence

\{(\begin{matrix}1 & 0 & 0 \\ a & 1 & 0 \\ b & 0 & 1 \end{matrix}) | a, b ∈ GF (q)\} = AGL(3, q)_{([1,0,0])}.

Let

U = \{(\begin{matrix}1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1 \end{matrix}) | a, b, c ∈ GF (q)\} .

Since |U| = q^{3} then U ∈ Syl_{p}(P GL(3, q)), where q = p^{e}, with U fixing a unique flag ((0, 0, 1), [1, 0, 0]). Hence

N_{P GL(3,q)}(U) = \{(\begin{matrix}λ & 0 & 0 \\ a & μ & 0 \\ b & c & ν \end{matrix}) | a, b, c ∈ GF (q), λ, μ, ν ∈ GF (q)^{∗}\}
= P GL(3, q)_{(0,0,1),[1,0,0]}.

The subgroup of Aut P G(2, q) generated by the collection of all elations is P SL(3, q).

P GL(3, q) is transitive on antiflags. As the centre and axis of a homology is an antiflag, we can without loss of generality let the antiflag be ((1, 0, 0), [1, 0, 0]). Then

\{(\begin{matrix}λ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) | λ ∈ GF (q)^{∗}\}

is all such homologies. All homologies of order r with r |q − 1 are conjugate.

Example (Generalised Homologies in P G(2, q)). Let α ∈ Aut GF (q) and α ≠ id. Then

g: (\begin{matrix}x \\ y \\ z \end{matrix}) ↦ (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & λ \end{matrix}) (\begin{matrix}x^{α} \\ y^{α} \\ z^{α} \end{matrix})

with λ ≠ 1 is a generalised homology (except when it’s planar for α = 2 and q = 4) with pseudo-axis [0, 0, 1] and pseudo-centre (0, 0, 1) with the fixed points on the axis being

{(1, t, 0) | t^{α} = t} ∪ {(0, 1, 0)}.

So g has 2 + q_{0} fixed points and 2 + q_{0} fixed lines if

α: x ↦ x^{q_{0}}.

Claim: All generalised homologies with more than three fixed points in P G(2, q) are conjugate to one of these. Proof: Without loss of generality let h have axis [0, 0, 1] and centre (0, 0, 1), and greater than three fixed points. Since there exist three collinear fixed points we have h^{[0,0,1]} ∈ P Γ L(2, q) fixing at least three points. Without loss of generality (1, 0, 0), (0, 1, 0), (1, 1, 0) are fixed (by the results on page 32). Hence h^{[0,0,1]} : (x, y, 0) ↦ (x^{α}, y^{α}, 0) for some α ∈ Aut GF (q) and since h is not a homology then α ≠ 1. We have

ker P Γ L(3, q)_{(0,0,1),[0,0,1]}^{[0,0,1]} = \{(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & λ \end{matrix}) | λ ∈ GF (q)^{∗}\} .

So

h: (\begin{matrix}x \\ y \\ z \end{matrix}) ↦ (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & λ \end{matrix}) (\begin{matrix}x^{α} \\ y^{α} \\ z^{α} \end{matrix}) ,

for some λ ∈ GF (q)^{∗} and λ ≠ 1.

Theorem 73 ([37]). In P G(2, q). We have D_{2(q+1)} ≤ P GL(2, q).

Theorem 74 ([84]). Let K be a (q +1)-arc in P G(2, q), q even, and suppose that K has a transitive homography stabiliser. Then it is a nondegenerate conic.

We make use of the below theorem constantly through Section 8, p. 55.

Lemma 75 ([84]). Let J be a cyclic subgroup of order q +1, q even, of P Γ L(3, q). Suppose that J fixes a point and fixes a line not on the point. Then the orbits of J on the points of P G(2, q) are the fixed point, the fixed line and q − 1 nondegenerate conics.

Proof. The stabiliser K in P GL(3, q) of a non-incident point-line pair in P G(2, q) is isomorphic to GL(2, q). Now a cyclic subgroup of GL(2, q) of order q + 1 induces a cyclic subgroup of P GL(2, q) of order q + 1, as (q − 1, q + 1) = 1. By inspection of Dickson’s list of subgroups of P GL(2, q) [37] we see that any two cyclic subgroups of order q + 1 of P GL(2, q) are conjugate. Since q is even, this forces any two cyclic subgroups of order q + 1 of GL(2, q) to be conjugate. For if K_{1} = ⟨k_{1}⟩ and K_{2} are subgroups of order q +1, conjugate in P GL(2, q), there exists g ∈ P GL(2, q), k_{2} ∈ K_{2} and λ ∈ GF (q) such that g^{−1}k_{1}g = λk_{2}. Since k_{1} and k_{2} have order q + 1, it follows that λ^{q+1} = λ^{2} = 1 so that λ = 1. Since P GL(3, q) is transitive on antiflags, any two antiflags have conjugate stabilisers. Hence any two cyclic subgroups of order q + 1 of P GL(2, q) which fix an antiflag are conjugate, so any two cyclic subgroups of order q + 1 of K are conjugate (since K ≅ GL(2, q)).

It is therefore enough to display a cyclic subgroup of order q + 1 of P GL(3, q), q even, whose orbits on P G(2, q) are a point, a line and q − 1 nondegenerate conics. We consider the following pencil of conics:

λ(x^{2} + axy + y^{2}) + μz^{2} = 0

for λ, μ ∈ GF (q), not both zero and trace(1/a) = 1. The degenerate conic given by μ = 0 is the point (0, 0, 1) and the degenerate conic given by λ = 0 is the line z = 0. The other q − 1 conics of the pencil are nondegenerate. The q + 1 conics of the pencil are pairwise disjoint and cover P G(2, q).

Let F be the following set of homographies:

f = \{(\begin{matrix}f & g & 0 \\ g & f + ag & 0 \\ 0 & 0 & 1 \end{matrix}) | f, g ∈ GF (q), not both zero\} .

A conic of the pencil is fixed by an element of F provided the matrix has determinant f^{2}+afg+g^{2} = 1. Now F is a cyclic subgroup of order q^{2}−1 of GL(3, q), since the 2×2 submatrices formed by the first two rows and columns form a field under addition and multiplication of matrices (adjoining the zero matrix), so they form a cyclic group under multiplication. But these 2 × 2 matrices determine the multiplicative structure of F. Thus F ∩ SL(3, q) is a cyclic subgroup of order q + 1 of GL(3, q) (as it is the kernel of the homomorphism defined by the determinant map, which is onto GF (q)^{∗} as can be seen by considering the matrices with g = 0). So F ∩ SL(3, q) gives rise to a cyclic subgroup of order q + 1 of P GL(3, q). Also, J has orbits on the points of P G(2, q) a point, a line and q − 1 nondegenerate conics, as required.

Lemma 76 (T. Penttila, private communication). Up to conjugacy, there is a unique cyclic subgroup of P GL(3, q) of order q + 1 for q odd. It’s orbits are: a point C, two orbits of length (q + 1)/2 together forming a line l, and q − 1 nondegenerate conics, with respect to each of which C and l are the pole-polar relationship.

Proof. Existence: The stabiliser of a nondegenerate conic in P GL(3, q) is isomorphic to P GL(2, q), which has a cyclic subgroup of order q + 1 by Singer’s Theorem 54, p. 27.

Uniqueness: As q + 1 is even, so a cyclic subgroup G of that order contains and centralises an element of order 2 which must be a homology. Hence it fixes the centre C and the axis l of the homology. Since (q − 1, q + 1) = 2, the kernel of the action on l has order 2, so the group induced on l has order (q + 1)/2. By Dickson’s List [37], there is, up to conjugacy, a unique cyclic subgroup of P GL(2, q) of order (q + 1)/2, and it follows that G is determined up to conjugacy.

Orbits: The stabiliser of an internal point in the stabiliser of a conic is dihedral of order 2(q + 1) and so contains a cyclic subgroup of order q + 1, which is transitive on the conic, fixes the internal point C and its polar line l and has two orbits of length (q + 1)/2 on that polar line.

Identifying GF (q^{2}) with AG(2, q) we see that there is a cyclic subgroup of P GL(3, q) of order q^{2} − 1, fixing a point C, a line l and transitive on the points of the plane other than C and l. Hence all orbits of a cyclic group of order q + 1, other than C and the two orbits on l, are nondegenerate conics.

3.3. Fixed configurations for P GL(3, q)

Let g be a non-trivial homography of P G(2, q) and let |g| = r. If there exists a triple of collinear fixed points of g then by Theorem 71, p. 32, g has an axis, and so centre. If the centre is on an axis then g is an elation and no further fixed points as then it would fix a quadrangle. If the centre is not on the axis then we have a homology. Otherwise, no three fixed points are collinear, and dually no three fixed lines are concurrent. So there are at most three fixed points, else we have a fixed quadrangle.

If we have three fixed points then we have the fixed configuration of Figure 1, a triangle element, and hence r |q − 1.

If we have two fixed points then we have the configuration of Figure 2, and hence r |q and r |q − 1. Hence the order of the collineation cannot be a prime.

If we have one fixed point then we have the configuration of Figure 3. If we have no fixed points then we have the empty configuration and hence r |q^{2} + q + 1.

Note. The above requires the use of homographies, as collineations have the counterexample of a subplane.

In Section 8, p. 55 we use the above results extensively.

3.4. Subgroups of P GL(3, q)

Let Y be a set of points of P G(2, q) and let H = P Γ L(3, q)_{Y}, then P Γ L(3, q)_{gY} = gHg^{−1} for g ∈ P Γ L(3, q). Since g can be thought of as a change of coordinates, if we only study points up to a change of coordinates, then we only study their group of symmetries up to conjugation. If r ||H|, r prime, we have by Cauchy’s Theorem 14, p. 5, there exists R ≤ H with |R| = r.

Suppose there are t conjugacy classes of subgroups of P Γ L(3, q) of order r with representatives R_{1}, …, R_{t}. Then R = g^{−1}R_{i}g for some i. Hence gY has stabiliser gHg^{−1} with subgroup gRg^{−1} = R_{i}. So by changing coordinates we can assume that

[figure] Figure 1.

[figure] Figure 2.

Figure 3.

Y has a stabiliser with subgroup R_{i} for some i. Note the above can be replaced by P GL(3, q) since it is normal in P Γ L(3, q).

Now suppose we have T ≤ H. If we change coordinates by an element g we need to know if T ≤ gHg^{−1}. Since it is possible that T may equal H we can only change coordinates with an element g such that gT g^{−1} = T, that is, by elements of N_{P Γ L(3,q)}(T). See Section 8, p. 55, for motivation of the above few paragraphs.

Theorem 77. If r is a prime, r ≠ 3, dividing q^{2} + q + 1 then there is a unique conjugacy class of subgroups of P GL(3, q) of order r^{i} for all i with r^{i} ||P GL(3, q)|.

Proof. From Singer’s Theorem 54, p. 27, there exists a cyclic subgroup of P GL(3, q) of order q^{2} + q + 1 which is regular on points (and lines) of P G(2, q). Hence a Sylow r-subgroup of P GL(3, q) is cyclic, since r | (q^{2} + q)q^{2}(q − 1)^{2} as r ≠ 3. If R_{1} and R_{2} are subgroups of P GL(3, q) with |R_{1}| = |R_{2}| = r^{i} for some i, then R_{1} ≤ S_{1} and R_{2} ≤ S_{2} with S_{1}, S_{2} being Sylow r-subgroups of P GL(3, q). So by Sylow’s Theorem there exists a g ∈ P GL(3, q) with gS_{1}g^{−1} = S_{2}. Hence gR_{1}g^{−1} ≤ S_{2} with |gR_{1}g^{−1}| = r^{i}. Since S_{2} is cyclic it has a unique subgroup of order r^{i}, so gR_{1}g^{−1} = R_{2}.

Corollary 78. If r is a prime dividing q + 1, r ≠ 2, then there is a unique conjugacy class of subgroups of order r^{i} for all i with r^{i} ||P GL(3, q)|.

Proof. By Singer’s Theorem 54, p. 27, there exists a cyclic subgroup of P GL(2, q) of order q + 1 and hence one of P GL(3, q) of order q^{2} − 1 and so has a subgroup of order q + 1. Hence a Sylow r-subgroup is cyclic since r | (q^{2} + q + 1)q^{2}(q − 1)^{2} as r ≠ 2. Then proceed similar to the last part of the proof of Theorem 77.

Theorem 79. If r is a prime dividing q −1 then there is a unique conjugacy class of subgroups generated by homologies of order r^{i} for r^{i} |q − 1.

Proof. Without loss of generality we denote the centre as (1,0,0) and the axis as [1,0,0], this gives the homology, g, as

g = (\begin{matrix}λ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) ,

with λ^{r^{i}} = 1 and λ^{n} ≠ 1 for n < r^{i}. Since GF (q)^{∗} is cyclic this determines ⟨g⟩.

Note. If r is a prime dividing q−1 then there are about (r−2)/6 conjugacy classes of subgroups of order r with generators having three fixed points with representatives of the form

g_{i} = (\begin{matrix}1 & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ^{i} \end{matrix}) ,

with λ^{r} = 1, λ ≠ 1 and i ≠ 1, 0 (mod r). We have

g_{i}, g_{1−i}, g_{1/i}, g_{i/(i−1)}, g_{1/(1−i)}, g_{(i−1)/i},

generating groups conjugate to this, hence explaining the dividing by six part of the above note.

Theorem 80. (i) If r = 3|q^{2}+q+1 then there exists a unique conjugacy class of subgroups of P GL(3, q) of order 3 with generator having no fixed points.

(ii) If r = 2 | q − 1 then there exists a unique conjugacy class of subgroups of

P GL(3, q) of order 2; this subgroup is generated by a homology. (iii) If r = p = 2 then there exists a unique conjugacy class of subgroups of

P GL(3, q), where q = p^{e}, of order 2; it is generated by an elation. (iv) If r = p ≠ 2 then there exists two subgroups of P GL(3, q), where q = p^{e}, of

order p; one is generated by an elation and the other by an element having one fixed point.

3.5. Polarities

A correlation of a projective plane is a permutation of the lines and points of the plane. A correlation is a collineation if it maps points to points and lines to lines, else it is a duality of the projective plane. If a duality has order two then it is a polarity.

Theorem 81 ([8]). A polarity Δ of a finite projective plane has absolute points. If the plane has order q then the number of absolute points is (q + 1) + r\sqrt{q} for some r ∈ ℤ.

Proof. Label the points of the projective plane of order q as P_{1}, …, P_{q^{2}+q+1} and the lines ΔP_{1}, …, ΔP_{q^{2}+q+1}. With this ordering let A be the incidence matrix of the plane so

A_{ij} = \begin{cases}1 & if P_{i} I ΔP_{j} \\ 0 & if P_{i} I̸ ΔP_{j} \end{cases}

Then A is symmetric since P_{i} I ΔP_{j} ⟺ ΔP_{i} I Δ^{2}P_{j} ⟺ ΔP_{i} I P_{j}. The number of absolute points of Δ is trace(A). We find

trace(A) = q + 1 + (q^{2} + q − 2d)\sqrt{q}

where d is the multiplicity of the eigenvalue of −\sqrt{q}.

Theorem 82 ([8]). Let r ≥ 0. If q is not a square then r = 0. If r = 0 then (i) If q is even the absolute points are the points of a non-absolute line l and the

absolute lines are the lines concurrent to Δl.

4. Hyperovals in P G(2, q)  41

(ii) If q is odd then the absolute points are the points of an oval and the absolute lines are the tangent lines to the oval.

Proof. Firstly we can not have two absolute points on an absolute line. Suppose P and Q are absolute points on an absolute line l. Let Z I̸ l. Then Z I ΔP and Z I ΔQ, so P I ΔZ and Q I ΔZ this implies that ΔZ I l, but l is absolute and Δl ≠ Z.

Let P be an absolute point so P I ΔP. Let l be a line through P. Then Δl is on ΔP, so l is not absolute. Hence every absolute line lies on exactly one absolute point and dually.

Now we want to prove that the number of non-absolute points incident with a non-absolute line is even. Let l be a non-absolute line. Define a map, g, on the points of l by g(P) = ΔP ∩ l where P I l. Then g is a permutation of order 2, since g(P) I ΔP implies P I Δg(P) we have P = Δg(P) ∩ l = g(g(P)) = g^{2}(P). The fixed points of g are the absolute points on l. So we have that there is an even number of non-absolute points on an non-absolute line.

Let q be even. Then every line has at least one absolute point. Hence there are at least q + 1 absolute points in the plane. Suppose there are q + 1 absolute points.

Let l be a line with a point P which is not absolute. Since q is even every line has at least one absolute point on it, different from P, and the lines have no absolute point in common. Hence every line on P contains exactly one absolute point. So if any point of a line is not absolute then that line only has one absolute point on it. Since two absolute points are on a line this line has all its points absolute. As this line has q + 1 absolute points on it there are no other absolute points. So for q even the set of absolute points is a line.

Let q be odd. By Theorem 81, p. 40, there exists an absolute point P. Let l be a line on P different from ΔP. Since l has one absolute point already and l is non-absolute and q +1 is even then l has another absolute point. So there are at least q + 1 absolute points.

Suppose there are q + 1 absolute points. Then every line l on an absolute point P such that l ≠ ΔP is on exactly two absolute points. As P was arbitrary we have that the set of absolute points is an oval, with the absolute lines being the tangent lines.

4. Hyperovals in P G(2, q)

Since P Γ L(3, q) is transitive on 4-arcs we can map a hyperoval H′ to a hyperoval H containing the points {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1)}, the fundamental quadrangle. We say any two sets, X_{1}, X_{2}, of points are equivalent if one is the image of another under some element of P Γ L(3, q), and we write X_{1} ≅ X_{2}.

Theorem 83 ([55]). A hyperoval, H, in P G(2, q), q even, q > 2, is equivalent to

D(f) = {(1, t, f(t)) | t ∈ GF (q)} ∪ {(0, 0, 1), (0, 1, 0)},

such that f is a permutation polynomial with f(0) = 0 and f(1) = 1 (see Section 5.1, p. 43).

When we can describe a hyperoval by such an f we say that f is an o-polynomial and the hyperoval is denoted D(f). We say o-polynomials are equivalent when their respective hyperovals are equivalent, and we write f ≅ g if f and g are equivalent.

Theorem 84 ([55]). Let f be a permutation polynomial of GF (q) with f(0) = 0 and f(1) = 1. For s ∈ GF (q) define a polynomial, f_{s}, on GF (q) as f_{s}(0) = 0 and f_{s}(x) = (f(x + s) + f(s))/x for x ∈ GF (q)^{∗}. Then f is an o-polynomial when f_{s} is a permutation polynomial for all s ∈ GF (q).

Theorem 85 ([55]). If D(f) with f(x) = ∑_{i=1}^{q−2} a_{i}x^{i} is a hyperoval in P G(2, q), q even, q > 2, then a_{i} = 0 for i odd, that is

f(x) = ∑_{j=1}^{\frac{q−2}{2}} a_{2j}x^{2j}.

Theorem 86 ([29]). Given f an o-polynomial, then the set of points D(f), under the following list of images of S_{4} implicitly define o-polynomials equivalent to f where the maps are; (x, y, z) maps to:

(a) (x + y, x, x + z) (b) (y + z, x + z, z) (c) (x + y, y, y + z)
(d) (x, x + z, x + y) (e) (x + z, x + y, x) (f) (y + z, x + y, y)
(g) (y, y + z, x + y) (h) (x + z, z, y + z) (i) (x, x + y, x + z)
(j) (x + y, x + z, x) (k) (z, x, y)  (l) (x + y, y + z, y)
(m) (y + z, z, x + z) (n) (z, y, x)  (o) (y, x + y, y + z)
(p) (x + z, x, x + y) (q) (x, z, y)  (r) (y + z, y, x + y)
(s) (y, z, x)  (t) (z, x + z, y + z) (u) (z, y + z, x + z)
(v) (y, x, z)  (w) (x + z, y + z, z) (x) (x, y, z)

The maps (k), (n), (q), (s), (v) fix the point (1, 1, 1); the maps (a), (c), (i), (o), (v) fix (0, 0, 1); the maps (e), (n), (u), (w) fix (0, 1, 0); the maps (c), (h), (l), (q), (w) fix (1, 0, 0). The maps (c), (i), (n), (q), (v) and (w) are involutions. In particular, the following polynomials g are o-polynomials equivalent to a given o-polynomial f:

(i) g(x) = f^{−1}(x), by map (q), (ii) g(x) = xf(x^{−1}), g(0) = 0, by map (v), (iii) g(x) = f(x + 1) + 1, by map (i), (iv) g(x) = (x + 1)f((x + 1)^{−1}) + 1, g(1) = 1, by map (o),

(v) g(x) = x + xf(1 + x^{−1}), g(0) = 0, by map (a), (vi) g(x) = x + (x + 1)f(x(x + 1)^{−1}), g(1) = 1, by map (c). as well as with each of the above with f replaced with f^{−1}, which are arrived by the maps (x), (k), (d), (t), (p) and (h), respectively.

When an o-polynomial f(x) = x^{k} for some k we denote the hyperoval of f as D(k).

Theorem 87 ([55]). If D(k) is a hyperoval then (k, q − 1) = (k − 1, q − 1) = 1 and

D(k) ≅ D(1 − k) ≅ D(k^{−1}) ≅ D(1 − k^{−1}) ≅ D((1 − k)^{−1}) ≅ D(k(k − 1)^{−1}).

Proof. k, 1 − k, k^{−1}, 1 − k^{−1}, (1 − k)^{−1}, k(k − 1)^{−1} are integers modulo q − 1. As (k, q − 1) = 1, k^{−1} is well-defined, and as (k − 1, q − 1) = 1, then (1 − k)^{−1} and (k − 1)^{−1} are well-defined.

From Theorem 86 use the maps (v), (q), (k), (s), and (n).

Corollary 88 ([55]). Let D(k) be a hyperoval. Then it is regular if and only if k = 2, 1/2 or −1.

Theorem 89 ([81]). When q ≥ 8 the q + 2 equivalent o-polynomials representing a regular hyperoval are exactly

(i) the 3 monomials x^{2}, x^{1/2}, x^{−1}, (ii) the q − 1 polynomials

g_{s}(x) = ∑_{i=1}^{(q−2)/2}(s + 1)s^{2i−1}x^{2i},

for s ∈ GF (q)^{∗}.

Corollary 90 ([55]). The only ovals of P G(2, 8) are translation ovals (in fact, conics and pointed conics).

Lemma 91. A hyperoval in P G(2, q), q = 2^{e}, is represented by an o-polynomial written over GF (q_{0}), q_{0} = 2^{h}, if and only if it is stabilised by the group of order e/h generated by

(x, y, z) ↦ (x^{q_{0}}, y^{q_{0}}, z^{q_{0}}).

5. Glynn’s Inequality

5.1. Lagrange Interpolation

Let g: GF (q) → GF (q) be any function. Then g corresponds to a unique polynomial f over GF (q) and of degree less than or equal to q − 1. Then f(t) = ∑_{0}^{q−1} a_{i}t^{i} with a_{i} ∈ GF (q), where a_{0} = g(0), a_{r} = − ∑_{x∈GF (q)^{∗}} g(x)x^{−r} for 1 ≤ r ≤ q − 2, and a_{q−1} = − ∑_{x∈GF (q)} g(x). This is known as Lagrange interpolation.

We use the techniques of Lagrange interpolation to generate polynomial functions. The concept of delta functions is used to construct the polynomials from the given data (used in Section 9, p. 62). To think of a polynomial as a function is to think of a ring homomorphism, ϕ, from GF (x)[x] to GF (q)^{GF (q)} with kernel (x^{q} − x) (see Section 5, p. 9, Chapter I). Now f(t) = 0 for all t if and only if x − t | f(x) for all t if and only if ∏_{t∈GF (q)}(x − t) | f(t). If h(x) = x^{q} − x then h(t) = 0 for all t ∈ GF (q) so h is in the kernel of ϕ. So ∏_{t∈GF (q)}(x − t) | h(t). Hence x^{q} −x = ∏_{e∈GF (q)}(x−t) = x ∏_{e∈GF (q)^{∗}}(x−t) gives that x^{q−1} −1 = ∏_{t∈GF (q)^{∗}}(x−t), hence we have ∏_{t∈GF (q)^{∗}}(−t) = −1.

Define the delta function as

δ_{b}(t) = \begin{cases}0 & if t ≠ b, \\ 1 & if t = b. \end{cases}

Now g(x) = (x^{q} − x)/(x − b) implies that g(t) = 0 if t ≠ b. From above, we have

g(b) = ∏_{\begin{matrix}t∈GF (q) \\ t≠b \end{matrix}}(b − t) = ∏_{c∈GF (q)^{∗}} c = −1.

Hence δ_{b} = −g, that is,

δ_{b}(x) = − (\frac{x^{q} − x}{x − b}) .

For q even, if b ≠ 0 we have

δ_{b}(x) = x^{q−1} + bx^{q−2} + ⋯ + b^{q−2}x,

and if b = 0 then

δ_{0}(x) = x^{q−1} + 1.

5.2. Glynn’s inequality

Define a partial ordering ≾ on the set of integers N_{q} = {n | 0 ≤ n ≤ q − 1}, with q = 2^{e}, with a, b ∈ N_{q} as

a ≾ b ⟺ a = ∑_{0}^{r−1} a_{i}2^{i}, b = ∑_{0}^{r−1} b_{i}2^{i}
with a_{i}, b_{i} ∈ {0, 1} for all i, and a_{i} ≤ b_{i} for all i.

In other words a ≾ b if and only if all terms appearing in the binary expansion of a also appear in the binary expansion of b. This is Glynn’s [44] partial ordering.

Result 92. If k ∈ ℤ then the expansion of

(1 + t)^{k} (mod t^{q} − t) over GF (q)

is calculated as follows:

1st: If k = 0 then (1 + t)^{k} = 1.

2nd: If k ≠ 0, let k = k (mod q − 1) so 1 ≤ k ≤ q − 1, then

(1 + t)^{k} = ∑_{\begin{matrix}c≾k \\ c∈N_{q} \end{matrix}} t^{c} (mod t^{q} − t).

Theorem 93 ([44]). D(k) is a hyperoval of P G(2, q) if and only if d ≾̸ kd, for all d ∈ {1, 2, …, q − 2}, where dk is reduced modulo q − 1 to lie in N_{q}.

Proof. Let k ∈ {1, 2, …, q − 2} and (k, q − 1) = (k − 1, q − 1) = 1. Then
D(k) is a hyperoval of P G(2, q)
⟺ f_{k}(t) = \frac{(t + 1)^{k} + 1}{t} is a permutation polynomial
⟺ [f_{k}(t)]^{r} (mod t^{q} − t) has zero coefficient for t^{q−1}
for all r ∈ {1, …, q − 2}
⟺ ∑_{λ∈GF (q)^{∗}}((λ + 1)^{k} + 1)λ^{−1})^{r} = 0 for all r ∈ {1, …, q − 2}
⟺ coefficients of t^{r} in ((t + 1)^{k} + 1)^{r} is zero for all r ∈ {1, …, q − 2}
⟺ ∑_{d≾r}(t + 1)^{kd} has zero coefficient for t^{r} for all r ∈ {1, …, q − 2}
⟺ ∑_{d≾r} ∑_{g≾kd} t^{g} has zero coefficient for t^{r} for all r ∈ {1, …, q − 2}
⟺ ∑_{d≾r≾kd} t^{r} = 0 for all r ∈ {1, …, q − 2}
⟺ |{d | d ≤ r ≤ kd}| ≅ 0 (mod 2) for all r ∈ {1, …, q − 2}
⟺ d ≾̸ kd for all d ∈ {1, …, q − 2}.

Now d ≾̸ kd for all d ∈ {1, …, q − 2} implies that (k, q − 1) = (k − 1, q − 1) = 1, since if (k, q − 1) = l ≠ 1, then m = (q − 1)/l satisfies m ∈ {1, …, q − 2} and mk ≅ q − 1 (mod q − 1). Hence m ≾ mk. Similarly if (k − 1, q − 1) = n ≠ 1 then p = (q − 1)/n satisfies p(k − 1) ≅ 0 (mod q − 1). Hence p ≅ pk (mod q − 1) and so p ≾ pk, p ∈ {1, …, q − 2}.

Theorem 94 ([44]). A polynomial f of degree at most q − 2 satisfying f(0) = 0 and f(1) = 1 is an o-polynomial if and only if the coefficient of x^{c} in f(x)^{b} (mod x^{q} − x) is zero for all pairs of integers (b, c) satisfying 1 ≤ b ≤ c ≤ q − 1, b ≠ q − 1, and b ≾ c.

Theorem 95 ([81]). Let f(x) = ∑_{i=1}^{q−2} a_{x}x^{i} and suppose that ∑_{i=1}^{q−2} a_{i} = 1. Then f is an o-polynomial if and only if for all b with 1 ≤ b ≤ q − 2 and all c with b ≾ c

∑ ∏_{k=1}^{n} a_{i_{k}}^{α_{k}} = 0,

where b = α_{1} + ⋯ + α_{n}, the α_{j} are distinct powers of 2 and the sum is over all i_{k} with c = ∑_{k=1}^{n} i_{k}α_{k} (mod q − 1).

Proof. Straightforward, use Theorem 94.

6. Known Hyperovals

We list the known hyperovals immediately and go into further detail later. Regular: Discussed in Section 2, p. 18, and on page 46. Translation: See page 47. Lunelli-Sce: See page 47. Segre-Bartocci: See page 49. Glynn: See page 50. Payne: See page 50. Cherowitzo: See page 51. O’Keefe-Penttila: See page 53. Irregular hyperovals in P G(2, 64): See page 54 for the two constructed by

Penttila and Pinneri and page 67 for the hyperoval constructed by Penttila and Royle.

Subiaco: See page 108 and page 149. Cyclic q–clan: See page 68 for the one constructed in P G(2, 256) by Penttila

and Royle, and page 121 for those constructed by cyclic q–clans in P G(2, q) for q = 256, 1024, 4096 and 16384.

We now go into some detail for the above hyperovals.

6.1. Regular hyperovals

We continue our investigation of the regular hyperovals from Section 2, p. 18.

Theorem 96 ([81]). If q = 2, 4 or 8 then every hyperoval is a regular hyperoval.

Proof. For q = 2 see Theorem 41 and for q = 4 see Theorem 42, p. 21. We prove that every hyperoval in P G(2, 8) is regular by showing each o-polynomial

describes a regular hyperoval. For q = 8 an o-polynomial will have degree being 2, 4 or 6 by Theorem 85, p. 42.

For f(x) = a_{2}x^{2} we have f(1) = 1 so a_{2} = 1, hence f(x) = x^{2}, and so equivalent to D(2).

For f(x) = a_{2}x^{2} + a_{4}x^{4} with a_{4} ≠ 0 we have as f(x) = 0 implies x = 0 so a_{2} = 0; with f(1) = 1 we have a_{4} = 1.

For f(x) = a_{2}x^{2} + a_{4}x^{4} + a_{6}x^{6} with a_{6} ≠ 0 we have a_{2} + a_{4} + a_{6} = 1. Consider the equations of Theorem 95, for the cases b = 3, c = 3 and b = 3, c = 7, which are

a_{4}^{2}a_{2} + a_{2}^{2}a_{6} = 0 and a_{6}^{2}a_{2} + a_{4}^{2}a_{6} = 0

respectively. There are 10 solutions (a_{6}, a_{4}, a_{2}) with a_{6} + a_{4} + a_{2} = 1 to this system of equations. This can be seen by noting that the system can be rewritten as

(a_{4}^{2} + a_{2}a_{6})a_{2} = 0 and (a_{4}^{2} + a_{2}a_{6})a_{6} = 0

so that the solutions are the point (0, 1, 0) together with the points of the conic a_{4}^{2} = a_{2}a_{6}. Since Theorem 89, p. 43, constructs 10 o-polynomials representing regular hyperovals, it follows that they are the only o-polynomials for P G(2, 8).

6.2. Translation hyperovals

The first irregular hyperovals to be found were the translation hyperovals [117] equivalent to D(2^{i}) where (i, e) = 1 with 1 < i and since D(2^{i}) ≅ D(2^{e−i}), we have i < \frac{e}{2}. A translation hyperoval has a stabiliser in P Γ L(3, q), q = 2^{e}, of order q(q−1)e.

Note that this is a translation hyperoval under the definition on page 27, since the map

(x, y, z) ↦ (x, y + cx, z + c^{2^{i}}x),

for all c ∈ GF (q) for the translation hyperoval D(2^{i}), is an elation fixing the translation hyperoval.

Theorem 97 ([84]). Let q ≥ 8 be even, let H be a hyperoval of P G(2, q) and let S be a Sylow 2-subgroup of the homography stabiliser H of H. If |S| = q then H is a translation hyperoval.

Corollary 98 ([84]). Let K be a q-arc in P G(2, q), with q ≥ 8 even, and suppose that it has a transitive homography stabiliser. Then K is a translation q-arc.

Theorem 99 ([84]). An irregular translation hyperoval in P G(2, q), q = 2^{e} ≥ 8, has homography stabiliser of order q(q−1) isomorphic to AGL(1, q), that is, C_{2}^{e}⋊C_{q−1}. The orbits on the hyperoval have lengths 1, 1 and q.

6.3. Lunelli-Sce hyperoval

In 1958 the Lunelli-Sce [75] hyperoval was found in P G(2, 16) with the use of a computer. The hyperoval is equivalent to D(f) with f(x) = x^{12} + x^{10} + η^{11}x^{8} + x^{6} + η^{2}x^{4} + η^{9}x^{2}, where η is a primitive element of GF (16) satisfying η^{4} = η + 1. This was the first case of a hyperoval not being equivalent to one described by a monomial o-polynomial.

Theorem 100 ([32, 47]). The Lunelli-Sce hyperoval in P G(2, 16) has a transitive homography stabiliser of order 36 isomorphic to C_{3}^{2} ⋊ C_{4}. The full collineation stabiliser has order 144 which is isomorphic to ((C_{3} × C_{3}) ⋊ C_{8}) × C_{2}.

The derived group of the full collineation group of the Lunelli-Sce hyperoval is the unique Sylow 3-subgroup, which is elementary abelian of order 9, with all elements of order 3 having a fixed configuration being a triangle with no fixed points on the hyperoval; there are 9 Sylow 2-subgroups each of which are isomorphic to C_{8} × C_{2}. In a Sylow 2-subgroup, there are four involutions: two Baer and one an elation. The Baer involutions have no fixed points on the hyperoval. The whole group has centre of order 2, which is generated by a Baer involution. Thus the full group is that stated in the above theorem.

We have classified all hyperovals with transitive stabiliser by the below theorem.

Theorem 101 ([68]). Let H be a hyperoval of P G(2, q). Suppose G the collineation stabiliser of H is transitive on H. Then H is either a regular hyperoval of P G(2, 2) or P G(2, 4) or it is the Lunelli-Sce hyperoval of P G(2, 16).

Proof. Let q = 2^{e}. Since G is transitive on H, then q + 2||G|, and also divides |P Γ L(3, q)| = (q^{2} + q + 1)(q − 1)^{2}q^{3}(q + 1)e. By Lemma 6, p. 4, we have q + 2 divides ((−2)^{2} − 2 + 1)(−3)^{2}(−2)^{3}(−1) which means q + 2 divides 3^{3} · 2^{3}e = 216e. Hence 2^{e} + 2 ≤ 216e so e ≤ 11. Since 2^{h} + 2 | 216e we have e = 1, 2 or 4. Every hyperoval for e = 1 and e = 2 is regular. For e = 4 there are only the regular and the Lunelli-Sce hyperoval (see below), and only the Lunelli-Sce hyperoval has a transitive stabiliser.

A more general theorem to the one above can be stated not just for Desarguesian projective planes but for projective planes of Lenz type greater than I, that is, a projective plane of order 2^{e} where a flag (P, l) with elation group with centre P and axis l has order 2^{e}.

Theorem 102 ([106]). If a projective plane of Lenz type greater that I and order a power of two contains a transitive hyperoval then the plane has order 2, 4 or 16.

Proof. Suppose not. Let H be a G-transitive hyperoval in a projective plane of order 2^{e} > 16 (we know that the order is not 8, as the unique projective plane P G(2, 8) of order 8 contains no transitive hyperovals). Let p be a primitive prime divisor of 2^{2e−2} − 1 (this exists, as e > 4). Then p divides 2^{e−1} + 1, so divides 2^{e} + 2 = |H|. Thus there is an element g of G of order p and p is not 3, as e > 4. So p is coprime to 2^{2e} + 2^{e} + 1, 2^{e} + 1, 2^{e} and 2^{e} − 1. It follows that g is planar. Since the plane has Lenz type greater than I, it follows from a theorem of Roth [115] that the order of the fixed subplane of g is 2^{i} for some i. So p divides 2^{e} − 2^{i}, giving p divides 2^{e−i} − 1, contrary to being a primitive prime divisor.

Hall [47] by computer and O’Keefe-Penttila [80] without the use of a computer proved that the only hyperovals in P G(2, 16) are the regular and Lunelli-Sce hyperovals.

6.4. Segre-Bartocci hyperoval

The Segre-Bartocci (announced in [120], proved in [121]) is the hyperoval equivalent to D(x^{6}) when q is not a square and q > 8. The order of the stabiliser of these hyperovals in P Γ L(3, q), q = 2^{e}, is 3(q − 1)e for e = 5 and (q − 1)e for e ≥ 7.

Theorem 103. We have D(6) being a hyperoval in P G(2, q), q = 2^{e}, whenever e is odd.

Proof. We use the results from Chapter III, p. 81. We first show that the matrices

A_{t} = (\begin{matrix}t & t^{3} \\ 0 & t^{5} \end{matrix}) ,

with t ∈ GF (q), q = 2^{e}, e odd, form a q–clan. That is, we prove that

trace (\frac{(t + u)(t^{5} + u^{5})}{(t^{3} + u^{3})^{2}}) = 1, for all t ≠ u.

We have

\frac{(t + u)(t^{5} + u^{5})}{(t^{3} + u^{3})^{2}} = \frac{t^{6} + tu^{5} + t^{5}u + u^{6}}{(t^{3} + u^{3})^{2}}

= 1 + \frac{tu^{5} + ut^{5}}{(t^{3} + t^{3})^{2}}

= 1 + \frac{t^{2}u + tu^{2}}{t^{3} + u^{3}} + \frac{t^{4}u^{2} + t^{2}u^{4}}{t^{6} + u^{6}}

So trace ((t + u)(t^{5} + u^{5})/(t^{3} + u^{3})^{2}) = trace(1), and trace(1) = 1 if and only if e is odd. Hence A_{t} is a q–clan if and only e is odd.

Following Section III.11, p. 104: Let a = (1, 0) then (aa, g_{t}(aa), t^{3}aa) = a(a, t^{1/2}, t^{3}a) where g_{t}(a) = \sqrt{aA_{t}a^{T}}. In the basis (a, 0, 0), (0, 1, 0) and (0, 0, a) we have a(a, t^{1/2}, t^{3}a) being equivalent to the point (1, t^{1/2}, t^{3}) in P G(2, q) and the set of points {(1, t^{1/2}, t^{3}) | t ∈ GF (q)} ∪ {(0, 1, 0), (0, 0, 1)} being a hyperoval. This set of points is equivalent to D(6), hence D(6) is a hyperoval.

6.5. Glynn hyperovals

Independently Hirschfeld in 1975 [54] and Eich and Payne [38] in 1972 constructed a hyperoval equivalent to D(x^{20}) in P G(2, 128). The order of its stabiliser is 2667.

Glynn constructed two infinite families of hyperovals in 1983 [42]. They are equivalent to the hyperovals

(I): D(x^{3σ+4}), where e ≥ 9 with σ^{2} ≡ 2 (mod q − 1), and

(II): D(x^{σ+λ}) where e ≥ 7 with σ^{2} ≡ 2 (mod q − 1) and λ^{2} ≡ σ (mod q − 1), with q = 2^{e} not a square, where σ and λ are automorphisms of GF (q) . The order of the stabiliser in P Γ L(3, q), q = 2^{e}, for the Glynn hyperoval I is (q − 1)e and for the Glynn hyperoval II is 3(q − 1)e for e = 7 and (q − 1)e for e > 7. Note the Glynn hyperoval II generalises the Hirschfeld-Eich-Payne hyperoval. A proof of the Glynn hyperovals using flocks of translation oval cones is given in [27].

6.6. Payne hyperoval

Payne [93] constructed a hyperoval equivalent to D(x^{1/6} + x^{3/6} + x^{5/6}), for q = 2^{e} with e ≥ 5 odd and where the exponents are read modulo q − 1. The order of the stabiliser for the Payne hyperoval in P G(2, q), q = 2^{e}, is 2e [131] (see also [87]).

Theorem 104 ([93]). Let q = 2^{e} with e ≥ 5 odd, the Payne hyperovals are an infinite family.

Proof. For e odd, (6, q − 1) = 1 so x ↦ x^{1/6} exists. Now D(1/6) and D(5/6) are equivalent to the hyperoval D(6). We use the ideas of Chapters III, p. 81 and IV, p. 141, to prove D(x^{1/6} + x^{1/2} + x^{5/6}) is a hyperoval. We let f(x) = x^{1/6} and g(x) = x^{5/6}. We show

trace (\frac{(x^{1/6} + y^{1/6})(x^{5/6} + y^{5/6})}{x + y}) = 1, for all x ≠ y.

We let X = x^{1/6} and Y = y^{1/6}, so we have

trace (\frac{(X + Y)(X^{5} + Y^{5})}{X^{6} + Y^{6}}) = 1 + trace (\frac{XY^{5} + X^{5}Y}{X^{6} + Y^{6}} )

= 1 + trace (\frac{XY (X^{4} + Y^{4})}{X^{6} + Y^{6}})

= 1 + trace (\frac{XY (X^{2} + Y^{2})}{(X^{2} + XY + Y^{2})^{2}} )

Now

\frac{XY (X^{2} + Y^{2})}{(X^{2} + XY + Y^{2})^{2}} = \frac{XY}{X^{2} + XY + Y^{2}} + (\frac{XY}{X^{2} + XY + Y^{2}} )^{2} ,

so this has trace zero. Hence for e odd, since trace(1) = 1, we have D(x^{1/6}+x^{1/2}+x^{5/6}) being a hyperoval.

6.7. Cherowitzo hyperoval

Let f_{C}(x) = x^{σ} + x^{σ+2} + x^{3σ+4}. Then we have the Cherowitzo ([29] and [27]) hyperovals equivalent to D(f_{C}), where q = 2^{e} with e odd and σ is an automorphism of GF (q) with σ^{2} ≡ 2 (mod q − 1). The order of the stabiliser in P Γ L(3, q), q = 2^{e}, for e = 5 is e [85] and for e ≥ 7 is divisible by e [29]. It should be added that only recently has D(f_{C}) been proved to be a hyperoval for all e odd [27].

We will prove that the Cherowitzo hyperovals are new; the Theorem below will be central to that proof.

Theorem 105 ([87]). Let q = 2^{e} where e ≥ 7 is odd. Let H be a Cherowitzo hyperoval. A collineation which fixes H and which fixes either (0, 1, 0) or (0, 0, 1) must be an automorphic collineation.

Lemma 106 ([85]). For q = 32 the Cherowitzo hyperoval is not equivalent to the listed hyperovals.

The proof below is quite long.

Theorem 107 ([106]). The Cherowitzo hyperovals in P G(2, q), q = 2^{e}, e odd, are new for e > 3.

Proof. We show the Cherowitzo hyperovals, C = D(f_{C}), are new by first showing that for any known non-Cherowitzo hyperoval, H, in P G(2, 2^{e}), e odd, if C ≅ H then the equivalence is a homography stabilising the fundamental quadrangle (FQ), and then using Theorem 105.

By Theorem 89, p. 43 the only o-polynomials over GF (2) representing regular hyperovals are x^{2}, x^{1/2} and x^{2} + x^{4} + ⋯ + x^{q−2}. Hence C is not equivalent to a regular hyperoval, for q > 8.

Let H be an irregular translation hyperoval. Then the stabiliser, H, of H in P Γ L(3, q) is isomorphic to AΓ L(1, q) (Theorem 99, p. 47). Since the group has orbits on H of lengths 1, 1 and q, and is 2-transitive on the orbit of length q, it follows that all quadrangles on H containing the two fixed points are equivalent under the stabiliser of H.

Now suppose that f is an o-polynomial over GF (2) representing a hyperoval D(f) equivalent to H. Then the stabiliser, H′, of D(f) is conjugate in P Γ L(3, q) to H. Since H′ contains the automorphic collineations, which fix only the points of the fundamental quadrangle, H must contain a group A conjugate to the automorphic collineations fixing a quadrangle on H, and no further points. Thus A fixes a quadrangle on the fixed points of H. Without loss of generality, H = D(α), for some generator α ∈ Aut GF (q). By the transitivity on quadrangles containing the fixed points noted above, without loss of generality, A fixes the FQ. Thus the equivalence between D(f) and D(α) stabilises the FQ and, without loss of generality, is a homography. If C ≅ H then C is stabilised by an involution fixing the FQ and two points of the FQ. Hence by Theorem 105, p. 51 we have the homography being the map (x, y, z) ↦ (x, z, y), that is, we have f_{C} = f_{C}^{−1}. By lemma 18 in [27] we have that if f(x) = x + x^{σ−1} + x^{σ+1} then

f^{−1}(x) = \frac{x^{σ+1}}{1 + x^{2} + x^{σ}} .

Now g(x) = x^{σ+2} then we have f_{C} = f ∘ g. Hence f_{C}^{−1} = g^{−1} ∘ f^{−1} so

f_{C}^{−1}(x) = (\frac{x^{σ+1}}{1 + x^{2} + x^{σ}} )^{\frac{1}{σ+2}}

= (\frac{x^{σ+1}}{1 + x^{2} + x^{σ}} ) (\frac{1 + x^{2} + x^{σ}}{x^{σ+1}} )^{\frac{1}{σ}}

= \frac{x^{σ+1}(1 + x^{σ} + x)}{(1 + x^{2} + x^{σ})(x^{1+ \frac{1}{σ}})}

= \frac{x^{σ− \frac{1}{σ}} (1 + x + x^{σ})}{1 + x^{2} + x^{σ}}.

Hence

f_{C}^{−1}(x) = \frac{x^{1/σ}(1 + x + x^{σ})}{1 + x^{2} + x^{σ}}.

So if f_{C}(x) = f_{C}^{−1}(x) then

x^{1/σ} + x^{1+1/σ} + x^{σ+1/σ} = x^{σ} + x^{2σ} + x^{σ+4} + x^{2σ+2} + x^{3σ+4} + x^{3σ+6} + x^{4σ+4}.

Hence the right hand side must collapse to three terms. That is, we require two more cancellations. It is easy to verify that the required cancellations only occur when q ≤ 8.

Now suppose H is a Payne hyperoval. Then by [131] [85] the stabiliser of the Payne hyperoval in P G(2, q), q = 2^{e}, e odd is cyclic of order 2e. The group has an involution fixing the FQ and moving two points. If C ≅ H then by [87] the involution is (x, y, z) ↦ (x, z, y) so we have f_{C} = f_{C}^{−1}. From above we see that this only occurs when q ≤ 8.

Let H be the Subiaco hyperoval. We are only interested in the case when q = 2^{e}, e odd. We argue as for the Payne hyperoval.

Let H be a known monomial, non-translation hyperoval. In [84] the group H of any known monomial non-translation hyperoval H in P G(2, q) is shown to be isomorphic to Γ L(1, q) for q > 128. Since this group has orbits on H of lengths 1, 1, 1, q − 1, and is transitive on the orbit of length q − 1, it follows that all quadrangles on H containing the three fixed points are equivalent under the stabiliser of H.

Now suppose that f is an o-polynomial over GF (2) representing a hyperoval D(f) equivalent to H. Then the stabiliser H′ of D(f) is conjugate in P Γ L(3, q) to H. Since H′ contains the automorphic collineations, which fix only the points of the FQ, H must contain a group A conjugate to the automorphic collineations fixing a quadrangle on H, and no further points. Thus A fixes a quadrangle on the fixed points of H. Without loss of generality, let H = D(n) where n is 6, σ + λ or 2σ + 4 where σ and λ are defined as for the Glynn hyperoval (page 50). By transitivity on quadrangles containing the fixed points, as noted above, then without loss of generality, A fixes the fundamental quadrangle. Thus the equivalence between D(f) and D(n) stabilises the FQ, and without loss of generality, is a homography. So if C ≅ H then there is a homography taking the FQ to the FQ. But D(n) has homography group C_{q−1} fixing three points of the FQ. Thus there exists a nontrival homography of C fixing three points of the FQ. Hence at least one of (0, 1, 0) and (0, 0, 1) is fixed contrary to Theorem 105.

For q = 32 and q = 128 the stabiliser, H, of some non-translation monomial hyperoval has order 3(q−1) [84]. This hyperoval, H, is equivalent to D(6) for q = 32 and equivalent to D(20) for q = 128. Now suppose that f is an o-polynomial over GF (2) representing a hyperoval D(f) equivalent to H for q = 32 or 128. As in the previous paragraph, H must contain a group A conjugate to the automorphic collineations fixing a quadrangle on H, and no further points. Thus A fixes a quadrangle on the fixed point of H. We have |A| being 5 or 7 for q = 32 or q = 128, respectively, so the set of four fixed points of H must be those fixed by A. Now 31 and 127 are prime, and 3 is coprime to 5 and 7, so any subgroup conjugate to A is a Sylow 5-subgroup or Sylow 7-subgroup for q = 32 or q = 128, respectively. For h ∈ H the fixed points of hAh^{−1} is equal to h(fixed points of A) which is equal to

h(orbit of length 3) ∪ {P } = (orbit of length 3) ∪ {P′}

for points P, P′ on H. We have A ≤ H′ = P Γ L(3, q)_{D(f)}. For g ∈ P Γ L(3, q) we have gAg^{−1} ≤ H. Since for A ∈ Syl_{e}(H) then gAg^{−1} ∈ Syl_{e}(H) for e = 5, 7. This implies that gAg^{−1} = hAh^{−1} for some h ∈ H. So if C ≅ H we have that there exists a nontrival homography of C fixing 3 points of the FQ. Hence at least one of (0, 1, 0) and (0, 0, 1) is fixed, contrary to Theorem 105.

6.8. O’Keefe-Penttila hyperoval

The O’Keefe-Penttila [82] hyperoval in P G(2, 32) is equivalent to D(f), where f(x) = x^{4} + ω^{11}x^{6} + ω^{20}x^{8} + ω^{11}x^{10} + ω^{6}x^{12} + ω^{11}x^{14} + x^{16} + ω^{11}x^{18} + ω^{20}x^{20} + ω^{11}x^{22} + ω^{6}x^{24} + ω^{11}x^{26} + x^{28}, with ω being a primitive root of GF (32) satisfying ω^{5} = ω^{2} + 1. The stabiliser of the O’Keefe-Penttila hyperoval in P Γ L(3, 32) has order 3 [82].

Penttila-Royle [110] have done an exhaustive computer search in q = 32 and have found that there are no further hyperovals in P G(2, 32). The hyperovals are the regular, translation, Segre-Bartocci, Payne, Cherowitzo, and O’Keefe-Penttila hyperovals.

6.9. Irregular hyperovals in P G(2, 64)

The smallest q for which hyperovals have not been classified is 64. In fact for many years an open question (see for example [55]) was whether or not there existed irregular hyperovals in P G(2, 64), as the only hyperovals in the list above for q = 64 are the regular ones.

The next sections will resolve this question by describing two inequivalent irregular hyperovals in P G(2, 64). The two hyperovals and the order of their stabiliser in P Γ L(3, 64) are:

(I): equivalent to D(f) with f(x) = ω^{42}x^{2} + ω^{21}x^{4} + ω^{42}x^{6} + x^{8} + ω^{21}x^{10} + x^{12} + ω^{21}x^{14}+ω^{21}x^{16}+x^{20}+x^{22}+ω^{42}x^{26}+ω^{42}x^{28}+ω^{21}x^{30}+ω^{42}x^{32}+ω^{42}x^{36}+ω^{21}x^{38}+ ω^{42}x^{40}+x^{42}+ω^{21}x^{44}+ω^{21}x^{48}+x^{52}+ω^{21}x^{54}+ω^{21}x^{56}+ω^{21}x^{58}+ω^{21}x^{60}+ω^{21}x^{62} with order 15, and

(II): equivalent to D(f) with f(x) = ω^{21}x^{4} + ω^{42}x^{6} + ω^{21}x^{8} + ω^{21}x^{10} + ω^{42}x^{12} + ω^{21}x^{14} + ω^{21}x^{16} + ω^{42}x^{18} + ω^{42}x^{20} + x^{24} + ω^{42}x^{26} + x^{30} + ω^{42}x^{32} + ω^{21}x^{34} + ω^{42}x^{36}+ω^{21}x^{38}+ω^{21}x^{40}+ω^{42}x^{42}+ω^{21}x^{44}+ω^{21}x^{46}+ω^{42}x^{48}+ω^{42}x^{50}+ω^{21}x^{52}+ ω^{21}x^{54} + ω^{21}x^{58} + ω^{21}x^{60} + x^{62} with order 60,

where ω is a primitive element of GF (64) satisfying ω^{6} = ω + 1.

7. Fano Configuration

Every 4-arc on a hyperoval H determines a Fano plane, containing six lines secant to H of the 4-arc, and a further 7th line. This line is called the diagonal line of the 4-arc. Now, this diagonal line can either be external to H or secant to H, by Theorem 28, p. 16, in which case we refer to the 4-arc itself as being external or a secant 4-arc, respectively. Now for each point P on H we can count the number of secant 4-arcs on P, and call this the height of P. This associates a number with each point, in such a way that points in the same orbit of the stabiliser of H must be assigned the same number. Therefore these numbers induce a partition on H which respects the orbit partition, but may be coarser. This multiset of numbers is called a profile of H.

8. Irregular Hyperovals in P G(2, 64)  55

The above result yields a quick method for distinguishing hyperovals, though profiling a hyperoval is a order q^{4} computation (since there are \binom{q+2}{4} quadrangles on H). It turns out that hyperovals (so far) are in fact distinguished by the coarser invariant consisting of the total number of external 4-arcs and secant 4-arcs.

Since any automorphism of a hyperoval preserves the partition of the points of H by their heights, by working in the stabiliser of one of the smaller cells, it is often possible to identify the automorphism group of the hyperoval.

8. Irregular Hyperovals in P G(2, 64)

In this section we will talk about the methods used to construct the first irregular hyperovals in P G(2, 64).

The overall picture is this: Assume that the hyperoval has symmetry. We make this assumption as weak as possible by assuming the symmetry stems from a subgroup of prime order. So by Cauchy’s Theorem page 5 we investigate each prime that divides the order of P Γ L(3, q). Now the hyperovals are a union of orbits of this prime subgroup, the orbits either having length the prime or one (fixed points). We conveniently start with the largest prime and then the next largest and so on. Firstly, we determine if the hyperoval can be a union of orbits and fixed points of the prime subgroup. If so, we go onto the next step, which involves determining conjugacy classes of the prime subgroup etc. These techniques we call “prime at a time.”

Now

|P Γ L(3, 64)| = 73.19.13.7^{2}.5.3^{5}.2^{24}.

So the feasible primes are 73, 19, 13, 7, 5, 3, 2.

Immediately we discard 73, since 73 > 66, which is the size of a hyperoval in P G(2, 64), and it’s impossible to fix 66 points.

We have 19 | q^{2} + q + 1, so the subgroup of order 19 is generated by a power of a Singer cycle and so has no fixed points, so all orbits are of length 19 on P G(2, 64). But 19 | 6̸6 so a hyperoval can not be the union of orbits of length 19.

We find that the primes 13, 7, 5, 3 and 2 can be used to construct hyperovals. Below we give a detailed account of the primes 13, 7 and 5, and a summary of the work done on 3, 2 and the no symmetry hypothesis.

For the rest of this section let ω be a primitive element of GF (64) satisfying ω^{6} = ω + 1.

8.1. Hyperovals, H, in P G(2, 64) with |P Γ L(3, 64)_{H}| divisible by 13.

An element of order 13 in P Γ L(3, 64) has one fixed point and (hence) one fixed line by Section 3.3, p. 37. The hyperoval H is a union of five orbits of length 13 and contains the fixed point.

Now 13 | |P Γ L(3, 64)| but 13^{2} | |̸P Γ L(3, 64)|, so by Sylow’s Theorem all subgroups of order 13 are conjugate in P Γ L(3, 64). Hence we can study our favourite subgroup of order 13.

Let

T = ⟨[\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & ω^{58} \end{matrix}]⟩ ,

then |T | = 13.

Let H denote the hyperoval we are looking for. Suppose 13||P GL(3, 64)_{H}|. Then without loss of generality we can say T ≤ P GL(3, 64)_{H}, applying the remarks in Section 3.4, p. 38.

Theorem 108. If 13 divides the order of the stabiliser of a hyperoval in P G(2, 64) then the hyperoval is regular.

Proof. By computer. Note that by the remarks above we need only determine the hyperovals fixed by T. Also since N_{P Γ L(3,64)}(T) is transitive on the 13-arcs fixed by T, we need only determine the hyperovals fixed by T on a given 13-arc fixed by T. In fact a quick computation shows that all 39-arcs in P G(2, 64) stabilised by a collineation of order 13 lie on a regular hyperoval.

8.2. Hyperovals, H, in P G(2, 64) with |P Γ L(3, 64)_{H}| divisible by 7.

Suppose 7 divides the stabiliser of the hyperoval H. Then the hyperoval is constructed from nine orbits of length 7 and 3 fixed points, a total of 66 points.

An element of order 7 has a fixed configuration of 3 fixed points and 3 fixed lines forming a triangle. Since P Γ L(3, 64) is transitive on triangles, without loss of generality, we can assume that the fixed points are (0, 0, 1), (0, 1, 0) and (1, 0, 0). Then we can construct a matrix

g = [\begin{matrix}1 & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ^{i} \end{matrix}] , where λ^{7} = 1, λ ≠ 1, and i ∈ ℤ_{7},

(by the note after Theorem 79, p. 39, since 7|q − 1), so |g| = 7.

We investigate the conjugacy classes of the group generated by g.

Let S be the subgroup of P Γ L(3, 64) of order 7 generated by g. Then S = ⟨g⟩ = ⟨g^{j}⟩ for all j with 1 ≤ j ≤ 6. So for some j, we have,

g^{j} = g_{i} = [\begin{matrix}1 & 0 & 0 \\ 0 & ω^{9} & 0 \\ 0 & 0 & ω^{9i} \end{matrix}]

with i ∈ ℤ_{7} since ⟨λ⟩ is the unique subgroup of order 7 of GF (64)^{∗}, we have λ^{j} = ω^{9}, for some j. As a non-trivial homology cannot stabilise a hyperoval i cannot be 0 or 1.

The fixed points of S are (0, 0, 1), (0, 1, 0) and (1, 0, 0). By permuting these fixed points , we see that g_{i} is conjugate to g_{j} where

j ≡ \frac{1}{i} ≡ 1 − i ≡ \frac{i}{i − 1} ≡ \frac{1}{1 − i} ≡ \frac{i − 1}{i} (mod 7).

Hence ⟨g_{2}⟩ = ⟨g_{6}⟩ and ⟨g_{3}⟩ = ⟨g_{4}⟩ = ⟨g_{5}⟩ so without loss of generality we can say i = 2 or 3.

Let a point of P G(2, 64) be called admissible if {P }∪{(1, 0, 0), (0, 1, 0), (0, 0, 1)} is a 4-arc. There are (64 − 1)^{2} such points. We have N_{P Γ L(3,64)}(S) being transitive on admissible points, since

N_{P Γ L(3,64)}(S) = C_{P Γ L(3,64)}(S) ≥ \{(\begin{matrix}1 & 0 & 0 \\ 0 & μ & 0 \\ 0 & 0 & ν \end{matrix}) | μ, ν ∈ GF (64)^{∗}\} ,

with the (64−1)^{2} matrices acting regularly on the admissible points. So N_{P Γ L(3,64)}(S) permutes the orbits of S, hence transitive on the orbits of S, so all orbits, not on the sides of the triangle (0, 0, 1), (0, 1, 0) and (1, 0, 0), are equivalent, so equivalent to the orbit containing (1, 1, 1). Now since 7 = 2^{3} − 1 we then have the orbit

{(1, ω^{9}, ω^{9·3}) | |ω^{9}| = 7}

being contained in P G(2, 8). Thus the orbit of (1, 1, 1) is an arc if and only if

D(i) = {(1, t, t^{i}) | t ∈ GF (8)} ∪ {(0, 1, 0), (0, 0, 1)}

is a hyperoval of P G(2, 8). This can only occur if i is congruent to 2, 4 or 6 modulo 7 (see Section 88, p. 43). Hence S = ⟨g_{3}⟩ can not stabilise a hyperoval.

Theorem 109. If 7 divides the order of the stabiliser of a hyperoval in P G(2, 64) then the hyperoval is regular.

Proof. By computer. Note that by the remarks above we need only determine the hyperovals fixed by S where i = 2. Also since N_{P Γ L(3,64)}(S) is transitive on the 7-arcs fixed by S, we need only determine the hyperovals fixed by S on a given 7-arc fixed by S. A computation shows that all 31-arcs in P G(2, 64) stabilised by a collineation of order 7 lie on a regular hyperoval.

8.3. Hyperovals, H, in P G(2, 64) with |P Γ L(3, 64)_{H}| divisible by 5.

An element of order 5 in P Γ L(3, 64) has one fixed point and one fixed line. By Lemma 75, p. 36 the fixed line and fixed point form an antiflag.

As for a stabiliser of order divisible by 13, we can construct a group

F = ⟨[\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & ω^{21} \end{matrix}]⟩ ,

which has order 5. Subgroups of P Γ L(3, 64) of order 5 are conjugate to F by Sylow’ Theorem. Now F has fixed point P = (1, 0, 0) and fixed line l : x = 0.

(We note that

(\begin{matrix}ω^{21} & ω^{21} & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix})^{−1} (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & ω^{21} \end{matrix}) (\begin{matrix}ω^{21} & ω^{21} & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) = (\begin{matrix}0 & 1 & 0 \\ 1 & ω^{21} & 0 \\ 0 & ω^{42} & 1 \end{matrix}) = g

where g ∈ P GL(3, 64) is the element of order 5 used on page 113, Chapter III.) Notice that {0, 1, ω^{21}, ω^{42}} ⊆ GF (64) is isomorphic to GF (4). Let C = C_{P GL(3,64)}(F), the centraliser of F in P GL(3, 64). Then C permutes

the fixed points and fixed lines of F, and so fixes the point (1, 0, 0) and fixes the line x = 0.

We now determine the normaliser of F in P Γ L(3, 64) by using the Samosa Lemma (Theorem 20, p. 7). Let the group P of the Samosa Lemma be F ≅ C_{5}. Firstly we need to determine d of the Samosa Lemma. We have

64^{m} − 1 ≡ 4^{m} − 1 (mod 5) ≡ (−1)^{m} − 1 (mod 5) ≡ 0 (mod 5) if and only if 2|m.

Hence 5 | 64^{m} − 1 implies that 2 | m So for F ≤ P GL(3, 64) we have 5 | 6̸4 − 1 but 5 | 64^{2} − 1 giving d = 2. Then we have a representation T′: C_{5} → GL(3, 64). Let W = GF (q)^{3}, then W = U ⊕V where U is a fixed one-dimensional subspace and V is a fixed two-dimensional subspace, that is, a fixed point and a fixed line, respectively. Now C_{GL(W)}(T′(C_{5})) fixes U and V. Let C_{5} = ⟨g⟩. Then

T′(g) = ( \begin{matrix}1 & 0 & & 0 \\ {0 \atop 0} & & T \end{matrix} )

with T: G → GL(V) irreducible. Also

C_{GL(W)}(T′(C_{5})) = \{( \begin{matrix}λ & 0 & & 0 \\ {0 \atop 0} & & A \end{matrix} ) | A ∈ C_{GL(V)}(T (C_{5}))\} .

So C_{GL(W)}(T′(C_{5})) = GF (64)^{∗} × GF (64^{2})^{∗} ≥ {λI | λ ∈ GF (64)^{∗}}, hence

C_{P GL(W)}(T′(C_{5})) = \frac{GF (64)^{∗} × GF (64^{2})^{∗}}{\{λI | λ ∈ GF (64)^{∗}\}} .

So C_{5} fixes l (the subspace V). Choose coordinates so that l = [0, 0, 1] and P = (0, 0, 1) (the one-dimensional subspace U). Now C_{5} fixes l which implies that C_{5} ≤ AGL(2, 64), identified with P GL(2, 64)_{l} as usual. Also C_{5} fixes P implying C_{5} ≤ GL(2, 64). Hence C_{GL(2,64)}(C_{5}) is regular on P G(2, 64) − (l ∪ {P }).

Let N = N_{P Γ L(3,64)}(F) then by Theorem 20, p. 7, we have

N ≅ GF (64^{2})^{∗} ⋊ Aut GF (64^{2})
≅ C_{4095} ⋊ C_{12}

with C ◁ N acting regularly on the points of P G(2, 64) not on l and not equal to P. We identify C with GF (64^{2})^{∗} = ⟨η⟩, where η^{2} = ω^{22}η + ω^{10}. Since C acts regularly on points of P G(2, 64) not on l and not equal to P, we identify points not on l with elements of GF (64^{2}), with P corresponding to 0. Then F corresponds to the group generated by multiplication by η^{819} and C is generated by multiplication by η. Hence

N ≅ C ⋊ ⟨η^{i} ↦ η^{2i}⟩

As C is transitive on orbits of F not on l, not equal to P, we may suppose that 1 ∈ H. Explicitly, Aut GF (64^{2}) = ⟨x ↦ x^{2}⟩ and η^{i} ↦ η^{2i} takes orbit

{η^{i}, η^{i+819}, η^{i+1638}, η^{i+2457}, η^{i+3276}} ↦ {η^{2i}, η^{2i+1638}, η^{2i+3276}, η^{2i+819}, η^{2i+2457}}

Identifying the orbit of η^{i} under F with i for i ∈ ℤ_{819}, the map is now i ↦ 2i with i ∈ ℤ_{819}.

There is a unique subgroup of C of order 63, consisting entirely of homologies, namely

(\begin{matrix}1 & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ \end{matrix})

with λ ∈ GF (64)^{∗}.

Let G = P Γ L(3, 64)_{H} with H irregular. Then |C ∩ G| = 5. (Note |C| = 13.5.7.3^{2}. If g ∈ C, g non-trivial, and |g||63, then g is a homology which by Theorem 52, p. 27, does not stabilise a hyperoval. Alternatively, g ∈ C and |g| = 13, then we have shown from Theorem 108, p. 56, that g does not stabilise an irregular hyperoval. Hence |g| = 5 or 1.)

Now C ∩ G = C_{G}(F), and N ∩ G = N_{G}(F). We have

\frac{N_{G}(F)}{C_{G}(F)} ≅ \frac{N ∩ G}{C ∩ G} ≅ \frac{(N ∩ G)C}{C} ≲ \frac{N}{C} ,

so

\frac{N_{G}(F)}{C_{G}(F)}

is cyclic of order dividing 12. Hence |N_{G}(F)| divides 60.

Theorem 110. If H is a hyperoval in P G(2, 64), G = P Γ L(3, 64)_{H}, F ≤ G with |F | = 5 and 60 divides |N_{G}(F)|, then H is a regular hyperoval or the hyperoval H_{1} equivalent to

f(x) = ω^{21}x^{4} + ω^{42}x^{6} + ω^{21}x^{8} + ω^{21}x^{10} + ω^{42}x^{12} + ω^{21}x^{14} + ω^{21}x^{16}
+ ω^{42}x^{18} + ω^{42}x^{20} + x^{24} + ω^{42}x^{26} + x^{30} + ω^{42}x^{32} + ω^{21}x^{34}
+ ω^{42}x^{36} + ω^{21}x^{38} + ω^{21}x^{40} + ω^{42}x^{42} + ω^{21}x^{44} + ω^{21}x^{46} + ω^{42}x^{48}
+ ω^{42}x^{50} + ω^{21}x^{52} + ω^{21}x^{54} + ω^{21}x^{58} + ω^{21}x^{60} + x^{62}

Conversely, H_{1} is a hyperoval having a subgroup F of order 5 with |N_{G}(F)| = 60.

Proof. By the arguments above, if H is irregular, then |N_{G}(F)| divides 60. So we conclude |N_{G}(F)| = 60. Now N has only one conjugacy class of cyclic subgroups of order 12, so without loss of generality,

N_{G}(F) = F ⋊ Aut GF (64^{2}).

The orbits of N_{G}(F) on the set of orbits of F are therefore {i, 2i, 4i, …, 2^{11}i}, i ∈ ℤ_{819}. A computation shows that, H, the only irregular hyperoval stabilised by N_{G}(F) has H − {P } = {0} ∪ {819k | 1 ≤ k ≤ 4} ∪ {51.2^{j} + 819k | 0 ≤ j ≤ 11, 0 ≤ k ≤ 4}, that is,

H = {0} ∪ {η^{819k} | 0 ≤ k ≤ 4} ∪ {η^{51.2^{j}+819k} | 0 ≤ j ≤ 11, 0 ≤ k ≤ 4}.

Also, a computation shows that H is equivalent to H_{1}.

Theorem 111. If H is a hyperoval in P G(2, 64), G = P Γ L(3, 64)_{H}, F ≤ G, with |F | = 5 and 15 divides |N_{G}(F)|, then H is equivalent to H_{1}, the regular hyperoval, or the hyperoval H_{2} equivalent to

f(x) = ω^{42}x^{2} + ω^{21}x^{4} + ω^{42}x^{6} + x^{8} + ω^{21}x^{10} + x^{12} + ω^{21}x^{14}
+ ω^{21}x^{16} + x^{20} + x^{22} + ω^{42}x^{26} + ω^{42}x^{28} + ω^{21}x^{30} + ω^{42}x^{32}
+ ω^{42}x^{36} + ω^{21}x^{38} + ω^{42}x^{40} + x^{42} + ω^{21}x^{44} + ω^{21}x^{48}
+ x^{52} + ω^{21}x^{54} + ω^{21}x^{56} + ω^{21}x^{58} + ω^{21}x^{60} + ω^{21}x^{62}

Conversely, H_{2} is a hyperoval having a subgroup F of order 5 with |N_{G}(F)| = 15.

Proof. Similar to last proof. Without loss of generality

N_{G}(F) = F ⋊ ⟨η ↦ η^{16}⟩.

Now we need one orbit of length 5, so by similar arguments, computation gives only

H = {0} ∪ {η^{819k} | 0 ≤ k ≤ 4} ∪ {η^{79.16^{j}+819k} | 0 ≤ j ≤ 2, 0 ≤ k ≤ 4}

∪ {η^{174.16^{j}+819k} | 0 ≤ j ≤ 2, 0 ≤ k ≤ 4} ∪ {η^{332.16^{j}+819k} | 0 ≤ j ≤ 2, 0 ≤ k ≤ 4}

∪ {η^{511.16^{j}+819k} | 0 ≤ j ≤ 2, 0 ≤ k ≤ 4}.

Also, a computation shows that H is equivalent to H_{2}.

Theorem 112. If H is a hyperoval in P G(2, 64), G = P Γ L(3, 64)_{H}, 5 dividing |G| then H is equivalent to H_{1}, H_{2}, or the regular hyperoval, and conversely.

Proof. Similar to first proof. Here we use N to assume that 1 ∈ H and then to only search for hyperovals on 1 and i, for one i from each orbit of Aut GF (64^{2}). This is a long search, but nothing new has been found.

Remark on the proofs: All three proofs are proved by computer.

Of course Theorem 112, p. 61, implies Theorem 111, p. 60, which implies Theorem 110, p. 59. However, as they are computer-dependent results, with each requiring more computing time, it is of advantage to prove them independently.

Note that the fact that the two new hyperovals are hyperovals is not really computer dependent, as it can be readily verified by a trivial program. There is a computer free proof given in Chapter IV.

Now to the size of the stabilisers of our hyperovals. Let C be a conic in P G(2, 4) with nucleus P and define R = C ∪ {P }, and let H be a hyperoval in P G(2, 64) such that P G(2, 4) ∩ H = R.

We know these facts from our above computations:

(i) |G| = 60.

(ii) G fixes P G(2, 4) and so fixes R.

(iii) G fixes the line l : x = 0 and the point P = (1, 0, 0). Hence G ≤ P Γ L(3, 64)_{R,l,P}. Now

G_{(R)} ≤ G_{(P G(2,4))} ≤ P Γ L(3, 64)_{(P G(2,4))} = ⟨σ⟩,

where σ ∈ Aut GF (64) of order 3 defined as σ: (x, y, z) ↦ (x^{4}, y^{4}, z^{4}). But equality occurs throughout as our o-polynomials are over GF (4).

Since P Γ L(3, 4)_{R,P} ≅ P Γ L(2, 4) with |P Γ L(2, 4)| = 120 with P Γ L(3, 4)_{R,P} being transitive on the six external lines to R (see the Note on page 23). Now

|P Γ L(3, 64)_{R,l,P}^{P G(2,4)}| = |P Γ L(3, 4)_{R,l,P} | = 20,

since P Γ L(3, 4)_{R,l,P} has index 6 in P Γ L(3, 4)_{R,P}. Hence

G = P Γ L(3, 64)_{R,l,P} = P Γ L(3, 64)_{C,l}.

For q even the only subgroups of P GL(2, q) of order 2(q + 1) are dihedral and they are conjugate [37].

Let H = P GL(3, 64)_{C,l}. Then H has index 6 in G, so |H| = 10, hence H ≅ D_{10}. Now G normalises H, so by [37] C_{q^{3}+1} ∩ G = F, so G = F ⋊ C_{12}. (We note that C_{5} ⋊ C_{12} ≤ AΓ L(1, 64^{2}).)

Alternatively by the method of construction it is clear that

F_{1} ⋊ Aut GF (64^{2})

is a subgroup of the stabiliser of H_{1}, and

F_{2} ⋊ Gal(GF (64^{2})/GF (16))

is a subgroup of the stabiliser of H_{2}, for some subgroups F_{1} and F_{2} of P Γ L(3, 64) of order 5, conjugate to F.

To see that the groups are the full stabilisers, we used an idea introduced in Section 7, p. 54. For each quadrangle contained in a hyperoval H, the diagonal line is either secant or external to H. For each point P of H, we count the number of quadrangles in H on P with diagonal line external to H and call this the height of P. The stabiliser of H must permute the points of H with the same height.

A computation shows that H_{1} has one point (P_{1}) of height 20700, 5 points (the orbit, C_{1}, of 1 under F_{1}) of height 21432, and 60 points of height 21373. It follows that the stabiliser of H_{1} is contained in the stabiliser of P_{1} and C_{1}, which is easy to compute, and has a (small) order of 360. So by explicitly calculating with the stabiliser of P_{1} and C_{1}, we can see that |P Γ L(3, 64)_{H_{1}}| = 60. In more detail, let g_{i} ∈ P Γ L(3, 64)_{H_{i}} so g_{i}P_{i} = P_{i} and g_{i}C_{i} = C_{i} for i = 1, 2. Then g_{i} fixes the Baer subplane containing P_{i} and C_{i}. We can say this Baer subplane is P G(2, 4) by our canonical labeling of P_{i} and C_{i}. Hence g_{i} ∈ (P Γ L(3, 4) ⋊ Gal(GF (64)/GF (4)))_{P_{i},C_{i}}. Since C_{i} is a conic in P G(2, 4) with nucleus P_{i} we have g_{i} ∈ P Γ O(3, 4) ⋊ Gal(GF (64)/GF (4)) which is g_{i} ∈ P Γ L(2, 4) ⋊ Gal(GF (64)/GF (4)), by Theorem 45, p. 23. Note |P Γ L(2, 4) ⋊ Gal(GF (64)/GF (4))| = 120.3. Since Gal(GF (64)/GF (4)) fixes H_{i} we need only concern ourselves with the elements of P Γ L(2, 4).

Similarly, a computation shows that H_{2} has one point (P_{2}) of height 21565, 5 points (the orbit, C_{2}, of 1 under F_{2}) of height 21436, 15 points of height 21254, 15 points of height 21508, 15 points of height 21505, and 15 points of height 21382. So again P Γ L(3, 64)_{H_{1}} is a subgroup of P Γ L(3, 64)_{{P_{2},C_{2}}}, so we can explicitly see that we have |P Γ L(3, 64)_{H_{2}}| = 15.

(Of course, when computing with the stabiliser of P_{i} and C_{i}, it is best to choose a point Q_{i} on H_{i}, not P_{i}, not in C_{i}, and compute with the stabiliser of P_{i}, Q_{i} and C_{i} as we know the length of the orbit of Q_{i} under P Γ L(3, 64)_{H_{i}} is 60 if i = 1 and 15 if i = 2.)

9. Searching Algorithms for Hyperovals

In this section we describe the algorithms and techniques used to search for hyperovals in P G(2, 64), in particular, hyperovals stabilised by F (the group of order 5 defined in Section 8.3, p. 57).

Let ω be a primitive element of GF (64) satisfying ω^{6} = ω + 1.

We define a k-arc to be nice if it is stabilised by F. Our aim is to find nice 66-arcs.

We determine the nice hyperovals by constructing graphs and finding cliques of suitable size. For each of the three theorems in Section 8.3, we determine a graph Γ_{i} with respect to a set of points X_{i} for i = 1, 2, 3.

By the results in Section 8.3 we can label the points of P G(2, 64) not on the fixed line of F by GF (64^{2}). Let η be a primitive element of GF (64^{2}) satisfying η^{2} = ω^{22}η + ω^{10}.

For Theorem 110, p. 59, we let X_{1} be a nice 6-arc defined as

X_{1} = {0} ∪ {η^{819k} | 0 ≤ k ≤ 4}.

The vertices of the graph Γ_{1} are orbits O_{x} where

O_{x} = {η^{x2^{j}+819k} | 0 ≤ j ≤ 11, 0 ≤ k ≤ 4},

is a nice 60-arc and X_{1} ∪ O_{x} is a nice hyperoval. When the program is ran it finds only two vertices: X_{1} ∪O_{53} which is equivalent to the regular hyperoval, and X_{1} ∪O_{51} which is the hyperoval H_{1}. The program took 14 seconds to run on a Sun Sparc IPC Workstation.

For Theorem 111, p. 60, we let

X_{2} = {0} ∪ {η^{819k} | 0 ≤ k ≤ 4},

with X_{2} being a 6-arc. The vertices of Γ_{2} are orbits O_{x} where

O_{x} = {η^{x16^{j}+819k} | 0 ≤ j ≤ 2, 0 ≤ k ≤ 4},

is a nice 15-arc and X_{2} ∪ O_{x} is a nice 21-arc. Edges are defined as:

O_{x} ∼ O_{y} ⟺ O_{x} ∪ O_{y} ∪ X_{2} is a nice 36-arc.

Hence a hyperoval in Γ_{2} is a clique of size 4. We find that Γ_{2} has very few edges (60 in total). We observe that the vertices of sufficient valency naturally fall into a few cliques by observation of the list of edges. The cliques are:

(i) X_{2} ∪ O_{63} ∪ O_{126} ∪ O_{252} ∪ O_{441}.

(ii) X_{2} ∪ O_{51} ∪ O_{102} ∪ O_{204} ∪ O_{408}.

(iii) X_{2} ∪ O_{71} ∪ O_{203} ∪ O_{348} ∪ O_{451}.

(iv) X_{2} ∪ O_{83} ∪ O_{142} ∪ O_{406} ∪ O_{452}.

(v) X_{2} ∪ O_{87} ∪ O_{166} ∪ O_{284} ∪ O_{665}.

(vi) X_{2} ∪ O_{79} ∪ O_{174} ∪ O_{332} ∪ O_{511}. Hyperoval (i) is regular, (ii) is H_{1}, and (iii), (iv), (v), (vi) are equivalent to H_{2}. Notice (iii) ↦ (iv) ↦ (v) ↦ (vi) ↦ (iii) under the map η^{i} ↦ η^{2i}. The program took eight and a half minutes to run on a Sun Sparc IPC Workstation.

The exhaustive search for hyperovals stabilised by an element of order 5 is slightly more complicated.

We set about trying to construct a sparsely edged graph Γ_{3}. We do this by noticing that any nice 21-arc must be on at least 9 nice 26-arcs for the 21-arc to complete to a nice hyperoval. We call such nice 21-arcs live. Now we can define Γ_{3}.

Let

X_{3} = {0} ∪ {η^{819k}, η^{s+819k} | 0 ≤ k ≤ 4}

where s is one of 516 numbers such that X_{3} is a nice 11-arc. Vertices of Γ_{3} are

O_{x} = {η^{x+819k} | 0 ≤ k ≤ 4},

with O_{x} being nice 5-arcs such that X_{3} ∪ O_{x} is a nice 16-arc. Edges are defined as:

O_{x} ∼ O_{y} ⟺ X_{3} ∪ O_{x} ∪ O_{y} is a live 21-arc.

As before, for each s, Γ_{3} is graph with few edges and so by inspection the cliques of size 11 can be found. The long search of going through each s gives no new hyperovals, but does find the regular hyperovals and the hyperovals H_{1} and H_{2} as expected. For each s, the program takes 16 minutes to complete on a Sun Sparc IPC Workstation. In all it took roughly three weeks to prove Proposition 112, p. 61.

To check that the set of points of the cliques of orbits is a hyperoval we employ two methods. Firstly, we use a program that determines that the set of 66 points has no three collinear. Secondly, we use the results of Section 5.1, p. 43, to construct a permutation polynomial from the set of 66 points and then use Theorem 84, p. 42, to determine that it is an o-polynomial.

Of interest: H_{1} has 36 o-polynomials over GF (4) and 6 of these have stabiliser fixing the point (0, 0, 1). Similarly, H_{2} has 144 o-polynomials over GF (4) with 24 of these have stabiliser fixing the point (0,0,1). The external lines to H_{i} ∩ P G(2, 4) for i = 1, 2 of P G(2, 4) are [1, 1, ω^{21}], [1, 1, ω^{42}], [1, ω^{21}, 1], [1, ω^{42}, 1], [ω^{21}, 1, 1], [ω^{42}, 1, 1].

Of particular interest is that H_{1} is equivalent to the hyperoval described by the o-polynomial

f = ω^{42}x^{4} + ω^{21}x^{6} + ω^{21}x^{8} + ω^{42}x^{10} + ω^{42}x^{14} + ω^{21}x^{16} + ω^{21}x^{18}
+ ω^{42}x^{20} + ω^{42}x^{24} + ω^{21}x^{26} + ω^{21}x^{28} + ω^{42}x^{30} + x^{32} + ω^{42}x^{34}
+ ω^{21}x^{36} + ω^{21}x^{38} + ω^{42}x^{40} + ω^{42}x^{44} + ω^{21}x^{46} + ω^{21}x^{48} + ω^{42}x^{50}
+ ω^{42}x^{54} + ω^{21}x^{56} + ω^{21}x^{58} + ω^{42}x^{60}

This o-polynomial’s coefficients are periodic except for the term x^{1/2} = x^{32}; this form of periodicity is referred to as “pseudo-periodicity.” By the use of the simple ideas of geometric series we can see that H_{1} is equivalent to the hyperoval described by the o-polynomial

f(x) = \frac{ω^{42}x^{4} + ω^{42}x}{(x^{2} + ω^{21}x + 1)^{2}} + x^{1/2},

under the map (x, y, z) ↦ (z, y + ω^{21}z, x + ω^{42}).

By observing the GF (4) o-polynomials of H_{2} we find that it is equivalent to the o-polynomial

f(x) = ω^{21}x^{2} + ω^{21}x^{6} + x^{8} + x^{10} + ω^{21}x^{12} + ω^{21}x^{16} + x^{18}
+ x^{20} + ω^{21}x^{22} + ω^{21}x^{26} + x^{28} + x^{30} + ω^{21}x^{32} + ω^{21}x^{36}
+ x^{38} + x^{40} + ω^{21}x^{42} + ω^{21}x^{46} + x^{48} + x^{50} + ω^{21}x^{52}
+ ω^{21}x^{56} + x^{58} + x^{60} + ω^{21}x^{62}

This o-polynomial shows the characteristic of “pseudo-periodicity”, that is, periodicity of the coefficients except for the x^{1/2} = x^{32} term. Again, we see that (from the above o-polynomial) H_{2} is equivalent to

f(x) = \frac{x^{4} + ω^{21}x^{3} + ω^{21}x^{2}}{(x^{2} + ω^{21}x + 1)^{2}} + ω^{42}x^{1/2},

under the map (x, y, z) ↦ (z, y + ω^{21}z, x + ω^{42}).

9.1. More on Hyperovals in P G(2, 64)

Theorem 113. Any Sylow 3-subgroup, S, of the homography stabiliser of a hyperoval in P G(2, 64) has order at most 9.

Proof. Since 3 ≡ 66 (mod 9), S must stabilise a triangle. Without loss of generality let the triangle be (0, 0, 1), (0, 1, 0) and (1, 0, 0). The pointwise stabiliser T of the triangle in P GL(3, 64) is just the homographies with diagonal matrices. Hence U = T ∩ S contains no non-identity homologies, so elements of U have the form

(\begin{matrix}1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t^{i} \end{matrix})

for some i such that (i, 9) = 1 with t having order 1, 3, or 9.

Now U has index 1 or 3 in S. If U has index 3 then a homography of the form

(\begin{matrix}0 & a & 0 \\ 0 & 0 & b \\ c & 0 & 0 \end{matrix})

normalises U which forces U to be either trivial or of order 3 and equal to the set of all homographies of the form

(\begin{matrix}1 & 0 & 0 \\ 0 & u & 0 \\ 0 & 0 & u^{2} \end{matrix})

where u has order 1 or 3. Hence S has order at most 9.

From now on we label the hyperoval H_{1} from the previous Section whose stabiliser is of size 60 by H_{60}, and the hyperoval H_{2} whose stabiliser is of order 15 by H_{15}.

The results in the rest of this section are from [111].

There are four conjugacy classes of elements of order 3 in P Γ L(3, 64), these are:

(i) A 1387’th power of a Singer cycle, g_{1}, represented by the matrix,

(\begin{matrix}0 & 0 & ω \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) .

(ii) A triangle element, g_{2}, represented by the matrix,

(\begin{matrix}1 & 0 & 0 \\ 0 & ω^{21} & 0 \\ 0 & 0 & ω^{42} \end{matrix}) .

(iii) An element g_{3}, of order 3 that is in Aut GF (64), represented by

(x, y, z) ↦ (x^{4}, y^{4}, z^{4}).

(iv) A homology, g_{4}, of order 3.

Since g_{4} is a homology, this element can not stabilise a hyperoval, by Theorem 52, p. 27.

The element g_{1} has no fixed points, so the hyperoval it stabilises is constructed by gluing together 22 orbits. The normaliser is transitive on orbits of ⟨g_{1}⟩ so one can pick their favourite orbit first. When the search is completed no hyperovals are found.

The element g_{2} has (0, 0, 1), (0, 1, 0), (1, 0, 0) as the fixed points. We can have the hyperoval containing all the fixed points or containing no fixed points; hence two searches need to be made. The normaliser of ⟨g_{2}⟩ is transitive on orbits not on the sides of the triangle. So we take the orbit of (1, 1, 1) under ⟨g_{2}⟩ as the first orbit, and then commence the two separate searches. Hyperovals with the three fixed points give only the regular hyperoval, and there are no hyperovals without the three fixed points.

The element ⟨g_{3}⟩ has as P G(2, 4) as a fixed configuration. Every line of P G(2, 64) that is a line of P G(2, 4) meets the hyperoval in either 0 or 2 points. Hence the hyperoval is disjoint with P G(2, 4) or its intersection is a hyperoval of P G(2, 4). If the hyperoval and P G(2, 4) are disjoint the search finds no hyperoval. Now assume that the hyperoval meets P G(2, 4) in a 6-arc (a hyperoval of P G(2, 4)), which is the regular hyperoval D(x^{2}) in P G(2, 4). A total of 366 hyperovals are constructed; 6 are regular, 36 isomorphic to H_{60}, 144 isomorphic to H_{15}, and 180 are isomorphic to a new hyperoval, Penttila-Royle, D(f) where

f(x) = ω^{42}x^{2} + ω^{42}x^{4} + x^{6} + ω^{42}x^{8} + x^{10} + x^{12} + ω^{21}x^{20} + ω^{21}x^{22}
+ ω^{42}x^{24} + ω^{42}x^{28} + x^{32} + ω^{42}x^{36} + ω^{42}x^{40} + ω^{21}x^{42}
+ ω^{21}x^{44} + x^{52} + x^{54} + ω^{42}x^{56} + x^{58} + ω^{42}x^{60} + ω^{42}x^{62}.

The hyperoval has a full collineation stabiliser of order 12 and we denote it by H_{12}.

For elements of order 2 of P Γ L(3, 64) we use Theorem 56, p. 28 to determine that we have either:

(i) An elation g_{1} represented by

(\begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}) ,

or, (ii) the field automorphism g_{2} represented by

(x, y, z) ↦ (x^{8}, y^{8}, z^{8}).

By Theorem 58, p. 28, the fixed line x = 0 of the elation g_{1} is secant to the hyperoval and the centre is not on the hyperoval. The normaliser N of ⟨g_{1}⟩ acts doubly transitively on the points on the axis not equal to the centre so we can select the two points (0, 0, 1) and (0, 1, 0) to be in the hyperoval. The stabiliser of these two points in N is still transitive on all the points off the axis and therefore a further orbit of length two can be chosen, in this case, (1, 0, 0) and (1, 1, 1). From this initial configuration the search found 2 regular hyperovals, 64 hyperovals isomorphic to H_{12}, and 64 hyperovals isomorphic to H_{60}.

The element g_{2} has fixed configuration P G(2, 8). As before, we do searches when the hyperoval is disjoint from P G(2, 8) and searches where they intersect; the intersection being a 10-arc of P G(2, 8) (without loss of generality, D(2)). When the hyperoval and P G(2, 8) are disjoint the search found nothing; when they intersect the 10-arc completes to a regular hyperoval.

Theorem 114 ([111]). There are precisely four isomorphism classes of hyperovals in P G(2, 64) stabilised by any non-trivial automorphism. They are the regular hyperoval D(x^{2}) and the three hyperovals H_{12}, H_{15} and H_{60}.

With current computing power under a no symmetry hypothesis (that is, trivial stabiliser) the search is infeasible.

10. A Quick Look in P G(2, q), q > 64

Searches for hyperovals (or related structures implying the existence of hyperovals) have been made in P G(2, q), for some values of q > 64.

For q = 128 we have

|P Γ L(3, 128)| = 337 · 127^{2} · 43 · 7^{3} · 3 · 2^{21}.

By using the “prime at a time” techniques one can immediately rule out the prime 337.

Due to the large orbits obtained from the groups of order 127 and 43, naive exhaustive searches are possible. Only the regular hyperovals are found stabilised by a group divisible by 127 or by a group divisible by 43.

Searches for hyperovals in P G(2, 128) and P G(2, 256) stabilised by the element

(x, y, z) ↦ (x^{2}, y^{2}, y^{2})

that is, o-polynomials with coefficients over GF (2), have been made in [111]. There are no new GF (2) o-polynomials in P G(2, 128) and P G(2, 256). The authors speculate that GF (2) o-polynomials are rare in planes of square order (the o-polynomials belonging to the regular, translation or the Subiaco hyperovals as described in Chapters III and IV).

Further searches have been done by Penttila and Royle (see [111] and [101]). As of the writing of this work (September 1995), new hyperovals have been found in P G(2, 256), P G(2, 1024), P G(2, 4096), and P G(2, 16384). These are associated with cyclic q–clans and we shall talk about these in Chapter III, Section 15.1, p. 121.

The new hyperoval found in P G(2, 256) has o-polynomial of the form f_{2} =: ω^{112}x^{2} + ω^{211}x^{4} + ω^{130}x^{6} + ω^{80}x^{8} + ω^{84}x^{10} + ω^{57}x^{12} + ω^{250}x^{14} + ω^{144}x^{16} + ω^{27}x^{18} + ω^{47}x^{20} + ω^{135}x^{22} + ω^{194}x^{24} + ω^{143}x^{26} + ω^{111}x^{28} + ω^{214}x^{30} + ω^{107}x^{32} + ω^{95}x^{34} + ω^{83}x^{36} + ω^{79}x^{38} + ω^{51}x^{40} + ω^{155}x^{42} + ω^{45}x^{44} + ω^{200}x^{46} + ω^{185}x^{48} + ω^{55}x^{50} + ω^{115}x^{52} + ω^{209}x^{54} + ω^{124}x^{56} + ω^{85}x^{58} + ω^{64}x^{60} + ω^{94}x^{62} + ω^{35}x^{64} + ω^{153}x^{66}+ω^{181}x^{68}+ω^{54}x^{70}+ω^{106}x^{72}+ω^{82}x^{74}+x^{76}+ω^{164}x^{78}+ω^{206}x^{80}+ω^{32}x^{82}+ ω^{69}x^{84} + ω^{183}x^{86} + ω^{200}x^{88} + ω^{21}x^{90} + ω^{22}x^{92} + ω^{239}x^{94} + ω^{228}x^{96} + ω^{53}x^{98} + ω^{31}x^{100} + ω^{206}x^{102} + ω^{62}x^{104} + ω^{171}x^{106} + ω^{122}x^{108} + ω^{190}x^{110} + ω^{18}x^{112} + ω^{204}x^{114} + ω^{66}x^{116} + ω^{172}x^{118} + ω^{169}x^{120} + ω^{177}x^{122} + ω^{9}x^{124} + ω^{254}x^{126} + x^{128} +ω^{254}x^{130} +ω^{9}x^{132} +ω^{177}x^{134} +ω^{169}x^{136} +ω^{172}x^{138} +ω^{66}x^{140} +ω^{204}x^{142} + ω^{18}x^{144}+ω^{190}x^{146}+ω^{122}x^{148}+ω^{171}x^{150}+ω^{62}x^{152}+ω^{206}x^{154}+ω^{31}x^{156}+ω^{53}x^{158}+ ω^{228}x^{160}+ω^{239}x^{162}+ω^{22}x^{164}+ω^{21}x^{166}+ω^{200}x^{168}+ω^{183}x^{170}+ω^{69}x^{172}+ω^{32}x^{174}+ ω^{206}x^{176}+ω^{164}x^{178}+x^{180}+ω^{82}x^{182}+ω^{106}x^{184}+ω^{54}x^{186}+ω^{181}x^{188}+ω^{153}x^{190}+ ω^{35}x^{192}+ω^{94}x^{194}+ω^{64}x^{196}+ω^{85}x^{198}+ω^{124}x^{200}+ω^{209}x^{202}+ω^{115}x^{204}+ω^{55}x^{206}+ ω^{185}x^{208}+ω^{200}x^{210}+ω^{45}x^{212}+ω^{155}x^{214}+ω^{51}x^{216}+ω^{79}x^{218}+ω^{83}x^{220}+ω^{95}x^{222}+ ω^{107}x^{224} + ω^{214}x^{226} + ω^{111}x^{228} + ω^{143}x^{230} + ω^{194}x^{232} + ω^{135}x^{234} + ω^{47}x^{236} + ω^{27}x^{238}+ω^{144}x^{240}+ω^{250}x^{242}+ω^{57}x^{244}+ω^{84}x^{246}+ω^{80}x^{248}+ω^{130}x^{250}+ω^{211}x^{252}+

ω^{112}x^{254}

where ω is a primitive element of GF (256) satisfying ω^{8} + ω^{4} + ω^{3} + ω^{2} + 1 = 0.

11. Applications of Hyperovals

11.1. Generalised Quadrangles

A large portion of Chapter III, p. 81, is dedicated to the connections between generalised quadrangles (see Section 9, p. 11 for the definition of a generalised quadrangle) and hyperovals.

11.2. Codes

In a point line geometry if we let A be the line-point incidence matrix of the geometry then the code is the span of the rows over ℤ_{p} for some prime p.

Usually the code is either the whole space of the geometry or a hyperplane. Only if p is well-chosen is the code interesting. For instance, in a projective plane of order q, the code is interesting if and only if p divides q [71][48].

We define the extended code, C, of projective plane of even order to be the code generated by

( \begin{matrix}& 1 \\ A & ⋮ \\ & 1 \end{matrix}) ,

where A is the line-point incidence matrix.

Let q be even, so p = 2. Let r_{1}, r_{2} be rows of A. Since any two distinct lines meet in a unique point then r_{1} · r_{2} = 1 so

(r_{1}, 1) · (r_{2}, 1) ≡ 0 (mod 2).

Also in a projective plane of order q, r_{1} · r_{1} = q + 1, so

(r_{1}, 1) · (r_{1}, 1) = q + 2 ≡ 0 (mod 2).

So the rows of the extended matrix are orthogonal, that is, C ⊆ C^{⊥}, where C is the extended code of the extended matrix A. Studying C is equivalent to studying C^{⊥} (since C = C^{⊥⊥}). We call C^{⊥} the dual of the extended binary code of the plane. We define the weight of a codeword, r, of the extended code as the number of ones in r. Let w_{i} be the number of words of weight i. The weight enumerator of C^{⊥} is the polynomial

w(x, y) = ∑_{i=0}^{q^{2}+q+2} w_{i}x^{i}y^{q^{2}+q+2−i}.

We will try to give some geometric interpretations of C^{⊥}, q even:

Let (u, 0) ∈ C^{⊥}. Then we define the support of u to be

support(u) = { points P | the component of u corresponding to P = 1}.

It will also be convenient to identify a line l with the row vector corresponding to it.

Let U = support(u). Then U has even type, that is, class [0, 2, 4, …, q^{2} + q], (see Section 1, p. 15, Chapter I). (Since for any line l, |l ∩ U| = (l, 1) · (u, 0) ≡ 0 (mod 2), as (u, 0) ∈ C^{⊥} with C^{⊥} = ⟨(l, 1) | l⟩^{⊥}. So |l ∩ U| is even.)

Let (u, 1) ∈ C^{⊥}. Let U = support(u). Then U has odd type, that is, class [1, 3, 5, …, q^{2} + q + 1]. (Since for a line l, |l ∩ U| = (l, 1) · (u, 1) ≡ 1 (mod 2) as (u, 1) ∈ C^{⊥}. So |l ∩ U| is odd.)

So studying C^{⊥} is equivalent to studying all sets of even type and all sets of odd type in the plane.

The minimum weight of C^{⊥} is the minimum weight of the nonzero code words in C^{⊥}.

Theorem 115 ([3]). A word, w, of C^{⊥} has weight q + 2 if and only if the word is w = (l, 1), for a line l, or support(w) is a hyperoval.

Proof. Recall that a nonempty set of even type has greater than or equal to q + 2 points with equality if and only if it is a hyperoval.

By the results of Theorem 27, p. 15 (and the notation used there); for a k-set of even type we have

t_{0} + t_{2} + ⋯ + t_{q} = q^{2} + q + 1,
2t_{2} + 4t_{4} + ⋯ + qt_{q} = k(q + 1), and
2t_{2} + 4.3t_{4} + ⋯ + q(q − 1)t_{q} = k(k − 1).

Hence we have (via the last two equations)

4.2t_{4} + ⋯ + q(q − 2)t_{q} = k(k − (q + 2)).

If 0 < k ≤ q + 2 then we have the right hand side of the above equation being less than zero while the left hand side is always positive. So k ≥ q + 2, with equality if and only if t_{4} = t_{6} = ⋯ = t_{q} = 0. Hence if the k-set has class [0, 2] and is non-empty then it must be that the k-set is a hyperoval.

Let K be a set of odd type. Let P ∉ K, then for each of the q + 1 lines on P they must meet K in at least one point. Hence |K| ≥ q + 1. Now a line is a set of type (1, q + 1), so is of odd type. Suppose | support(u)| = q + 1 = |K| for some k-set K but is not a line. Let l be a secant line to K. Let Q ∉ K be a point on l. Then by the previous result every line on Q meets K in at least one point, implying |K| > q + 1. Hence support(u) is a line for (u, 1) ∈ C^{⊥}. Thus a set of odd type has greater than or equal to q + 1 points with equality if and only if it is a line.

Thus the minimum weight of C^{⊥} is q + 2 and

w_{q+2} = (q^{2} + q + 1) + the number of hyperovals in the plane.

A binary code C is self-orthogonal if and only if C ⊆ C^{⊥}, and is self-dual if and only C = C^{⊥}. The extended binary code of a projective plane of even order q is self-dual if and only if q ≡ 2 (mod 4). We know only one such plane: P G(2, 2).

The above results have been used by Lam et al. [70] to show that if a projective plane of order 10 existed it would have no hyperovals and by investigating the code (amongst other things!) Lam et al. [69] proved the non-existence of a projective plane of order 10.

11.3. Ovoids

A k-cap in P G(3, q) is a set of k points no three collinear.

An ovoid is a (q^{2} + 1)-cap of P G(3, q) if q > 2; an ovoid of P G(3, 2) is a set of 5 points, no 4 coplanar.

An excellent introduction to the relationship between ovoids and ovals is [108]. There are only two known classes of ovoids in P G(3, q) up to equivalence:

(i) Elliptic Quadric: This is the set of all totally singular one-dimensional subspaces of GF (q)^{4} with respect to a non-singular quadratic form of Witt index one.

(ii) Tits Ovoid: This is the set of all absolute points of a polarity of W (q) (see

Chapter III) where q = 2^{e} and e is odd, e > 1. By the action of P Γ L(4, q) on the ovoids we can describe the set of points of the

ovoids by our favourite canonical form:

Elliptic Quadric: Let a ∈ GF (q) with x^{2} + x + a being irreducible over GF (q), then we can describe the elliptic quadric as:

{(t^{2} + st + as^{2}, 1, s, t) | s, t ∈ GF (q)} ∪ {(1, 0, 0, 0)}.

The stabiliser in P GL(4, q) of an elliptic quadric, namely P O^{−}(4, q), acts 3– transitively on the points of the elliptic quadric.

Tits Ovoid: Let q = 2^{e} with e odd, e > 1, and let σ ∈ Aut GF (q) such that σ^{2} ≡ 2 (mod q − 1) then the Tits ovoid can be described as:

{(t^{σ} + st + s^{σ+2}, 1, s, t) | s, t ∈ GF (q)} ∪ {(1, 0, 0, 0)}.

The stabiliser in P GL(4, q) of a Tits ovoid, namely the Suzuki group Sz(q), acts 2-transitively on the points of the Tits ovoid.

For a Tits ovoid, as defined above with σ, we can define a Tits oval as an oval equivalent to D(σ) − {(0, 1, 0)} and a Tits hyperoval as a hyperoval equivalent to D(σ).

Theorem 116. Every plane meets an ovoid in either 1 point (tangent plane) or q + 1 points (secant plane).

A plane section of an ovoid in P G(3, q) is an oval in P G(2, q).

Every plane section of an elliptic quadric is a conic. Every plane section of a Tits ovoid is a Tits oval. (The Tits oval is a translation oval and so has a unique axis; these axes form the Lüneburg spread [133] [74].)

Theorem 117 (External Lines Lemma). Let O be a translation oval of P G(2, q), q even, with nucleus N, and let l be an axis of O. Let P be a point of O on l and let Q be another point of l, distinct from P and N. Suppose that O′ is a translation oval containing P such that its nucleus N′ is a point on l distinct from Q. If every line on Q external to O is also external to O′ then l is an axis of O′.

For α ∈ Aut GF (q) and a function d: GF (q) → GF (q), let Ω(α, d) denote the set

Ω(α, d) = {t^{α} + st + d(s), 1, s, t) | s, t ∈ GF (q)} ∪ {(1, 0, 0, 0)}.

With this definition we can say the elliptic quadric is the ovoid Ω(2, s ↦ as^{2}) where trace(a) = 1, and Ω(σ, σ + 2) with σ^{2} = 2 describes the Tits ovoid.

Lemma 118 ([108]). Suppose that Ω is an ovoid in P G(3, q), where q = 2^{e} > 2, and that π is a secant plane such that π ∩ Ω is a translation oval with associated automorphism α. Let l be an axis of π ∩ Ω. Suppose that each secant plane to Ω on l meets Ω in a translation oval. Then there is a collineation of P G(3, q) mapping Ω to Ω(α, d) where d: GF (q) → GF (q) satisfies d(0) = 0 and

d(a) + d(b) ∉ (a + b)^{α/(α−1)}K (2)

for all a, b ∈ GF (q) with a ≠ b, where K is the kernel of the trace map. Conversely, if α is a generator of Aut GF (q) and d ∈ GF (q) → GF (q) has d(0) = 0 and satisfies Eq. 2 for all a ≠ b, then Ω(α, d) is an ovoid.

Theorem 119 ([108]). Suppose that Ω is an ovoid in P G(3, q), where q is even, q > 2, and that π is a secant plane such that π ∩ Ω is a translation oval. Let l be an axis of π ∩ Ω. Then all secant planes to Ω on l meet Ω in translation ovals if and only if Ω is an elliptic quadric or a Tits ovoid.

Proof. By Lemma 118, p. 72, there is a generator α of Aut GF (q) and a function d such that Ω = Ω(α, d):

Let

M_{u} = (\begin{matrix}1 & u^{α} & u & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & u & 0 & 1 \end{matrix}) ,

for u ∈ GF (q). Then

M_{u} (\begin{matrix}t^{α} + st + d(s) \\ 1 \\ s \\ t \end{matrix}) = (\begin{matrix}(t + u)^{α} + s(t + u) + d(s) \\ 1 \\ s \\ t + u \end{matrix}) ,

so M_{u}Ω = Ω. Also, M_{u} is a translation of the inversive plane I (see below) associated with Ω. Moreover, M = {M_{u} | u ∈ GF (q)} is transitive on the points other than (0, 0, 0, 1) of the oval π_{s} ∩ Ω of the secant plane π_{s}: x_{3} = sx_{2} to Ω on l for each s ∈ GF (q). Hence I does not have Hering type I.1. So, by the results of Glynn [43], Ω is an elliptic quadric or a Tits ovoid.

Corollary 120 ([108]). Since the only ovals of P G(2, 8) are translation ovals (see page 46) the only ovoids of P G(3, 8) are the elliptic quadric and the Tits ovoid.

Theorem 121 ([56]). The only ovoids of P G(3, 2) and P G(3, 4) are the elliptic quadrics.

Let O_{1} and O_{2} be ovals of P G(2, q) and let P be a point of P G(2, q) not on either of the ovals and distinct from their nuclei. Then O_{1} and O_{2} are compatible at P if:

(i) they have the same nucleus, (ii) they have a point Q in common, (iii) the line P Q is a tangent line to each oval, and (iv) every secant line to O_{1} on P is external to O_{2}.

Hence we have that every external line to O_{1} on P is a secant line to O_{2}.

A pencil of an ovoid Ω is the set of secant plane sections to Ω for planes on a fixed tangent line.

The following theorem was found independently by D. Glynn and T. Penttila [105].

Theorem 122 (The Plane Equivalence Theorem). An ovoid of P G(3, q) is equivalent to a set of q ovals O_{s}, s ∈ GF (q), of P G(2, q) all with nucleus (0, 1, 0) satisfying

O_{s} ∩ O_{t} = {(0, 0, 1)} for all s ≠ t in GF (q),

and such that O_{s} and O_{t} are compatible at (0, 1, s + t). Moreover, each pencil of the ovoid Ω gives rise to such a set, and for each plane π of the pencil there is a parameterisation of the planes π_{s} of the pencil, s ∈ GF (q), such that π_{0} = π and there is a homography M_{s} taking π_{s} ∩ Ω to O_{s} and the tangent line of the pencil to the line [1, 0, 0].

The image under a collineation of such a set of ovals is called a fan of ovals and the image of the line [1, 0, 0] is called the common tangent.

Let O_{1} and O_{2} be ovals and P_{1} and P_{2} be points of P G(2, q). We say that (P_{1}, O_{1}) matches with (P_{2}, O_{2}) if there is a collineation g such that gO_{1} and O_{2} are compatible at P_{2} and gP_{1} = P_{2}.

The Plane Equivalence Theorem leads us to study compatible ovals and therefore matching point-oval pairs. Hence a classification of hyperovals in P G(2, q) will result in a classification of ovoids in P G(3, q).

As all hyperovals are classified in P G(2, q) for q ≤ 32 (see Section 6, p. 46) then all matching point-oval pairs can be determined. Penttila and O’Keefe [79] [83] have shown that the only fans in P G(2, 16) are with conics and so every plane section of an ovoid in P G(3, 16) is a conic, hence all ovoids of P G(3, 16) are elliptic quadrics. O’Keefe, Penttila and Royle [86] have shown that in P G(3, 32) all matches are with translation ovals, hence by Theorem 119, p. 72, the only ovoids are Tits ovoids or elliptic quadrics.

11.4. Inversive Planes

An inversive plane is an incidence structure ℐ = (P, B, I) with elements of B called circles, satisfying,

(i) any three distinct points of P lie on exactly one circle, (ii) if P is a point and C_{1}, C_{2}, C_{3} are distinct circles such that C_{1} ∩C_{2} = C_{2} ∩C_{3} =

{P } then C_{1} ∩ C_{3} = {P }.

It is interesting to note that given a point P ∈ P, the points different from P and the circles incident with P form an affine plane [36]. The order of the inversive plane, if it is finite, is the order of the constructed affine plane (which is independent of the choice of P).

Let I = (P, B, I) be an incidence structure such that

(i) P = points of an ovoid O,

(ii) B = secant planes to O,

(iii) I is the natural incidence.

Then I is an inversive plane.

A bundle is a set of circles through two distinct points. The two distinct points associated with a bundle are called the carriers of a bundle.

A pencil is any maximal set of mutually tangent circles (that is, meeting in exactly one point) through a common point P, called the carrier of the pencil.

A flock is a set F of mutually disjoint circles of an inversive plane such that, with exception of precisely two points, the carriers of the flock, every point of the inversive plane is on a unique circle of F.

An inversive plane arising from an ovoid is called egglike, and any inversive plane of even order is egglike [36, 6.2.14]. From this we have:

Theorem 123. If two inversive planes of even order are isomorphic then their corresponding ovoids are isomorphic.

Proof. By Theorem 200, p. 143, Chapter IV, all flocks of ovoids in P G(3, q) are linear. Hence the incidence structure with points the points and circles of the inversive plane and lines the bundles, pencils and flocks of the inversive planes is isomorphic to the point-line structure of a projective plane. (The incidence here is natural: a circle is incident with a bundle, pencil or flock if it is in the bundle, pencil or flock, respectively; a point is incident with no flock and with only those bundles and pencils for which it is a carrier). Thus an isomorphism between two inversive planes of even order induces an isomorphism between the two associated projective spaces, carrying points of the first inversive plane to points of the second. Thus if the inversive planes are isomorphic so are the underlying ovoids.

From this theorem and the results in Section 11.3, p. 71, we can now say that there is exactly one inversive plane of each order 2, 4, 16. There are exactly two inversive planes of order 8 and 32.

11.5. Partial Geometries and Strongly Regular Graphs

A partial geometry is an incidence structure pg = (P, B, I) such that:

(i) Each point is incident with t + 1 lines (t ≥ 1) and two distinct points are incident with at most one line.

(ii) Each line is incident with s+1 points (s ≥ 1) and two distinct lines are incident

with at most one point. (iii) If P is a point and l is a line not incident with P, then there are exactly α

(α ≥ 1) points P_{1}, …, P_{α} and α lines l_{1}, …, l_{α} such that P I l_{i}, P_{i} I l_{i}, and P_{i} I l, for all i = 1, …, α

If |P| = v and |B| = b then v = \frac{1}{α}(1 + s)(st + α) and b = \frac{1}{α}(1 + t)(st + α) and we denote pg = (P, B, I) by pg(s, t, α).

Let ℐ = (P, B, I) be an incidence structure such that

(i) P = points of a hyperoval H,

(ii) B = secant lines to H,

(iii) I in the natural incidence. Then ℐ is a partial geometry pg(q − 2, q/2, (q/2) − 1).

Now every partial geometry defines a strongly regular graph. Let Γ be the point graph of the partial geometry pg(s, t, α) = (P, B, I). That is, the vertices of Γ are P and x ∼ y (adjacency) whenever x and y are collinear in pg. The graph Γ is regular since every point of pg is collinear with s(t + 1) points (not including itself). The adjacency graph of the partial geometry has:

(i) any two adjacent vertices have λ = s − 1 + t(α − 1) common neighbours, (ii) any two nonadjacent vertices have μ = α(t + 1) common neighbours.

Hence Γ is a strongly regular graph, with parameters

srg(ν, k, λ, μ) = srg(\frac{1}{α}(1 + s)(st + α), s(t + 1), s − 1 + t(α − 1), α(t + 1)).

11.6. B-ovals

A Buekenhout oval [21], or B-oval, of order q is a set M of q + 1 ≥ 3 points and a set F of permutations of M such that

(i) each f ∈ F has order at most 2;

(ii) F is quasi sharply 2-transitive in the sense that for any (a_{1}, a_{2}) and (b_{1}, b_{2}) ∈ M × M, with a_{i} ≠ b_{j} for i, j = 1, 2, there is a unique f ∈ F such that f(a_{1}) = a_{2} and f(b_{1}) = b_{2}

The identity permutation belongs to F if and only if q is even [21].

Every oval, O, of a projective plane, π, gives rise to a B-oval, whose points are those of the oval, O. The permutations, ϕ_{P}, are associated with each point, P, not on the oval, ϕ_{P} fixes the point X when |P X ∩ O| = 1, and maps X to X′ where |P X ∩ O| = 2 with P X ∩ O = {X, X′}. We call such a B-oval obtained from an oval in a projective plane a projective B-oval. Thus we have B-conics, a projective B-oval obtained from a conic.

Faina [39] has classified all B-ovals of order q ≤ 8 using some results of Mathon [76]. There are nine isomorphism classes.

11.7. Howell Designs

Let Γ be a graph with edges E(Γ) and vertices V (Γ). A graph Γ_{1} is a spanning subgraph of Γ if V (Γ_{1}) = V (Γ). A factorisation F = {F_{1}, …, F_{t}} of the graph Γ is a partition of E(Γ) into edge-disjoint spanning subgraphs, called factors. A subgraph Δ of Γ is suborthogonal to F if |E(Δ) ∩ E(F_{i})| ≤ 1 for 1 ≤ i ≤ t and orthogonal to F if |E(Δ) ∩ E(F_{i})| = 1 for 1 ≤ i ≤ t.

A k-factor, Δ, of Γ is a spanning subgraph of Γ with each vertex of Δ having valency k, that is, Δ is a k-regular spanning subgraph of Γ. (Note one-factors are sometimes called perfect matchings.) A partition of the edges of a k-regular graph into one-factors is called a one-factorisation.

Let K_{2n} be the complete graph on 2n vertices.

A Howell design H(s, 2n) is an s × s array, n ≤ s ≤ 2n − 1, in which cells are empty or contain an edge of K_{2n}, no edge occurs in more than one cell, and each row and column is a 1-factor of K_{2n}. Or, more conventionally [60], a Howell design H(s, 2n) is a square of side s, n ≤ s ≤ 2n − 1, such that each cell is empty or contains an unordered pair of integers from {1, 2, …, 2n} such that each integer from 1 to 2n appears exactly once in every row and column and every unordered pair of integers appears at most once in a cell of the square. A Howell design where s = 2n − 1 is sometimes called a Room design.

Example. Let H be a hyperoval. Define Γ to be a graph with V (Γ) = {points of H}. Let l and m be external lines to H with P = l ∩ m. Then the edges of Γ are the secants to H meeting l and m not on P. The vertices of Γ become the symbols for the Howell design. The edges of Γ become the pairs which occur together in the Howell design. A pair of orthogonal one-factorisations become the rows and columns of the Howell design.

Theorem 124 ([1]). Let H be a hyperoval in a projective plane, π, of order q. Then there is a Howell design H(q, q + 2).

Proof. Let X be a point of π not on H. Let Γ be the complete graph on q + 2 vertices minus a 1-factor where V (Γ) = {points of H} and adjacency is defined as

P ∼ Q ⟺ P Q is not a line on X.

Any external line l on X gives a 1-factorisation of Γ with 1-factors being the lines on Y where Y I l and distinct from X.

Any two external lines on X give an orthogonal 1-factorisation. So from the above Example we get a Howell design H(q, q + 2).

Corollary 125. If e ≥ 2 then there is a Howell design of type H(2^{e}, 2^{e} + 2).

12. End Notes

Notes on Section 1, p. 15. The results in this Section are well known.

Notes on Section 2, p. 18. For Theorem 45, p. 23, we use the easy fact that all sharply 4-transitive groups on a set of size six are classified (being A_{6} and S_{6}).

The results in this Section are well known. The proof to Theorem 36, p. 19, The theorems and proofs in this section are well-known. is nonclassical.

Notes on Section 3, p. 23. The definition of collineation here is the same as that of page 23 by Theorem 5, p. 4.

The results in this Section are well known. Work on generalised homologies and elations is new but not original; work by the author and Penttila (see [106]). Section 3.4, p. 38, is known though unpublished (proofs communicated by Penttila).

Notes on Section 3.1, p. 24. The results in this Section are classical.

Let g be a collineation of a linear space ℐ (page 14). The incidence structure (Fixed points of g, Fixed lines of g, I |_{restricted}) is a linear space. Bruijn and Erdös [34] has shown that for a linear space (generalised to other structures by Buekenhout [23] and Dembowski [36]) the number of lines is equal to or greater than the number of points. Hence for an automorphism of a linear space the number of fixed lines is equal to or greater than the number of fixed points.

Theorem 60, p. 29, was proved in [15] but may have been proved in an earlier paper. In [15] theorems stronger than Theorem 65, p. 30 and Theorem 66, p. 31 are proved from which they follow (with difficulty).

Notes on Section 3.5, p. 40. The study of a set of points no three collinear, especially when the set had size q + 1, may have originated via Theorem 82, p. 40.

A correlation, g, of a projective space is a permutation g of the subspaces of P G(d, q) which preserves equality of dimension (that is, dim x_{1} = dim x_{2} if and only if dim gx_{1} = dim gx_{2}) and incidence (with incidence being symmetrised inclusion).

A correlation either preserves or reverses the usual ordering ≤ on the dimensions on a flag (that is, if X I x and dim X ≤ dim x then either we have dim gX ≤ dim gx or dim gx ≤ dim gX). If g preserves ≤ then g is a collineation, else g is a duality. A duality of order two is a polarity.

Notes on Section 4, p. 41. The results in this Section are well known.

Notes on Section 5, p. 43. The results in this Section are well known.

Notes on Section 6, p. 46. The Lunelli-Sce hyperoval is sometimes referred to as the Hall hyperoval as it was rediscovered by Hall [47].

The results on the Lunelli-Sce hyperoval after Theorem 100, p. 48, have been calculated using the algebraic package CAYLEY [26].

The author would like to thank Bill Cherowitzo for working out the inverse of the Cherowitzo o-polynomial.

Theorem 102, p. 48, and Theorem 107, p. 51, are new and appear in [106] and are a collaborative effort between the author and Penttila. Theorem 103, p. 49, and Theorem 104, p. 50, take advantage of the herd model to be introduced in Chapter III. The rest of the Section is well known.

Notes on Section 6.8, p. 53. A rational form of the O’Keefe-Penttila opolynomial is

\frac{η^{11}x^{12} + η^{8}x^{10} + x^{9} + η^{6}x^{8} + η^{19}x^{7} + η^{19}x^{6}η^{20}x^{6} + η^{20}x^{5} + x^{4} + η^{27}x^{3} + η^{11}x}{x^{12} + x^{10} + x^{8} + x^{6} + x^{4} + x^{2}}

where η is a primitive element of GF (32) satisfying η^{5} = η^{2} + 1.

Notes on Section 7, p. 54. This Section is from [110].

Notes on Section 8, p. 55. This Section has been published in [107]. The work is a collaborative effort between the author and Penttila.

Notes on Section 8.1, p. 55. Elements of GF (q^{2}) satisfying x^{2} + ω^{58}x + 1 = 0 are of order 13. Hence T is the group generated by the companion matrix of this polynomial.

Notes on Section 8.2, p. 56. Let ω be a primitive element of GF (q), q = 2^{e}, with ⟨ω⟩ = GF (q)^{∗}. For

G_{p,n} = ⟨(\begin{matrix}1 & 0 & 0 \\ 0 & ω^{i} & 0 \\ 0 & 0 & ω^{ni} \end{matrix})⟩

where |ω^{i}| = p, p | q − 1, to stabilise a hyperoval, it is necessary that the orbit of (1, 1, 1) is an arc, since the centraliser of G_{p,n} in P GL(3, q) is transitive on points with all coordinates non-zero; thus all orbits on these points are equivalent.

If the prime p = 2^{j} − 1, that is, a Mersenne prime (so j is necessarily a prime) then the orbit {(1, ω^{i}, ω^{ni}) | |ω^{i}| divides p} is contained in P G(2, 2^{j}). Thus the orbit of (1, 1, 1) is an arc if and only if D(n) is a hyperoval in P G(2, 2^{j}).

Unfortunately this result only seems useful when all the monomial hyperovals are known in P G(2, 2^{j}).

Notes on Section 8.3, p. 57. Recall the Samosa Lemma 7, Chapter I.

The centraliser of an element g of order a q-primitive prime divisor of q^{n} − 1 can be bigger than just a Singer cycle: when the prime is not a p-primitive prime divisor, but rather divides p^{ni} − 1 for some divisor i of h, when q = 2^{h}, then the

Gal (\frac{GF (q^{n})}{GF (2^{ni})})

part of the normaliser of the Singer cycle in question centralises g. Hence we can not always assume that from Theorem 20, p. 7 that GF (q^{d})^{∗} part of the normaliser is the centraliser of the group G in P Γ L(3, q). In particular:

For q = 4^{e}, e odd, g is of order 5, we have i = 2, as the centraliser of an element of order 5 in P Γ L(2, q) has order (q +1)e and thus the centraliser of an element of order 5 in P Γ L(3, q) has order (q^{2} − 1)e. From this we have C_{P Γ L(3,64)}(C_{5}) has an element of order 3 that is not in C_{P GL(3,64)}(C_{5}). This is the reason why we concentrate on calculating in C_{P GL(3,64)}(C_{5}).

This is also of relevance to Theorem 186, p. 118, Chapter III.

Notes on Section 9, p. 62. The algorithms in this Section were implemented by the author and Penttila. Section 9.1, p. 65, is from [111].

Notes on Section 10, p. 68. The work in P G(2, 128) on the primes is by the author. The other searches are by Penttila and Royle ([111]).

Notes on Section 11, p. 69. The results in this Section are well known.

Notes on Section 11.6, p. 76. Pascal’s theorem states that the three points of intersection of opposite sides of any hexagon inscribed in a nondegenerate conic are collinear. Buekenhout in [22] proved that if Pascal’s theorem holds for any oval in a projective plane (not necessarily finite) then the plane is Pappian (equivalent to a plane constructed from a three dimensional vector space defined over a field) and the oval is a conic. This theorem shows how a local property of the plane can determine what type of plane it is. Buekenhout’s results relied on the improved classification of 3-transitive groups by Zassenhaus’ [137] work on 2-transitive groups.

The concept of B-oval arose from this work of Buekenhout.

CHAPTER III Generalised Quadrangles

Recall the definitions of generalised quadrangles in Section 9, p. 11, Chapter I. We usually label 𝒮 = (𝒫, ℬ, I) to be a GQ with points 𝒫 and lines ℬ. We say GQ(s, t) or 𝒮 (s, t) to mean a GQ of order (s, t). The reference [103] will be referred to as Payne and Thas. Note in this Chapter our groups are acting on the right!

1. Parameters and Regularity

Theorem 126 (Payne and Thas, 1.2.1). A finite generalised quadrangle of order (s, t) has (s + 1)(st + 1) points and (t + 1)(st + 1) lines.

Let 𝒮 = (𝒫, ℬ, I) be a generalised quadrangle of order (s, t). Let

E = {{x, y} | x, y ∈ 𝒫, x ∼ y, and x ≠ y},

then (𝒫, E) is a graph, the point graph (see page 75) of 𝒮.

Recall the definition and parameters of a strongly regular graph (SRG) in Section 11.5, p. 75.

Lemma 127 (Payne and Thas, 1.2.2). (𝒫, E) is a strongly regular graph with parameters v = (s + 1)(st + 1), where |𝒫| = v, k = s(t + 1), λ = s − 1, and μ = t + 1.

Theorem 128 (Payne and Thas, 1.2.2). A generalised quadrangle of order (s, t) has

(s + t)|st(t + 1)(s + 1) and (s + t)|s^{2}(st + 1)

Let 𝒮 = (𝒫, ℬ, I) be a generalised quadrangle. For x ∈ 𝒫 we define x perp to be

x^{⊥} = {y ∈ 𝒫 | y ∼ x}.

\underline{82 III. GQs}

We note that x ∈ x^{⊥} for all x ∈ 𝒫. For a set A ⊆ 𝒫 we define

A^{⊥} = ⋂_{x∈A}{x^{⊥}},

that is, A^{⊥}, is the set of points collinear with all points of A.

The trace of a pair of distinct points x and y is

trace(x, y) = {x, y}^{⊥} = x^{⊥} ∩ y^{⊥} = {z ∈ 𝒫 | z ∼ x and z ∼ y}.

Lemma 129. If x ∼ y, x ≠ y, then |{x, y}^{⊥}| = s + 1. If x ≁ y then |{x, y}^{⊥}| = t + 1.

We define the span of a pair of distinct points x and y to be

span(x, y) = {x, y}^{⊥⊥} = {u ∈ 𝒫 | u ∼ z for all z ∈ {x, y}^{⊥}},

that is, points collinear with all points of {x, y}^{⊥}. If x ≁ y then we call {x, y}^{⊥⊥} the hyperbolic line determined by x and y.

Lemma 130. If x ∼ y, x ≠ y, then |{x, y}^{⊥⊥}| = s + 1. If x ≁ y then 2 ≤ |{x, y}^{⊥⊥}| ≤ t + 1.

Proof. If x ∼ y then {x, y}^{⊥⊥} equals the line incident with x and y (there can be no other points by GQ2 (page 11)).

Let x ≁ y. Now x and y are collinear with all points of {x, y}^{⊥} so we have x, y ∈ {x, y}^{⊥⊥}, hence we have the lower bound. Let u, v ∈ {x, y}^{⊥}. Any point in {x, y}^{⊥⊥} must be collinear with u and v hence {x, y}^{⊥⊥} ⊆ {u, v}^{⊥}. As u ≁ v Lemma 129 implies |{u, v}^{⊥}| = t + 1.

Theorem 131 (Payne and Thas, 1.2.3). A finite generalised quadrangle of order (s, t) has t ≤ s^{2}.

Corollary 132. A finite generalised quadrangle of order (2, t) has t = 2 or 4.

Proof. From above t ≤ 4, the divisiblility condition disallows t = 3 and thickness implies t ≥ 2.

We denote a generalised quadrangle of order (2, 2) by Q(4, 2) and denote a generalised quadrangle of order (2, 4) by Q(5, 2) (following Payne and Thas). These generalised quadrangles correspond to a parabolic and elliptic orthogonal space of rank 2 over GF (2) denoted O(5, 2) and O^{−}(6, 2), respectively. A grid of type (2, 2) corresponds to a hyperbolic orthogonal polar space O^{+}(4, 2), which is a Payne and Thas generalised quadrangle of order (2, 1) (Q(3,2) in Payne and Thas notation).

Theorem 133 ([52][53]). In a generalised quadrangle of order (s, t) if t ≠ s^{2} then t ≤ s^{2} − s.

The pair (x, y) of distinct points is called regular if x ∼ y or if x ≁ y then |{x, y}^{⊥⊥}| = t + 1. A point x is regular if all distinct pairs (x, y) are regular. Hence for x ≁ y, this implies that {x, y}^{⊥⊥} = {v, w}^{⊥} for all v, w ∈ {x, y}^{⊥}. (The above definitions are easier to visualise in the dual setting.)

A triad of points is a triple of pairwise non-collinear points. Given a triad T = {x, y, z}, a centre of T is a point collinear to x, y and z, that is, a point of T^{⊥}. We say the triad T is acentric if |T^{⊥}| = 0, centric if |T^{⊥}| > 0 and unicentric if |T^{⊥}| = 1.

Theorem 134 (Bose-Shrikhande [17]). In a generalised quadrangle of order (s, t) we have:

t = s^{2} if and only if each triad has s + 1 centres

That is, any three pairwise non-collinear points has exactly s + 1 neighbours.

A point x is coregular provided each line incident with x is itself regular. The noncollinear pair (x, y) is antiregular provided |z^{⊥} ∩ {x, y}^{⊥}| ≤ 2 for all z ∈ 𝒫 − {x, y}. A point x is antiregular provided (x, y) is antiregular for all y ∈ 𝒫 − {x}.

Let s^{2} = t, so by Theorem 134, p. 83, for any triad (x, y, z) we have |{x, y, z}^{⊥}| = s+1. So we have |{x, y, z}^{⊥⊥}| ≤ s+1. We say (x, y, z) is 3-regular if |{x, y, z}^{⊥⊥}| = s + 1. The point x is 3-regular if each triad containing x is 3-regular.

An interesting property of 3-regular points is:

Theorem 135 (Payne and Thas, 1.3.3). Let 𝒮 be a GQ(s, s^{2}) and suppose that any triad contained in {x, y}^{⊥}, x ≁ y, is 3-regular. The incidence structure with pointset {x, y}^{⊥}, with lineset the set of elements {u, v, w}^{⊥⊥}, where u, v, w are distinct points in {x, y}^{⊥}, and with the natural incidence, is an inversive plane of order s (see Section 11.4, p. 74).

By Theorem 1.3.6 in Payne and Thas if s = t and each point in x^{⊥} − {x} is regular, then every point is regular.

2. Some examples of generalised quadrangles

2.1. Sylvester’s Syntheme-Duad geometry

The following construction was first discovered by Sylvester [123].

Let X = {1, 2, 3, 4, 5, 6}. Let

𝒫 = {x | x ⊆ X, |x| = 2}

ℬ = {L | L is a partition of X into 3 disjoint subsets of size 2}

and I is membership. Then ℐ = (𝒫, ℬ, I) is a generalised quadrangle of order 2.

2.2. T_{2}^{∗}(H)

Let H be a hyperoval in P G(2, q), q = 2^{e}, q > 2. Embed P G(2, q) in P G(3, q). Define:

The set of points, 𝒫 = points of P G(3, q) − P G(2, q).

The set of lines, ℬ = lines of P G(3, q) − P G(2, q) meeting H (uniquely).

incidence = incidence of P G(3, q).

Theorem 136 ([36]). For q > 2 we have T_{2}^{∗}(H) = (𝒫, ℬ, incidence) is a generalised quadrangle of order (q − 1, q + 1).

Proof. Any two points are on exactly one line of P G(3, q), (this line may meet H). So GQ1 is satisfied.

Any point is collinear with exactly (q + 2)q other points. When q > 2 the incidence of P G(3, q) shows that there are no points collinear with all other points so axiom ptGQ (page 11) is satisfied.

For q > 2 every line has q points on it, and every point has q + 2 lines on it, so we have thickness.

Let x be a point and L be a line not incident with x. So L is a line not incident with the point x. Let L meet H at a point y. The plane ⟨x, L⟩ will be a secant plane to H. Let the plane meet H at point y and another point z. There is a unique line M through z and x, which also meets the line L at a point w. So we have x I M I w I L.

Remark. For q = 2, T_{2}^{∗}(H) is a firm GQ, in fact, it is a grid of type (3, 3).

T_{2}^{∗}(H_{1}) ≅ T_{2}^{∗}(H_{2}) if and only if there exists an automorphism, g, of the space taking the plane π_{1} ⊇ H_{1} onto the plane π_{2} ⊇ H_{2} for which g: H_{1} ↦ H_{2}.

2.3. T_{2}(O)

Let O be an oval of P G(2, q). Embed P G(2, q) in P G(3, q). Define:

the set of points, 𝒫 =

(i): points of P G(3, q) − P G(2, q),

(ii): hyperplanes of P G(3, q) meeting O in one point, and (iii): a new symbol (∞), the set of lines, ℬ = (a): lines of P G(3, q) − P G(2, q) meeting O, and (b): points of O, and incidence as: points (i) are incident only with lines (a), defined by the incidence of P G(3, q).

points (ii) are incident with all lines (a) contained in it and meeting the same point of O.

(∞) is incident with no lines (a) and all lines (b).

Let T_{2}(O) = (𝒫, ℬ, incidence).

Theorem 137 ([36]). We have T_{2}(O) is a generalised quadrangle of order q.

Proof. The point (∞) is not collinear with any points of type (a). There exists points of type (i) not incident with points of type (ii). So ptGQ axiom is satisfied.

Let x, y be points of type (ii). The planes x and y will lie on two lines if the planes are secant to O, contradiction. Let x, y be points of type (i). These points are incident with lines of type (a) which is a line of P G(3, q) hence GQ1 is satisfied. Let y be a point of type (i), these points are never incident with the point (∞). Let y be a point of type (ii), then the plane y must be secant to O to lie on two lines with (∞), contradiction. Let x be a point of type (i) and y be a point of type (ii). There is only one line (line of P G(3, q)) incident with x and y else there will be two lines on x and the point of O which y is tangent to, contradiction. Hence GQ1 is satisfied.

A line of type (a) has q affine points (points of type (i)) on it and a point of type (ii) on it, so a total of q + 1 points. A line of type (b) has q points of type (ii) on it and is also incident with the point (∞). A point of type (i) has q + 1 lines on it (the cone with vertex the point and the base being O). A point of type (ii) has q lines (lines) on it from the tangent point and the tangent point itself (a line of type (b)). The point (∞) has the q + 1 lines of O. Hence for q ≥ 2 we have thickness.

Now we will prove that the GQ2 axiom is satisfied. We will split up the proof into the different types of anti-flags.

The point is (∞):

(i) Every line of type (b) is incident with (∞).

(ii) Every line, L, of type (a) is not incident with (∞). The line L is incident with a unique point, M, of O, a line of the GQ. There is a unique tangent plane, y, on M and the line L, this is a point of the GQ. Hence (∞) I M I y I L.

Let x be a point of type (ii), which intersects O at the point M:

(i) Let L be a line of type (a), that is a line of P G(3, q):

(a) L intersects O at M, but not in x: Then there is a unique plane, y, through M and L. So we have x I M I y I L.

(b) L does not intersect O at M: Then L intersects the plane x at a point y. There is a unique line (line), N, through M and y. Then x I N I y I L.

(ii) Let L be a line of type (b), that is point of O distinct from M: The point

(∞) is collinear with M and L, so x I M I (∞) I L. Let x be a point of type (i):

(i) Let L be a line of type (a), that is not incident with x, L intersecting O at R:

(a) We have ⟨x, L⟩ generating a secant plane, z to O, meeting it at the points R and S. The line M through x and S on the plane z meets L at a point (point) y. Hence x I M I y I L.

(b) We have ⟨x, L⟩ generating a tangent plane z to O, meeting at the point

R. Then there is a line M from x to R, so x I M I z I L.

(ii) L is a line of type (b): Let y be the unique tangent plane on x and L. Let M be the unique line on x and L, then x I M I y I L.

Theorem 138. If q is even then the point (∞) in T_{2}(O) is regular.

Proof. We need to show that all distinct pairs of points (x, (∞)) are regular. If x ∼ (∞) then those pairs are regular. We now assume x ≁ (∞). Let x be a point of type (i). Then {x, (∞)}^{⊥} is the set of planes on x meeting O at a tangent, that is, points of type (ii). There are q + 1 such planes. Since q is even all tangent planes meet in a line (with the line meeting P G(2, q) in the nucleus of O). So we can say {x, (∞)}^{⊥⊥} is the set of points on the line that is the intersection of the planes and the point (∞), which makes a total of q + 1 points. So (x, (∞)) is regular.

2.4. P (𝒮 , x)

To each regular point x of the GQ(s), 𝒮 = (𝒫, ℬ, I), there is associated a GQ P (𝒮 , x) = (𝒫′, ℬ′, I′) of order (s − 1, s + 1). The pointset is the set 𝒫 − x^{⊥}, the lineset is made from (a) lines not incident with x and (b) hyperbolic lines {x, y}^{⊥⊥} for x ≁ y. Incidence is defined as: If y ∈ 𝒫′ and L ∈ ℬ′ of type (a) then y I′ L if and only if y I L and if y ∈ 𝒫′ and L ∈ ℬ′ of type (b) then y I′ L if and only if y ∈ L. (Theorem 3.1.4 Payne and Thas.)

Theorem 139 ([91]). T_{2}^{∗}(H) is isomorphic to P (T_{2}(O), (∞)) with O = H − {x} and x ∈ H (hence O is an oval with nucleus x). But P (T_{2}(O), x) is not isomorphic to any T_{2}^{∗}(H) for any x ≠ (∞).

2.5. T_{3}(Ω)

Let Ω be an ovoid of P G(3, q). Embed P G(3, q) in P G(4, q). Define:

the set of points, 𝒫 =,

(i): points of P G(4, q) − P G(3, q),

(ii): hyperplanes of P G(4, q) meeting Ω in one point, and (iii): a new symbol (∞), the set of lines, ℬ =, (a): lines of P G(4, q) − P G(3, q) meeting Ω, and (b): points of Ω, and incidence is the incidence of T_{2}(O) with P G(3, q) replaced with P G(4, q)

Theorem 140 ([36]). Let Ω be an ovoid in P G(3, q), then T_{3}(Ω) is a GQ(q, q^{2}).

Theorem 141 (Payne and Thas, 3.3.1, 3.3.2). (i) Properties of T_{2}(O): all type (b) lines are regular; the point (∞) is regular or antiregular according as q is even or odd.

(ii) Properties of T_{3}(Ω): all type (b) lines are regular; the point (∞) is 3-regular.

2.6. Other GQs from arcs

Let q = 2^{e}, H = P G(2, q) and Σ = P G(3, q).

We can give two other constructions of GQs from H. We have H embedded in Σ as before, and define O_{x} = H − {x} for x ∈ H. The GQ T_{2}^{∗}(O_{x}) is defined as points being points of Σ − H and planes containing x, with H given a special treatment by being labelled by (∞), lines being tangent lines to O_{x}, and incidence being carried over from Σ. This GQ is in fact the GQ T_{2}(O) where O = O_{x}.

Let O_{xy} = H − {x, y} for distinct points x, y ∈ H. Let the points be (i) points of Σ − H, (ii) planes that contain x but not y, and (iii) planes that contain y but not x, and let the lines be all the lines of Σ − H meeting H at a point of O_{xy}. incidence is that of Σ. We denote this GQ of order (q + 1, q − 1) by T_{2}(O_{xy}).

Remark. Orbits of the stabilisers of H on pairs of points {x, y} of H correspond to isomorphism classes of T_{2}(O_{xy}).

3. Classical Generalised Quadrangles

Classical generalised quadrangles are generalised quadrangles from polar spaces of rank two. We define W (q) to be the space of Sp(4, q), that is, the polar space with respect to a symplectic polarity of P G(3, q). We define Q(4, q) to be the space of O(5, q) and define Q(5, q) to be the space of O^{−}(6, q). We define H(d, q^{2}) to be the polar space U(d + 1, q^{2}), the polar space from a nonsingular hermitian variety of P G(d, q^{2}) for d = 3 or d = 4.

For the classical generalised quadrangles W (q), Q(4, q) and Q(5, q) we have s = q and for H(3, q^{2}) and H(4, q^{2}) we have s = q^{2}. The values of t for these GQs are calculated below:

Let x be an absolute point with respect to the polarity Δ. Then the quotient space Δ(x)/x will replace the form ⟨ , ⟩ with ⟨ , ⟩′ where

⟨u + x, u + x⟩′ = ⟨u, v⟩,

similarly for the quadratic case. This form ⟨ , ⟩′ is nondegenerate and of the same type as ⟨ , ⟩.

For W (q): Δ(x)/x ≅ GF (q)^{2} with every point absolute (as the form is alternating), so q + 1 absolute points. So t = q.

For H(3, q^{2}): Δ(x)/x ≅ GF (q^{2})^{2} form is, without loss of generality, x_{1}x̄_{2} − x_{3}x̄_{4}, where x ↦ x̄ is conjugation. So q + 1 absolute points (in subfield, P G(1, q) ⊆ P G(1, q^{2})). So t = q.

For H(4, q^{2}): Δ(x)/x ≅ GF (q^{2})^{3}. The form gives a classical unital in P G(2, q^{2}) with q^{3} + 1 points. So t = q^{3}.

For Q(4, q): Δ(x)/x ≅ GF (q)^{3}. We have a conic in P G(2, q) with q + 1 points. So t = q.

For Q(5, q): Δ(x)/x ≅ GF (q)^{4}. We have an elliptic quadric in P G(3, q^{2}) with q^{2} + 1 points. So t = q^{2}.

Note: Q(3, q) (see Section 9, p. 11) is a Payne and Thas generalised quadrangle, it has s = q. We have Δ(x)/x ≅ GF (q)^{2} which has two points so t = 1.

3.1. Trace and span in W (q)

Generally, x^{⊥} = Δ(x) ∩ {absolute points}. Since all points are absolute for W (q) we have x^{⊥} = Δ(x).

Now

trace(x, y) = {x, y}^{⊥} = x^{⊥} ∩ y^{⊥} = Δ(x) ∩ Δ(y) = Δ(⟨x, y⟩).

For x ≁ y we have

span(x, y) = {x, y}^{⊥⊥} = Δ^{2}(⟨x, y⟩) = ⟨x, y⟩

Hence for x ≁ y a hyperbolic line ⟨x, y⟩ in W (q) is a line of P G(3, q).

3.2. Regularity of W (q)

In W (q) we have Δ(⟨x, y⟩) = Δ(x) ∩ Δ(y). All points of W (q) are regular if and only if all non-collinear pairs of points are regular. So we let x ≁ y. Then ⟨x, y⟩ is a line of P G(3, q) as all points of P G(3, q) are absolute under Δ. So we have the size of ⟨x, y⟩ being equal to q + 1. But |{x, y}^{⊥⊥}| = |⟨x, y⟩| = q + 1 = t + 1. Hence the pair (x, y) is regular. Hence all points of W (q) are regular.

In fact by theorem 3.3.3 in Payne and Thas all lines of W (q) are regular if and only if q is even

Theorem 142 (Payne and Thas, 5.2.1). A generalised quadrangle of order q is isomorphic to W (q) if and only if all its points are regular.

4. Ovoids and Spreads

Let 𝒮 = (𝒫, ℬ, I) be a generalised quadrangle. Then the set 𝒪 ⊆ 𝒫 of 𝒮 is an ovoid if every line of 𝒮 meets 𝒪 in exactly one point. A set 𝒟 ⊆ ℬ of 𝒮 is a spread if every point of 𝒮 is on exactly one line of the set 𝒟.

Theorem 143 (Payne and Thas, 1.8.1). If 𝒪 is an ovoid of a generalised quadrangle of order (s, t) then |𝒪| = st + 1. Dually, if 𝒟 is a spread of a generalised quadrangle of order (s, t) then |𝒟| = st + 1.

A polarity of a GQ(s) = (𝒫, ℬ, I) is an automorphism of the GQ of order two that maps Δ: 𝒫 ↦ ℬ and Δ: ℬ ↦ 𝒫.

Let Δ be a polarity of the generalised quadrangle 𝒮. We have absolute points if and only if s = t.

Theorem 144 (Payne and Thas, 1.8.2). The set of absolute points (lines) of a polarity of a generalised quadrangle of order s is an ovoid (spread) of the generalised quadrangle.

Proof. Let Δ be a polarity of the generalised quadrangle 𝒮. We will first show that each line is incident with at most one absolute point. Suppose not, that is, let x and y be distinct absolute points on a line L. Since x ∼ y we have Δ(x) ∼ Δ(y). Now Δ(x), Δ(y) and L can not be all distinct else we will get a triangle (since x I Δ(x) etc.) So without loss of generality we let L = Δ(x). Then y I L implies y I Δ(x) which implies x I Δ(y). Since y I Δ(y) and x I Δ(y) we have L = Δ(x) = Δ(y) by GQ1. But this implies x = y. Contradiction.

Now we show that every line has at least one absolute point. Let L be a line of the generalised quadrangle. First, suppose L is absolute. Then Δ(L) I L hence Δ(L) is an absolute point (on L). Now suppose L is not absolute. Then Δ(L) I̸ L so by GQ2 we have a unique point-line pair (x, M) with Δ(L) I M I x I L. As x I M we have Δ(M) I Δ(x), so by uniqueness of x and M we have Δ(x) = M and Δ(M) = x. Hence x I Δ(x) so x is absolute.

Theorem 145 (Payne and Thas, 1.8.4). Let 𝒮 be a generalised quadrangle of order s and having a regular pair (x, y) with x ≁ y. If 𝒪 is an ovoid of 𝒮 then

|𝒪 ∩ {x, y}^{⊥}| , |𝒪 ∩ {x, y}^{⊥⊥}| ∈ {0, 2},
and |𝒪 ∩ ({x, y}^{⊥} ∪ {x, y}^{⊥⊥})| = 2.

Theorem 146 (Payne and Thas, 1.8.4). If a generalised quadrangle of order s contains an ovoid 𝒪 and a regular point z ∉ 𝒪 then s is even.

Corollary 147. W (q) has ovoids only if q is even.

Proof. Use Theorem 146 and Theorem 142, p. 88.

Theorem 148 (Payne and Thas, 1.8.3). A generalised quadrangle of order (s, t) with t > s^{2} − s has no ovoids.

Corollary 149. A generalised quadrangle of order (s, s^{2}) has no ovoids.

In particular, T_{3}(Ω) has no ovoids.

Theorem 150 (Payne and Thas, 3.4.2). T_{2}(O) always has ovoids.

Proof. We give two constructions of ovoids of T_{2}(O), one of which works if O is a secant plane section of an ovoid Ω.

Let π be a plane with no point in common with O. There are q^{2} points of type (i) in π. These affine points of π and the point (∞) form an ovoid. Every line of type (b) is on (∞) only. Every line of type (a) meets π exactly once.

Suppose O is contained in an ovoid Ω of P G(3, q). Let O = {x_{0}, …, x_{q}}. Let π_{i} be the tangent plane of O at x_{i}, i = 0, …, q. The set (Ω − O) ∪ {π_{0}, …, π_{q}} is an ovoid of T_{2}(O). Every point x_{i} of O as a line meets only one point (∞) of the generalised quadrangle ovoid. So every line of type (b) meets the ovoid exactly once. Each q lines (lines) contained in π_{i} are tangent to Ω, so each of the q lines on x_{i} contains only the point π_{i}. Each of the q^{2} − q other lines on x_{i} but not in π_{i} meets Ω at two points, the point x_{i} and an affine point of Ω. So each line meets the ovoid in precisely one point.

Theorem 151 ([125]). An ovoid of W (q), q even, is an ovoid of P G(3, q).

Proof. Let 𝒪 be an ovoid of W (q). All points of W (q) are points of P G(3, q). Let x, y be points of W (q). If ⟨x, y⟩ is an absolute line, then it is a line of W (q) so meets 𝒪 at one point. Then ⟨x, y⟩ is a line of P G(3, q) that meets these points of 𝒪 in one point. Then ⟨x, y⟩ is a line of P G(3, q) that meets 𝒪 at one point.

If ⟨x, y⟩ is not an absolute line, then ⟨x, y⟩ is a hyperbolic line {x, y}^{⊥⊥}. But we have shown that |𝒪 ∩ {x, y}^{⊥⊥}| = 0 or 2. Hence the nonabsolute line ⟨x, y⟩ meets the set 𝒪 in P G(3, q) in 0 or 2 points. Since 𝒪 has size q^{2} + 1 then we have 𝒪 being an ovoid of P G(3, q).

Theorem 152 ([119]). An ovoid Ω of P G(3, q), q even, is an ovoid of a generalised quadrangle isomorphic to W (q).

Remark. The above theorem uses the fact that an ovoid determines a nondegenerate alternating form (up to similiarity) for q even. Hence the tangent lines of an ovoid are absolute with respect to the polarity induced by this form, (these lines form a linear complex of P G(3, q)).

For q even and a square only one type of ovoid of Q(4, q) is known. But for q even and not a square, two types of ovoids of Q(4, q) are known. Their projection from the nucleus of Q onto a P G(3, q) are the elliptic quadric and the Tits ovoid of P G(3, q) (see Section 11.3, p. 71). The corresponding spreads in W (q) are the regular spread and the Lüneburg spread of P G(3, q) [125]. W. M. Kantor [66] has constructed, for odd non-prime values of q, ovoids of Q(4, q) that are not contained in P G(3, q). These ovoids arise from flocks of the quadratic cone in P G(3, q). (Since the planes of the flock have a common point, an ovoid of O(5, q) arises, rather than just one of O^{+}(6, q).)

5. Isomorphisms of some GQs

We list some isomorphisms between the known GQs.

Theorem 153 (Payne and Thas, 3.2.1, 3.2.2, 3.3.3). (i) If T_{2}(O) has at least one regular pair of nonconcurrent type (a) lines on distinct points of O, then O is a conic and T_{2}(O) ≅ Q(4, q).

(ii) Q(4, q) is isomorphic to the dual of W (q). Moreover, Q(4, q) or W (q), is self-

dual if and only if q is even. (iii) The GQ T_{2}(O) is isomorphic to Q(4, q) if and only if O is an irreducible conic;

it is isomorphic to W (q) if and only if q is even and O is a conic. (iv) If T_{2}(O) has a regular type (ii) point, then q is even, O is a conic and T_{2}(O)

is isomorphic to Q(4, q).

(v) If T_{3}(Ω) has a regular type (a) line, then Ω is an elliptic quadric and T_{3}(Ω) is isomorphic to Q(5, q).

(vi) If T_{3}(Ω) has a 3-regular point other than (∞), then Ω is an elliptic quadric and T_{3}(Ω) is isomorphic to Q(5, q).

For q odd any oval is a conic and any ovoid is an elliptic quadric, so that T_{2}(O) ≅ Q(4, q) and T_{3}(Ω) ≅ Q(5, q). For q even, q > 4, there are always ovals O for which T_{2}(O) has a unique line of regular points. Also for q = 2^{e}, e odd and e > 2, the Tits ovoid, Ω, provides examples of T_{3}(Ω) not isomorphic to Q(5, q).

6. Subquadrangles

Let 𝒮 = (𝒫, ℬ, I) be a generalised quadrangle of order (s, t). We say 𝒮′ = (𝒫′, ℬ′, I′) is a subquadrangle (or subGQ) of order (s′, t′) of 𝒮 if 𝒮′ is a generalised quadrangle and 𝒫 ⊆ 𝒫′, ℬ ⊆ ℬ′ and I′ is a restriction of I to 𝒫′ × ℬ′. If 𝒮 ≠ 𝒮′ then 𝒮′ is a proper subGQ of 𝒮, so we have 𝒫′ ≠ 𝒫 or ℬ′ ≠ ℬ, hence s ≠ s′ or t ≠ t′. For convenience we will denote by 𝒮′(s′, t′) ⊆ 𝒮 (s, t) to mean that the generalised quadrangle 𝒮′ of order (s′, t′) is a subGQ of the generalised quadrangle 𝒮 of order (s, t) and that 𝒫, 𝒫′ belong to the appropriate generalised quadrangle. We denote collinearity in 𝒮′ by ∼_{sub} .

If two points x and y are non-collinear in the subGQ then they are non-collinear in the generalised quadrangle, (since, points and lines of 𝒮′ are subsets of 𝒮, so if we have x ∼ y but x ≁_{sub} y then we will contradict GQ2 in 𝒮).

Let L ∈ ℬ. Then we have

(i) L ∈ ℬ′ is a internal line of 𝒮′ or (ii) L ∉ ℬ′ and L is incident with a unique point x ∈ 𝒫′. We call L a tangent

line of 𝒮′ at x, or (iii) L ∉ ℬ′ and L is incident with no points of 𝒫′. We call L an external line

of 𝒮′. Notice that L meets 𝒮′ in either 0, 1 or all points and in no other number. Say L meet 𝒮′ in two points x and y, so L ∉ 𝒮′. Then x and y are not collinear in 𝒮′ but are collinear in 𝒮. Contradiction.

Dually, we can define a point x as a internal point, tangent point and external point.

Lemma 154. No tangent point of a subGQ 𝒮′ = (𝒫′, ℬ′, I′) ⊆ 𝒮 = (𝒫, ℬ, I)

is on a tangent line of a subGQ.

Proof. Suppose so. Let L be a tangent line of 𝒮′ and y be a tangent point of 𝒮′ with y I L, y ∉ 𝒫′, L ∉ ℬ′. Let x be the point that L meets 𝒮′ in, as x ∈ 𝒫′ we have x ≠ y. Let M be the line on the point y, M ∈ ℬ′, M ≠ L, so x I̸′ M. By GQ2 we have a unique pair (z, N) ∈ 𝒫 × ℬ such that x I′ N I′ z I′ M in 𝒮′. As ℬ′ ⊆ ℬ and 𝒫′ ⊆ 𝒫 we have a triangle.

Lemma 155. Let 𝒮′(s′, t′) ⊆ 𝒮 (s, t) with t > t′. Then there are no external lines of 𝒮′ if and only if t = st′.

Proof. Simply count the number of lines in 𝒮′ and compare with the number of lines in 𝒮.

Theorem 156 (Payne and Thas, 2.3.1). Let 𝒮 (s′, t′) be a proper subGQ of 𝒮 (s, t). Then either s = s′ or s ≥ s′t′ (or dually, t = t′ or t ≥ s′t′).

If s = s′ then each external point is collinear with 1 + s′t′ points of an ovoid of 𝒮′ and dually.

If s = s′t′ then each external point is collinear with exactly 1 + s′ points of 𝒮′ and dually.

Corollary 157. If 𝒮′(s, t′) ⊆ 𝒮 (s, t), with t′ < t, then 𝒮′ has ovoids. In fact, every pair of non-collinear points in 𝒮′ lies on an ovoid.

Theorem 158 (Payne and Thas, 2.2.2). Let 𝒮′(s, t′) ⊆ 𝒮 (s, t) with t′ < t. Then we have

(i) t′ ≤ s and if t′ = s then t = s^{2}.

(ii) \sqrt{s} ≤ t′ ≤ s and s^{3/2} ≤ t ≤ s^{2}.

(iii) If t = s^{3/2} then t′ = \sqrt{s}.

Corollary 159. If 𝒮′(s, t′) ⊆ 𝒮 (s, t), with t′ < t, then t > s.

Corollary 160 (Payne and Thas, 2.2.2). Let 𝒮′(s, t′) ⊆ 𝒮 (s, t) with t′ < t. There can be no proper subGQ 𝒮′′(s, t′′) of 𝒮′.

Remark. Though 𝒮′′(s, t′′) is a grid of type (s, 1), if t′′ = 1, and this can arise.

6.1. Rosette of ovoids

A rosette of ovoids, or a rosette, based at x in a generalised quadrangle 𝒮 (s, t) is a set ℛ of ovoids pairwise meeting at x such that

{𝒪 − {x} | 𝒪 ∈ ℛ},

partition the points of the generalised quadrangle not collinear with x. There are s^{2}t points not collinear with x, so |ℛ| = s.

Theorem 161. Let 𝒮′(s, t′) ⊆ 𝒮 (s, t) with t′ < t. Each tangent line of 𝒮′ meeting at a point x gives rise to (t−t′) rosettes based at x in 𝒮′. Also each external line of 𝒮′ gives a partition of 𝒮′ into ovoids.

Example. For q even,

W (q) ≅ Q(4, q) ⊆ Q(5, q).

Now O^{−}(6, q) = O^{−}(4, q)⊥O^{+}(2, q) where O^{+}(2, q) is a secant line. Now each ovoid of Q(4, q) is an ovoid of W (q) which is an ovoid of P G(3, q).

A point of Q(5, q) not in Q(4, q) gives rise to an elliptic quadric in Q(4, q). (Let H be a hyperplane containing the given Q(4, q), let x be an external point of Q(4, q) not on H. The set of points of Q(5, q) collinear with x form an elliptic quadric cone with vertex x, which meets H in an elliptic quadric). Conversely, let 𝒪 be an ovoid (elliptic quadric) in H. Let S be the span of the ovoid. The totally singular points are perpendicular to S, so the perp of S is a secant line to the ovoid. Hence each ovoid appears twice.

There are q^{2}(q^{2} −1)/2 elliptic quadrics in Q(4, q), each of which has q^{2} +1 tangent planes. The number of tangent planes in Q(4, q) is (q^{2} + 1)(q + 1)q(q − 1)/2. So there are q elliptic quadrics on a given tangent plane. Clearly these form a rosette (as the solids they span pairwise intersect in the tangent plane). So the number of rosettes is the number of tangent planes.

Each rosette appears twice. For a line gives rise to a rosette if and only if it is perpendicular to the tangent plane. The perp of a tangent plane is a plane containing a pair of intersecting lines. Hence we have a model of generalised quadrangle Q(5, q) as follows:

Points are of two types:

(i) Points of Q(4, q), and

(ii) (elliptic quadric of Q(4, q), ±).

Lines are of two types:

(i) Lines of Q(4, q), and

(ii) (elliptic quadric rosettes of Q(4, q), ±).

and incidence is natural except for the highly nontrivial complication of the signs.

6.2. Automorphisms

Theorem 162. Let 𝒮_{θ} = (𝒫_{θ}, ℬ_{θ}, I_{θ}) be the fixed substructure of an automorphism, θ, of a GQ of order (s, t). Then the substructure will be one of the following:

(i) B_{θ} = ∅ and P_{θ} is a set of pairwise noncollinear points.

(ii) P_{θ} = ∅ and B_{θ} is a set of pairwise noncollinear lines.

(iii) P_{θ} contains a point x such that every point of P_{θ} is collinear with x, and each

line of B_{θ} is incident with x. (iv) B_{θ} contains a line L such that every line of B_{θ} is concurrent with L, and each

point of P_{θ} is incident with L.

(v) 𝒮_{θ} is a grid.

(vi) 𝒮_{θ} is a dual grid. (vii) 𝒮_{θ} is a subquadrangle.

7. 4-gonal families and EGQs

The material in this section is the reason why groups act on the right in this chapter. It seemed to be ridiculous to rewrite all of this on the left, or to convert the rest of the thesis to the right.

Let 𝒮 = (𝒫, ℬ, I) be a GQ(s, t). A collineation g of 𝒮 is a whorl about a point x provided g fixes setwise each line incident with x.

Theorem 163 (Payne and Thas, 8.1.1). Let g be a nonidentity whorl about a point x. Then one of the following must occur:

(i) g fixes no point of 𝒫 − x^{⊥}.

(ii) There is a point, y ≁ x that is fixed by g. Let T = {x, y}^{⊥} and U = {x, y}^{⊥⊥}. Then, in the notation of Section 6.2, T ∪ {x, y} ⊆ 𝒫_{g} ⊆ T ∪ U and L ∈ ℬ_{g} if and only if L is incident with a point of T and a point of U ∩ 𝒫_{g}.

(iii) The substructure of fixed elements of g is a subquadrangle 𝒮_{g} of order (s′, t) where t ≥ s/t ≥ s′ implying s > t. This is a more general case of (ii).

If a whorl, g, about x is either the identity element or fixes no point of 𝒫 − x^{⊥} then g is an elation about x. If g fixes each point of x^{⊥}, that is, fixing each line on x pointwise, then g is a symmetry about x. From Theorem 163 any symmetry about x is also an elation about x.

The symmetries about x form a group. For each y ∼ x and y ≠ x, the group of symmetries about x acts semiregularly on the set {L ∈ ℬ | y I L, x I̸ L}, and so this group has order dividing t. The point x is called the centre of symmetry, (dually, axis of symmetry), provided its group of symmetries has order t, hence any such group acts regularly.

If there is a group, G, of elations about x acting regularly on 𝒫 − x^{⊥}, then we say the GQ is an elation generalised quadrangle (EGQ) with elation group G and base point x. If 𝒮 is an EGQ(s, t) with base point x and elation group G and if G contains a full group of s symmetries about each line through x, then 𝒮 is called a translation generalised quadrangle (TGQ) with base point x and translation group G.

Let 𝒮 be an EGQ(s, t) with base point x and elation group G. Let y be a fixed point of 𝒫 − x^{⊥} and L_{0}, …, L_{t} be the lines incident with x. Define the points z_{i} and the lines M_{i} by L_{i} I z_{i} I M_{i} I y for all i = 0, …, t. Put A(i) = {g ∈ G | M_{i}^{g} = M_{i}}, A^{∗}(i) = {g ∈ G | z_{i}^{g} = z_{i}} and ℱ = {A(i) | 0 ≤ i ≤ t}. Then |G| = s^{2}t, ℱ is a collection of 1 + t sugroups of G, each of order s, and each A^{∗}(i) is a subgroup of order st containing, as a subgroup, A(i). Importantly they satisfy two conditions:

K1: A(i)A(j) ∩ A(k) = {1} for distinct i, j, k.

K2: A^{∗}(i) ∩ A(j) = {1} for distinct i, j.

Conversely, let G be a group of size s^{2}t with 1 + t subgroups, A(i), each of order s and 1 + t subgroups, A^{∗}(i), of order st with A(i) ≤ A^{∗}(i) satisfying K1 and K2, then we can describe an EGQ(s, t) with base point (∞) as follows. Points are of three kinds:

(i) elements of G,

(ii) right cosets A^{∗}(i)g, g ∈ G, i = 0, …, t,

(iii) a symbol (∞). Lines are of two kinds:

(a) right cosets A(i)g, g ∈ G, i = 0, …, t,

(b) symbols [A(i)], i = 0, …, t.

Incidence is: a point of type (i), g, is incident with each line A(i)g, i = 0, …, t. A point of type (ii), A^{∗}(i)g is incident with [A(i)] and with each line A(i)h contained in A^{∗}(i)g. The point (∞) is incident with each line of type (b), [A(i)]. These are the only incidences.

Note that A^{∗}(i) = A(i) ∪ {g ∈ G | A(i)g ∩ A(j) = ∅ for 0 ≤ j ≤ t} and we sometimes say A^{∗}(i) is a tangent space to A(i) (see Payne and Thas p. 167).

Theorem 164 (Payne and Thas, 8.2.1). An EGQ 𝒮 with base point x and elation group G is isomorphic to some EGQ with A(i) and A^{∗}(i) being subgroups of the elation group G.

Let G be a group of order s^{2}t. Let ℱ = {A(i) ≤ G | 0 ≤ i ≤ t} with A(i) having order s satisfy K1 and let ℱ^{∗} = {A^{∗}(i) ≤ G | 0 ≤ i ≤ t} with A^{∗}(i) having order st satisfy K2 with A(i) ≤ A^{∗}(i), then we say ℱ is a 4-gonal family for G. From above we see that each 4-gonal family produces an EGQ(s, t) denoted 𝒮 (G, ℱ).

Theorem 165 (Payne and Thas, 8.2.2). (i) G acts by right multiplication as

a (maximal) group of elations about (∞). (ii) A(i) is the subgroup of G fixing the line A(i) of the EGQ. (iii) Any automorphism of G leaving ℱ invariant induces a collineation of the EGQ

fixing (∞).

Theorem 166. Let O be an oval. Then T_{2}(O) is self-dual if and only if O is a translation oval.

Proof. Let O be an oval in a hyperplane, H, of P G(3, q). Now if O is a conic it is also a translation oval for q even (but not for q odd). If O is a conic then T_{2}(O) is isomorphic to Q(4, q) which is self-dual for q even. Since all ovals are conics for q odd by Segre’s Theorem (page 20), we may assume q is even.

For q even and O is not a conic (∞) is regular and all type (b) lines are regular, but no type (a) lines are, so any duality of T_{2}(O) must take (∞) to a type (b) line. Now T_{2}(O) is a TGQ with base point (∞) hence there is an elation group with axis H. So if there is a duality ∧ taking (∞) to P, then T_{2}(O)^{∧} is a TGQ with base point P, that is, there is a group, G, of automorphisms of T_{2}(O) fixing P, (∞) and all points of type (ii) on P and is regular on the lines of P G(3, q) meeting H in a point of O − {P }.

Now by Theorem 141, p. 87, it follows that Aut T_{2}(O) fixes (∞) and so permutes the points and lines of a given type. We also have that the stabiliser of (∞) in Aut T_{2}(O) is the stabiliser of O in P Γ L(4, q). Hence G is a subgroup of P Γ L(4, q). Let l be the tangent line to P in H. Since G fixes l planewise, we have G being a subgroup of P GL(4, q) and as G is transitive on the lines meeting H in a point of O −{P }, we have G being transitive on O −{P }. So by Corollary 98, p. 47, it follows that O − {P } is a translation q-arc, and so O is a translation oval.

For the converse see the original proofs in Chapter 12 of Payne and Thas.

Theorem 167 (T. Penttila, personal communication). We have T_{3}(Ω)^{∧} is an EGQ if and only if Ω is an elliptic quadric or a Tits ovoid.

8. Constructing GQs of order (q^{2}, q)  97

8. Constructing GQs of order (q^{2}, q)

We now explicitly construct some EGQs of order (q^{2}, q), for q a prime power.

Let q be either odd or even. Let f: GF (q)^{2} × GF (q)^{2} → GF (q) be a given biadditive map. Let G = {(a, c, b) | a, b ∈ GF (q)^{2}, c ∈ GF (q)} with multiplication on G defined as

(a, c, b) · (a′, c′, b′) = (a + a′, c + c′ + f(b, a′), b + b′).

If f is nonsingular then the centre of G is

Z = {(0, c, 0) | c ∈ GF (q)}.

Under this multiplication

(a, c, b)^{−1} = (−a, −c + f(b, a), −b).

Suppose that for each t ∈ GF (q) there is an additive map d_{t}: GF (q)^{2} → GF (q)^{2} and a map g_{t}: GF (q)^{2} → GF (q). Then we can let

A(∞) = {(0, 0, b) ∈ G | b ∈ GF (q)^{2}}

and, for t ∈ GF (q),

A(t) = {(a, g_{t}(a), d_{t}(a)) ∈ G | a ∈ GF (q)^{2}}.

Now for each t ∈ GF (q) ∪ {∞} we have A(t) being a subgroup of G if and only if

g_{t}(a + b) − g_{t}(a) − g_{t}(b) = f(d_{t}(a), b) (3)

for all a, b ∈ GF (q)^{2}, t ∈ GF (q). Let A^{∗}(t) = A(t)Z for all t ∈ GF (q) ∪ {∞} and let ℱ = {A(t) | t ∈ GF (q) ∪ {∞}}. We would like to know when ℱ is a 4-gonal family.

Theorem 168 (Payne and Thas, 10.4.1). We have ℱ being a 4-gonal family for G providing the following hold:

(i) Equation (3) holds, (ii) The function d(t, u): a ↦ d_{t}(a) − d_{u}(a) is nonsingular for all t ≠ u, (iii) g_{t}(a) = g_{u}(a) for t ≠ u implies a = 0, and (iv)

g_{t}(d^{−1}(t, v)(a)) − g_{v}(d^{−1}(t, v)(a)) + g_{v}(d^{−1}(v, u)(−a)) − g_{u}(d^{−1}(v, u)(−a)) = 0

implies a = 0 for distinct t, u, v is true. Then ℱ and G yield a GQ of order (q^{2}, q) denoted (as in the previous section) 𝒮 (G, ℱ).

Suppose G_{1} and G_{2} are groups and ℱ a 4-gonal family of G_{1}. If θ: G_{1} → G_{2} is a group isomorphism or group anti-isomorphism then 𝒮 (G_{1}, ℱ) and 𝒮 (G_{2}, ℱ^{θ}) are isomorphic generalised quadrangles.

Note we have not explicitly stated what d_{t}, g_{t} and f are! We will need to develop some more machinery before we can do this. The next two Sections will help construct some 𝒮 (G, ℱ) and in doing so develop a connection between these GQs and ovals. In particular in Section 10, p. 102, we will look at specific forms for d_{t}, g_{t} and f when q is even.

8.1. Incidence of 𝒮 (G, ℱ) of order (q^{2}, q)

We shall now take a closer look at the incidence of 𝒮 (G, ℱ). This section is mostly for fun, but determining the coset representatives for the lines of 𝒮 (G, ℱ) is used in Section 17.2, p. 127.

It will be of benefit to write the coset representatives of A(t)(a, c, b) and A^{∗}(t)(a, c, b) for t ∈ GF (q) ∪ {∞}, c ∈ GF (q) and a, b ∈ GF (q)^{2}. It is easy to check that they are:

A(t)(0, c, b) for t ∈ GF (q), c ∈ GF (q) and b ∈ GF (q)^{2},
A(∞)(a, c, 0) for c ∈ GF (q) and a ∈ GF (q)^{2},
A^{∗}(t)(0, 0, b) for t ∈ GF (q) and b ∈ GF (q)^{2}, and
A^{∗}(∞)(a, 0, 0) for a ∈ GF (q)^{2}.

Each line has q^{2}+1 points incident with it. Below we denote the lines of 𝒮 (G, ℱ) as the set of point incident with them:

[A(∞)] = {A^{∗}(∞)(a, 0, 0) | a ∈ GF (q)^{2}} ∪ {(∞)}.
[A(t)] = {A^{∗}(t)(0, 0, b) | b ∈ GF (q)^{2}} ∪ {(∞)}, for each t ∈ GF (q).
A((∞))(a, c, 0) = {A^{∗}(∞)(a, 0, 0)} ∪ {(a, c + f(b, a), b) | b ∈ GF (q)^{2}},
for a ∈ GF (q)^{2}, c ∈ GF (q).
A(t)(0, c, b) = {A^{∗}(t)(0, 0, b)} ∪ {(a, g_{t}(a) + c, d_{t}(a) + b) | a ∈ GF (q)^{2}},
for t, c ∈ GF (q), b ∈ GF (q)^{2}.

The collinearity on points of 𝒮 (G, ℱ) is:

{A^{∗}(∞)}^{⊥} = {(0, c, b) | c ∈ GF (q), b ∈ GF (q)^{2}}
∪ {A^{∗}(∞)(a, 0, 0) | a ∈ GF (q)^{2}} ∪ {(∞)}.
{A^{∗}(t)}^{⊥} = {(a, g_{t}(a) + c, d_{t}(a)) | c ∈ GF (q), a ∈ GF (q)^{2}}
∪ {A^{∗}(t)(0, 0, b) | b ∈ GF (q)^{2}} ∪ {(∞)}, for all t ∈ GF (q).
{(a, c, b)}^{⊥} = {A^{∗}(t)(0, 0, b − d_{t}(a)) | t ∈ GF (q)} ∪ {A^{∗}(∞)(a, 0, 0)}
∪ {(d, c − g_{t}(a) + g_{t}(d), d_{t}(d) + b + d_{t}(a) |

a ≠ d ∈ GF (q)^{2}, t ∈ GF (q)} ∪ {(a, ¸+f(d, a), d) | d ∈ GF (q)^{2}}.

9. q–clan geometry  99

{A^{∗}(∞)(a, 0, 0)}^{⊥} = {(a, c + f(b, a), b) | c ∈ GF (q), b ∈ GF (q)^{2}}

∪ {A^{∗}(∞)(d, 0, 0) | d ∈ GF (q)^{2}} ∪ {(∞)}.

{A^{∗}(t)(0, 0, b)}^{⊥} = {(a, g_{t}(a) + d, d_{t}(a) + b) | d ∈ GF (q), a ∈ GF (q)^{2}}

∪ {A^{∗}(t)(0, 0, d) | d ∈ GF (q)^{2}} ∪ {(∞)}.

9. q–clan geometry

Following Payne [96][95][6], we investigate collections of 2 × 2 matrices over GF (q) that will later on be used to construct 4-gonal families yielding a GQ(q^{2}, q). We also see the use of the trace equation defined in Section 6, p. 9, Chapter I.

Let C = {A_{t} | t ∈ GF (q)} be a family of 2 × 2 matrices with entries in GF (q). We define the quadratic form Q_{st} as

Q_{st}(x, y) = (x y) (A_{s} − A_{t}) (\begin{matrix}x \\ y \end{matrix}) .

If Q_{st} is anisotropic for all s ≠ t we call C a q–clan. If C = {A_{t} | t ∈ GF (q)} is a q–clan so is C′ = {A_{t} − A_{0} | t ∈ GF (q)}, so without

loss of generality we let A_{0} equal the zero matrix 0. Also if \{A_{t} = (\begin{matrix}a & b \\ c & d \end{matrix}) | t ∈

GF (q)\} is a q–clan then so is \{A′_{t} = (\begin{matrix}a & b + c \\ 0 & d \end{matrix}) | t ∈ GF (q)\}, hence without loss

of generality each A_{t} is upper triangular. (We say two matrices A = (\begin{matrix}a & b \\ c & d \end{matrix}) and

B = (\begin{matrix}e & f \\ g & h \end{matrix}) are equivalent, denoted A ≡ B, if (\begin{matrix}a & b + c \\ 0 & d \end{matrix}) = (\begin{matrix}e & f + g \\ 0 & h \end{matrix}).) If

A_{t} = (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) then

Q_{st}(x, y) = (a_{s} − a_{t})x^{2} + (b_{s} − b_{t})xy + (c_{s} − c_{t})y^{2}.

As we shall only be concerned with fields of characteristic two Q_{st} is anisotropic if and only if

trace (\frac{(a_{s} + a_{t})(c_{s} + c_{t})}{(b_{s} + b_{t})^{2}}) = 1.

We will show how q–clans are related to other structures, for instance: elation generalised quadrangles from q–clans (Section 10, p. 102), flocks of quadratic cones from q–clans (Chapter IV), translation planes from flocks (Chapter IV, Section 2, p. 141).

The known q–clans for q even are presented in Chapter IV, Section 8, p. 147.

We now investigate further the entries of the matrices A_{t} for q even, and in doing so develop the theory of q–clans and ovals.

9.1. Herds and q–clans

If C is a q–clan we have b_{s} ≠ b_{t} for all s ≠ t. So t ↦ b_{t} is a permutation; we may relabel the subscript so that b_{t} = t^{1/2}. We have

C′ = \{A′_{t} = (\begin{matrix}a_{1}^{−1/2} & 0 \\ 0 & a_{1}^{1/2} \end{matrix}) A_{t} (\begin{matrix}a_{1}^{−1/2} & 0 \\ 0 & a_{1}^{1/2} \end{matrix}) | t ∈ GF (q)\} ,

is also a q–clan with A′_{0} = 0, A′_{t} = (\begin{matrix}a′_{t} & t^{1/2} \\ 0 & c′_{t} \end{matrix}) with A′_{1} = (\begin{matrix}1 & 1 \\ 0 & c′_{1} \end{matrix}). So without loss

of generality a_{1} = 1.

Let a = c′_{1} (since Q_{01} is anisotropic, trace(a) = 1). Define f: GF (q) → GF (q) by f(t) = a_{t} and g: GF (q) → GF (q) by g(t) = c_{t}/a. Then f(0) = g(0) = 0 and f(1) = g(1) = 1. Since Q_{st} is anisotropic for all s ≠ t, we have, with f(0) = g(0) = 0 and f(1) = g(1) = 1,

T_{a}(f, g) : trace (\frac{a(f(s) + f(t))(g(s) + g(t))}{s + t}) = 1, for all s ≠ t.

Conversely, if f and g satisfy T_{a}(f, g) and if A_{t} = (\begin{matrix}f(t) & t^{1/2} \\ 0 & ag(t) \end{matrix}), then C = \{A_{t} | t ∈

GF (q)\} is a q–clan with A_{0} = 0 and A_{1} = (\begin{matrix}1 & 1 \\ 0 & a \end{matrix}) where trace(a) = 1.

We now show that the permutation polynomials f and g of a q–clan

C = \{(\begin{matrix}f(t) & t^{1/2} \\ 0 & ag(t) \end{matrix}) | t ∈ GF (q)\}

are in fact o-polynomials (see Section 4, p. 41, Chapter II). Moreover, given f and g q − 1 other o-polynomials can be constructed.

Theorem 169. Let q be even. Let f, g: GF (q) → GF (q) with f(0) = g(0) = 0 and f(1) = g(1) = 1. Then T_{a}(f, g) is true if and only if g is an o-polynomial, f_{s} is an o-polynomial for all s ∈ GF (q) where

f_{s}(x) = \frac{f(x) + asg(x) + s^{1/2}x^{1/2}}{1 + as + s^{1/2}}

and trace(a) = 1.

Proof.

(⇒) Suppose T_{a}(f, g) is true. We also suppose that f(0) = 0 and f(1) = 1. The function f is one-to-one since if x ≠ y and f(x) = f(y) then a(f(x) + f(y))(g(x) + g(y))/(x + y) = 0, contradicting T_{a}(f, g). Let H be the set of points {(1, t, f(t)) | t ∈ GF (q)} ∪ {(0, 1, 0), (0, 0, 1)}. Since f is one-to-one no line on (0, 1, 0) meets H in more than two points. Clearly no line on (0, 0, 1) meets H in more than two points. We show that no three points of {(1, t, f(t)) | t ∈ GF (q)} are collinear.

Suppose x, y, z ∈ GF (q) are distinct and that the points (1, x, f(x)), (1, y, f(y)) and (1, z, f(z)) are collinear. Then

\frac{f(x) + f(y)}{x + y} = \frac{f(x) + f(z)}{x + z} = \frac{f(y) + f(z)}{y + z} = b, say.

Since trace is additive we have

trace(ab(g(x) + g(y))) + trace(ab(g(x) + g(z))) = trace(ab(g(y) + g(z))).

But this is contrary to T_{a}(f, g). So H is a hyperoval. As f(0) = 0 and f(1) = 1, f is an o-polynomial.

Since T_{a}(f, g) is true if and only if T_{a}(g, f) is true, g is also an o-polynomial. We now look at trace(b(f(x) + f(y))(f_{s}(x) + f_{s}(y))/(x + y)) where b = a + s^{−1} +

s^{−1/2}.

trace (\frac{b(f(x) + f(y))(\frac{f(x)+asg(x)+s^{1/2}x^{1/2}}{1+as+s^{1/2}} + \frac{f(y)+asg(y)+s^{1/2}y^{1/2}}{1+as+s^{1/2}} )}{x + y}) , x ≠ y

= trace (\frac{a(f(x) + f(y))(g(x) + g(y))}{x + y})

+ trace (\frac{1}{s} \frac{(f(x) + f(y))^{2}}{x + y} + \frac{1}{s^{1/2}} \frac{f(x) + f(y)}{(x + y)^{1/2}}) , x ≠ y

= trace (\frac{a(f(x) + f(y))(g(x) + g(y))}{x + y}) , x ≠ y, since trace(X^{2} + X) = 0.

So T_{b}(f, f_{s}) is true if and only if T_{a}(f, g) is true. Since f_{s}(0) = 0 and f_{s}(1) = 1 for all s ∈ GF (q), f_{s} is an o-polynomial. Putting x = 0 and y = 1 in T_{a}(f, g), we see that trace(a) = 1.

(⇐) Let

f_{s}(x) = \frac{f(x) + asg(x) + s^{1/2}x^{1/2}}{1 + as + s^{1/2}}

for some a with trace(a) = 1. Suppose that f_{s} is an o-polynomial for all s ∈ GF (q).

Fix x ≠ y. Then (1, x, f_{s}(x)), (1, y, f_{s}(y)) and (0, 1, 0) are not collinear for all s ∈ GF (q). So f_{s}(x) ≠ f_{s}(y), giving f_{s}(x) + f_{s}(y) ≠ 0, that is,

f(x) + f(y) + s(ag(x) + ag(y)) + s^{1/2}(x^{1/2} + y^{1/2}) ≠ 0 for all s ∈ GF (q).

The above equation is a quadratic in s^{1/2}. Hence this implies that

trace (\frac{(f(x) + f(y))(ag(x) + ag(y))}{x + y}) = 1.

Thus T_{a}(f, g) holds.

Hence for q even every q–clan in our particular form yields q + 1 ovals; this naturally leads to the following definition.

A herd of ovals in P G(2, q), q even, is a family of q + 1 ovals {O_{s} | s ∈ GF (q) ∪ {∞}}, each containing (1, 0, 0), (0, 1, 0) and (1, 1, 1) and with nucleus (0, 0, 1), with

O_{∞} = {(1, t, g(t)) | t ∈ GF (q)} ∪ {(0, 1, 0)},
O_{s} = {(1, t, f_{s}(t)) | t ∈ GF (q)} ∪ {(0, 1, 0)}, s ∈ GF (q), where

f_{s}(t) = \frac{f(x) + asg(x) + s^{1/2}x^{1/2}}{1 + as + s^{1/2}} ,

for some a where trace(a) = 1.

Thus the last theorem says that, for q even, a q–clan, C, gives rise to a herd of ovals of P G(2, q), which we shall denote by H(C), and conversely.

Remarks. 1. The only if part of the theorem is due to Payne [93], although not explicitly stated there. The proof given here is from [30], and, in particular, does not involve generalised quadrangles.

2. We have provided a proof of the existence of Payne’s [93] hyperovals that does not involve the use of generalised quadrangles, as desired by Cherowitzo [28].

3. This theorem is used by Penttila and Royle to classify 32-clans by computer [111]. Results are also obtained for q–clans, q even, q small.

4. The sufficiency half of the proof needs only the hypothesis that each f_{s} is a permutation.

10. Elation Generalised Quadrangles from q–clans

Let q be even and recall the definition of 4-gonal family in Section 7, p. 94, and the construction of a generalised quadrangle from a group G and a 4-gonal family ℱ in Section 8, p. 97. We are now ready to define f, g_{t} and d_{t}.

Let

G = {(a, c, b) | a, b ∈ GF (q)^{2}, c ∈ GF (q)}

with muliplication defined as

(a, c, b)(a′, c′, b′) = (a + a′, c + c′ + b ∘ a′, b + b′),

where

b ∘ a = \sqrt{bP a^{T}} with P = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) .

Let C = \{A_{t} | t ∈ GF (q)\} be a q–clan where A_{t} = (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) , a_{t}, b_{t}, c_{t} ∈ GF (q).

11. EGQs from q–clans  103

By taking g_{t}(a) = \sqrt{aA_{t}a^{T}} and d_{t}(a) = b_{t}a then g_{t}, d_{t} satisfy the conditions of Theorem 168, p. 97, yielding a 4-gonal family ℱ = {A(t) | t ∈ GF (q) ∪ {∞}} where

A(∞) = {(0, 0, b) ∈ G | b ∈ GF (q)^{2}},
A(t) = \{(a, \sqrt{aA_{t}a^{T}} , b_{t}a) | a ∈ GF (q)^{2}\}, t ∈ GF (q).

The centre of G is

Z = {(0, c, 0) | c ∈ GF (q)}.

For t ∈ GF (q) ∪ {∞} the tangent space at A(t) is

A^{∗}(t) = A(t)Z.

If q is even and C is a q–clan then the 4-gonal family ℱ yields an elation generalised quadrangle, GQ(C) = 𝒮 (G, ℱ), of order (q^{2}, q).

10.1. Ovoids from q–clans

We show that every GQ(C) for q even has ovoids.

We recall that for q even (a, c, b)^{−1} = (a, c + b ∘ a, b). Let H = (\begin{matrix}1 & 1 \\ 0 & d \end{matrix}) with

trace(d) = 1, hence H has no eigenvalues in GF (q). Let

𝒪 = {(a, c, aH) | a ∈ GF (q)^{2}, c ∈ GF (q)} ∪ {(∞)}.

Then 𝒪 is an ovoid of GQ(C).

Proof. Since |𝒪| = q^{3} + 1 we only need to show that any two points of 𝒪 are not collinear. That is, g = (a, c, aH)(a′, c′, a′H)^{−1} ∉ A(t) for all t ∈ GF (q) ∪ {∞} unless a = a′ and c = c′.

Now suppose, for t ∈ GF (q) ∪ {∞},

g = (a, c, aH)(a′, c′ + (a′H) ∘ a′, a′H)
= (a + a′, c + c′ + (a′H) ∘ a′ + (aH) ∘ a′, aH + a′H) ∈ A(t).

Then g ∈ A(∞) if and only if a = a′ and c = c′. For t ∈ GF (q) we have g ∈ A(t) if and only if

c + c′ + (a′H) ∘ a′ + (aH) ∘ a′ = \sqrt{(a + a′)A_{t}(a + a′)^{T}}

and

(a + a′)H = t^{1/2}(a + a′).

But since H has no eigenvalues in GF (q) this implies that a = a′ which implies that c = c′.

11. Subquadrangles of EGQs

We now show that there are q + 1 subquadrangles of GQ(C). With each of the q + 1 subquadrangles being associated with an oval.

Let q be even and let G the group of order q^{5} with multiplication defined as in the previous section and let ℱ be the 4-gonal family associated with G under this multiplication.

Let C = \{(\begin{matrix}x_{t} & y_{t} \\ 0 & z_{t} \end{matrix}) | t ∈ GF (q)\} be a q–clan. Let 𝒮 = 𝒮 (G, ℱ) = GQ(C).

For a ≠ 0 put G(a) = {(aa, c, ba) | a, b, c ∈ GF (q)} so

(aa, c, ba) · (a′a, c′, b′a) = ((a + a′)a, c + c′, (b + b′)a).

Notice G(a) is a elementary abelian group of order q^{3}. If we define

A_{a}(∞) = A(∞) ∩ G(a) A_{a}^{∗}(∞) = A^{∗}(∞) ∩ G(a)
A_{a}(t) = A(t) ∩ G(a) A_{a}^{∗}(t) = A^{∗}(t) ∩ G(a).

Then GQ(a) = ℱ (a) = {A_{a}(t) | t ∈ GF (q) ∪ {∞}} is a 4-gonal family for G(a) and we denote the associated GQ as GQ(a).

By [99] 𝒮 (a) = 𝒮 (G(a), ℱ (a)) can be thought of as a subquadrangle of order q of 𝒮. Also they showed that there are exactly q + 1 such subquadrangles since GQ(a) = GQ(b) if and only if ⟨a⟩ = ⟨b⟩. Most importantly it can be shown that each GQ(a) is isomorphic to the GQ T_{2}(O) for some oval O. We now determine what the oval is.

Since G(a) is elementary abelian it is isomorphic to GF (q)^{3} under componentwise addition. We define scalar multiplication as

d(aa, c, ba) = (daa, dc, dba) for a, b, c, d ∈ GF (q).

It is important to note that θ: (a, c, b) ↦ (da, dc, db) for 0 ≠ d ∈ GF (q) is an automorphism of G fixing each A(t) of ℱ. There is an isomorphism between G(a) and points of P G(2, q) by

(a, 0, 0) ↦ (1, 0, 0)
(0, 1, 0) ↦ (0, 1, 0)
(0, 0, a) ↦ (0, 0, 1).

We have ℱ (a) being a set of q + 1 points of P G(2, q) no three collinear. So

O_{a} = \{(1, \sqrt{aA_{t}a^{T}} , y_{t}) | t ∈ GF (q)\} ∪ \{(0, 0, 1)\}

is an oval of P G(2, q) with nucleus (0, 1, 0).

Hence GQ(a) is isomorphic to T_{2}(O_{a}) where O_{a} corresponds to the 4-gonal family ℱ (a).

Recall that we denote the group of all collineations of a GQ(C) fixing (∞) and (0, 0, 0) by G_{0}. Since each GQ(a) contains the points (∞) and (0, 0, 0) we can say that each element of G_{0} permutes G(a) and so O_{a} and GQ(a) among themselves. We note that no element of G can do this. Hence G_{0} is the full group of collineations of GQ(C) that acts on the set {G(a) | for q + 1 representatives for a}. (See [99] or [97] for details.)

12. A view of the Subiaco q–clan

We now define a q–clan, S, that will be a major part of this and later chapters. Let d ∈ GF (q), q even, such that d^{2} + d + 1 ≠ 0 and trace(1/d) = 1. Let

a = \frac{d^{2} + d^{5} + d^{1/2}}{d(1 + d + d^{2})} ,
f(x) = \frac{d^{2}(x^{4} + x) + d^{2}(1 + d + d^{2})(x^{3} + x^{2})}{(x^{2} + dx + 1)^{2}} + x^{1/2}, and
g(x) = \frac{d^{4}x^{4} + d^{3}(1 + d^{2} + d^{4})x^{3} + d^{3}(1 + d^{2})x}{(d^{2} + d^{5} + d^{1/2})(x^{2} + dx + 1)^{2}} + \frac{d^{1/2}}{d^{2} + d^{5} + d^{1/2}} x^{1/2}.

Then

S = S_{d} = \{A_{x} = (\begin{matrix}f(x) & x^{1/2} \\ 0 & ag(x) \end{matrix}) | x ∈ GF (q)\}

is a q–clan.

We call S the Subiaco q–clan. We prove S is a q–clan in Theorem 213, p. 154, in Chapter IV. The herd of the q–clan is related to the hyperovals H_{60} and H_{15} constructed by myself and Penttila.

Historically, the q–clan S first appeared in [30] and was investigated in the papers [6] [97] [100], mostly investigating the isomorphisms of GQ(S). In the following Sections we will give a summary of these results. The papers [98] [101] [27] are heavily influenced by the already mentioned papers.

13. Fundamental Theorem of q–clan Geometry

Let q be even. Let G = {(a, c, b) | a, b ∈ GF (q)^{2}, c ∈ GF (q)} with multiplication as defined in Section 10, p. 102. Recall that the 4-gonal family ℱ = {A(t) | t ∈ GF (q) ∪ {∞}} is defined from the q–clan

A(t) = \{(a, \sqrt{aA_{t}a^{T}} , b_{t}a) | a ∈ GF (q)^{2}\}
A(∞) = {(0, 0, b) | b ∈ GF (q)^{2}}.

We are now interested in determining the full collineation group G of GQ(C) constructed from a q–clan C. In particular, we are interested in the case of the Subiaco q–clan. Of extreme importance is The Fundamental Theorem, described below, which (usually) allows the full collineation group of GQ(C) to be determined by knowing how a collineation of GQ(C) maps the line [A(∞)] to another line on the point (∞). Hence to determine the full collineation group G of GQ(C) it suffices to determine the subgroup G_{0} fixing (0, 0, 0) (and G_{0} also fixes (∞)). Since the Subiaco GQ is not classical (as the ovals from the herd are not regular) we use:

Theorem 170 ([104]). Given a q–clan C then if 𝒮 (C) is not the GQ H(3, q^{2}) the point (∞) is fixed by the full collineation group of 𝒮 (C).

Right multiplication by elements of G induces a group of collineations of GQ(C) acting regularly on the points not collinear with (∞) and fixing each line through (∞).

Let C = \{A_{t} = (\begin{matrix}a_{t} & b_{t} \\ 0 & c_{t} \end{matrix}) | t ∈ GF (q)\} and C′ = \{A′_{t} = (\begin{matrix}x_{t} & y_{t} \\ 0 & z_{t} \end{matrix}) | t ∈ GF (q)\}

be q–clans with A_{0} = A′_{0} = 0. Now let GQ(C) and GQ(C′) be the corresponding GQs constructed from the q–clans from the same group G. Hence the points (a, c, b) and (∞) of GQ(C) and GQ(C′) are the same, but with the other points and lines labelled with A′(t) instead of A(t) and A′^{∗}(t) instead of A^{∗}(t). We now state the Fundamental Theorem which is used to determine the full collineation group of GQ(C), though we state it in the more general form.

Theorem 171 (Fundamental Theorem of q–clan Geometry [6, 96, 97, 100]). Let the map θ: GQ(C) → GQ(C′) be an isomorphism with θ: (∞) ↦ (∞), [A(∞)] ↦ [A′(∞)] and (0, 0, 0) ↦ (0, 0, 0). Then the following exist:

(i) D ∈ GL(2, q), (ii) λ ∈ GF (q), λ ≠ 0, (iii) σ ∈ Aut GF (q), and (iv) a permutation π: t ↦ t̄ of GF (q), for which A′ ; t̄ ≡ λD^{T} A_{t}^{σ}D + A′ ; ¯0 for all

t ∈ GF (q), and θ: GQ(C) → GQ(C′) is induced by an automorphism (also denoted by θ) of G of the following form:

θ = θ(σ, D, λ, π):
(a, c, b) ↦ (λ^{−1}a^{σ}D^{−T} , λ^{−1/2}c^{σ} + λ^{−1}\sqrt{a^{σ}D^{−T} A′ ; ¯0D^{−1}(a^{σ})^{T}} , b^{σ}P DP + λ^{−1}¯0^{1/2}a^{σ}D^{−T})

Conversely, given (i), (ii), (iii) and (iv) the automorphism of G θ = θ(λ, D, σ, π) induces an isomorphism from GQ(C) to GQ(C′) mapping (∞), [A(∞)] and (0, 0, 0) to (∞), [A′(∞)] and (0, 0, 0), respectively.

Note the Fundamental Theorem is related to the collineations determining when two flocks of a quadratic cone (Chapter IV, p. 141) are equivalent.

We define three maps on G and see how they map A_{t} to A′_{t}. We have A_{t} =

(\begin{matrix}f(t) & t^{1/2} \\ 0 & g(t) \end{matrix}).

Shift by s . For s ∈ GF (q) defined by

τ_{s}: (a, c, b) ↦ (a, c + \sqrt{aA_{s}a^{T}} , b + s^{1/2}a).

In the Fundamental Theorem we have t̄= t + s, λ = 1, D = I, A′ ; ¯0 = A_{s}, σ = id.

Scale by a . For a ≠ 0 ∈ GF (q).

σ_{a}: (a, c, b) ↦ (a, a^{1/4}c, a^{1/2}b).

Now σ_{a} leaves A(∞) and A(0) invariant and for t ∈ GF (q) we have A_{t} ↦ A′_{at} = a^{1/2}A_{t}. In the Fundamental Theorem it is λ = a^{−1/2}, D = a^{−1/2}I, σ = id, A′ ; ¯0 = 0 and t̄= at.

Flip .

φ: (a, c, b) ↦ (b, c + a ∘ b, a).

We have φ interchanging A(0) and A(∞) and for t ≠ 0 then A_{t} ↦ t^{−1}A_{t} = A′_{t^{−1}}. Note that the above maps, for t ∈ GF (q) ∪ {∞}, map

τ_{s}: t ↦ t + s
σ_{a}: t ↦ at
φ: t ↦ t^{−1}

(With infinity being treated conventionally.)

These maps provide all the possible Möbius transformations. Hence we can say P GL(2, q) is acting on P G(1, q) ≅ GF (q) ∪ {∞}. (Since P GL(2, q) acts 3transitively, and so, primitively, on P GL(1, q). So P GL(2, q)_{(∞)} = AGL(1, q) is a maximal subgroup of P GL(2, q). So

⟨τ_{s}, σ_{a} | s, a ∈ GF (q)⟩ = AGL(1, q).

Since φ ∉ AGL(1, q) we have

⟨τ_{s}, σ_{s}, φ | s, a ∈ GF (q)⟩ = P GL(2, q).)

Now we have:

Theorem 172 ([6], 2.3). Let θ: G → G be an automorphism of G obtained as a finite sequence of shifts, flips and scales. Moreover, suppose C and C′ are two normalised q–clans for which θ maps the 4-gonal family ℱ of C to the 4-gonal family ℱ′ of C′ in such a way that it effects the identity permutation on GF (q)∪{∞}. Then θ must have the form

θ: (a, c, b) ↦ (aa, ac, ab)

for some a ∈ GF (q) a ≠ 0 so ℱ = ℱ′.

14. Subiaco q–clans

In this section we will deal with the Subiaco q–clans which first appeared in [30] and were explicitly studied in Chapter IV. We will use the Subiaco q–clans as (re)defined by Payne.

14.1. Collineations of the Subiaco q–clan

We (re)define the Subiaco q–clan in the version that Payne constructed [6]. Let d ∈ GF (q) such that trace(d^{−1}) = 1. Define

f′(t) = d^{2}t^{4} + d^{3}t^{3} + (d^{2} + d^{4})t^{2}
g′(t) = (d^{2} + d^{4})t^{3} + d^{3}t^{2} + d^{2}t
k(t) = (v(t))^{2} = t^{4} + d^{2}t^{2} + 1

so we can define

f(t) = \frac{f′(t)}{k(t)} + d^{−1/2}t^{1/2}, and
g(t) = \frac{g′(t)}{k(t)} + d^{−1/2}t^{1/2}.

In Chapter IV we define the q–clans

S_{d} = {A_{t}^{(d)} | t ∈ GF (q)} for all q even
S′ = {A′_{t} | t ∈ GF (q)} for q = 4^{e} e odd
S′′ = {A′′_{t} | t ∈ GF (q)} for q = 2^{e} e odd

For B ∈ GL(2, q), C′ = {B^{T} A_{t}B | t ∈ GF (q)} is a q–clan whenever C = {A_{t} | t ∈ GF (q)} is (see [102]).

Now we show that the q–clans S_{d}, S′ and S′′ are equivalent to the q–clan

C_{d} = \{A_{t} = (\begin{matrix}f(t) & t^{1/2} \\ 0 & g(t) \end{matrix}) | t ∈ GF (q)\} ,

with f and g as defined above with trace(d^{−1}) = 1.

Now for all q even A_{t} is equivalent to

(\begin{matrix}1 & 1 \\ (\frac{d}{1+d+d^{2}} )1/2 & (\frac{d}{1+d+d^{2}} )^{1/2} + 1 \end{matrix}) A_{t}^{(d)} (\begin{matrix}1 & (\frac{d}{1+d+d^{2}} )1/2 \\ 1 & (\frac{d}{1+d+d^{2}} )^{1/2} + 1 \end{matrix}) .

14. Subiaco q–clans  109

For q = 4^{e}, e odd, let d = ω where ω ∈ GF (4) satisfying ω^{2} + ω + 1 = 0, then A_{t} is equivalent to

(\begin{matrix}ω^{2} & 0 \\ 0 & ω \end{matrix}) A′_{t} (\begin{matrix}ω^{2} & 0 \\ 0 & ω \end{matrix}) .

For q = 2^{e}, e odd, let d = 1 then A_{t} is equivalent to

(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) A′′_{t} (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) .

Let C be a Subiaco q–clan. In [6] it is shown that for θ(σ, D, λ, π) we can uniquely determine π from σ, D and λ. So we can now label θ by θ(σ, D, λ).

Recall that the subgroup G_{0} of the full collineation group G of GQ(C) is the subgroup that fixes (∞) and (0, 0, 0). Let H be the subgroup of G that fixes the line [A(∞)]. Let M be the subgroup of H that fixes [A(∞)] and [A(0)]. Then we have a theorem from [6].

Theorem 173 ([6], 6.4). Let C be a Subiaco q–clan and let GF (q_{0}) be the smallest subfield of GF (q) containing d. Let r = [GF (q) : GF (q_{0})] so r is odd. Then the subgroup M is given by

M = {θ(σ, aI, a^{−2}): (a, c, b) ↦ (aa^{σ}, ac^{σ}, ab^{σ}) |

0 ≠ a ∈ GF (q), σ ∈ Gal (\frac{GF (q)}{GF (q_{0})})\}.

So |M| = (q − 1)r.

We will now quote extensively from [100].

The results in Section 5 of [100] are used to show that:

Theorem 174 ([100], 5.1, 5.2). We have

|H| = 2e(q − 1) and |G_{0}| = 2e(q^{2} − 1).

We now investigate the action of the involutions (of lines on (∞)) where one fixes [A(0)] and one fixes [A(d)] where trace(1/d) = 1.

Let I_{0} = θ(1, B_{0}, id) with

B_{0} = (\begin{matrix}1 + d + d^{2} & d^{1/2} + d^{5/2} \\ d^{3/2} & 1 + d + d^{2} \end{matrix})

and t ↦ t/(dt + 1), be the involution fixing [A(0)].

Let I_{d} = θ(1, B_{d}, id) with

B_{d} = (\begin{matrix}1 + d^{4} + d^{6} + d^{7} & d^{1/2} + d^{9/2} + d^{13/2} \\ d^{15/2} & 1 + d^{4} + d^{6} + d^{7} \end{matrix})

and t ↦ (t(d^{2}+1)+d^{3})/(dt+d^{2}+1) be the involution fixing [A(d)]. Now if (I_{0}∘I_{d})^{j} = 1 then I_{0} ∘ I_{d} permutes the lines [A(t)] in a cycle of length q + 1 [100]. Importantly the action of I_{0} ∘ I_{d} on the q + 1 lines [A(t)] can be restricted to GF (q) ∪ {∞} ≅ P G(1, q) so the action is on points of P G(1, q). This action of I_{0} ∘I_{d} on P G(1, q) is determined by the matrix B_{0} ∘ B_{d}.

Now I_{0} ∘ I_{d} maps G(a) to G(aB_{0}B_{d}) and we have in general, (using the notation introduced in [100] and explored in further detail in [98]),

(B_{0} ∘ B_{d})^{j} = \frac{1}{d^{1/2}} (\begin{matrix}[10j + \frac{1}{2}] & [10j] \\ [10j] & [10j − \frac{1}{2}] \end{matrix})

where [x] = λ^{x} + λ^{−x} with λ ∈ GF (q^{2}) \ GF (q) such that λ^{q+1} = 1. Since λ^{q+1} = 1 is equivalent to λ = η^{j} for some j where η is a primitive element of GF (q^{2}), then λ ∉ GF (q) if and only if j ≡ 0 (mod q + 1). Hence λ^{q+1} = 1 is equivalent to j ≡ 0 (mod q − 1).

We have (B_{0} ∘ B_{d})^{j} leaving invariant some G(a) if and only if (λ^{10})^{j} ∈ GF (q). So if λ = η^{q−1}, (λ^{10})^{j} ∈ GF (q) if and only if 5j ≡ 0 (mod q + 1).

Theorem 175 ([100], 5.3). Let d = λ + λ^{−1} where λ = η^{q−1}, η a primitive element of GF (q^{2}). Then

(i) I_{0} ∘ I_{d} permutes the lines through (∞) in a cycle of length q + 1.

(ii) If q + 1 ≢ 0 (mod 5) then I_{0} ∘ I_{d} permutes the G(a) in a cycle of length q + 1. (iii) If q + 1 ≡ (mod 5) then I_{0} ∘ I_{d} permutes the G(a) in five cycles of length

(q + 1)/5.

Corollary 176 ([100], 5.4). If q = 2^{e} and q + 1 ≢ 0 (mod 5) there is a unique Subiaco oval. It has stabiliser of order 2e inherited from G_{0}.

Proof. Uniqueness follows from Theorem 4.4 of [100] and Theorem 175 (ii). Since |G_{0}| = 2e(q^{2} − 1), G_{0} is transitive on all q + 1 G(a)’s and the kernel induces the identity map on the plane of the oval O_{a}.

We now concentrate our energies on the case q+1 ≡ 0 (mod 5); that is for q = 4^{e}, e odd.

14.2. Subiaco q–clans, for q = 4^{e}, e odd

Let C be the Subiaco q–clan for q = 4^{e}, e odd. Then the 64–clan has H_{60} and H_{15} as elements of its herd. Since C is an infinite family q–clan we have generalised these hyperovals. But these hyperovals are inequivalent in P G(2, 64). The next question is are there two inequivalent ovals in H(C)? We (slowly) answer this question.

Let q = 2^{e} = 4^{r} r odd so e = 2r. Let d = ω and ω ∈ GF (q) such that ω^{2} + ω + 1 = 0. Recall that

g_{t}(a) = \sqrt{aA_{t}a^{T}} .

From [100] we define two involutions on the lines of (∞) by

\begin{aligned}I_{∞} & : (a, c, b) ↦ (aP, g_{ω}(a) + c, (b + ω^{2}a)P), \\ I_{1} & : (a, c, b) ↦ (bP, c + a ∘ b, aP), \end{aligned} (4)

and their compositions

\begin{aligned}I_{∞} ∘ I_{1} & : (a, c, b) ↦ (b + ω^{2}a, c + g_{ω}(a) + a ∘ b, a), \\ I_{∞} ∘ I_{1} & : (aa, c, ba) ↦ ((b + ω^{2}a)a, c + ag_{ω}(a), aa). \end{aligned}

Now I_{∞} ∘ I_{1} leaves invariant each G(a), a ∈ P G(1, q), and has order 5.

Lemma 177 ([100], 6.3). G_{0} has exactly q + 1 involutions each fixing a unique G(a).

In [100] we also show that the orbit containing G(a) has length at most 4(q+1)/5. Let Ω_{1} be the G_{0}-orbit containing G((1, 1)). Then by lemma 6.2 in [100] this orbit has length (q + 1)/5.

Note that I_{∞} fixes only G((1, 1)). For s ∈ GF (q) I_{s} fixes a unique G(a). Now (from [100]) I_{s} fixes G((0, 1)) if and only if s^{5} + s^{4} + ω^{2}s + 1 = 0 and I_{s} fixes

G((c(s), a(s) + 1)) if and only if c(s) ≠ 0,

where

a(s) = \frac{(s^{5} + 1)(1 + ω + ω^{2}) + (s^{4} + s)(1 + ω)}{(1 + ωs + s^{2})^{5/2}}, and
c(s) = \frac{(ω^{1/2} + ω^{5/2})s^{5} + ω^{3/2}s + ω^{1/2}s + ω^{3/2}}{(1 + ωs + s^{2})^{5/2}} .

Theorem 178 ([100], 6.4 also cf Theorem 7.6 of [6]). G_{0} has two orbits on the G(a), the short orbit Ω_{1} of length (q + 1)/5 and containing G((1, 1)) and a long orbit Ω_{2} of length 4(q + 1)/5.

Proof. Since G_{0} acts on Ω_{1} and |Ω_{1}| = (q + 1)/5 is odd, each of the q + 1 involutions I_{s} ∈ G_{0} must fix some G(a) ∈ Ω_{1}. Hence by Lemma 177 no I_{s} can fix any G(b) not in Ω_{1}. Suppose Ω_{2} is any G_{0}-orbit on the G(a)’s other than Ω_{1}. The stabiliser of any G(b) ∈ Ω_{2} must have odd order, implying that 4, the highest power of 2 dividing |G_{0}|, must divide |Ω_{2}|. By Theorems 4.4 of [100] and Theorem 175, p. 110, |Ω_{2}| = k(q + 1)/5, with 1 ≤ k ≤ 4. Since (q + 1)/5 is odd, it must be that k = 4.

14.3. Ovals for q = 4^{e}, e odd

For a given a the oval O_{a} obtained directly from G(a) is

O_{a} = {(1, g_{t}(a), t^{1/2}) ∈ P G(2, q) | t ∈ GF (q)} ∪ {(0, 0, 1)}.

The map (x, y, z) ↦ (x^{2}, z^{2}, y^{2}) on P G(2, q) replaces O_{a} with

D_{q+1}(h_{a}(t)) = D(h_{a}(t)) − {(0, 0, 1)} where h_{a}(t) = aA_{t}a^{T} .

(Using the o-polynomial notation of Chapter II, Section 4, p. 41.)

For a = (1, 1), h_{a}(t) = \frac{ω^{2}(t^{4} + t)}{t^{4} + ω^{2}t^{2} + 1} + t^{1/2},

in P G(2, 64) this o-polynomial is equivalent to the hyperoval H_{60}.

For a = (0, 1), h_{a}(t) = \frac{t^{3} + t^{2} + ω^{2}t}{t^{4} + ω^{2}t^{2} + 1} + ωt^{1/2},

in P G(2, 64) this o-polynomial is equivalent to the hyperoval H_{15}.

For a = (1, 1), O_{a} belongs to the short G_{0}-orbit Ω_{1} with length (q + 1)/5. For a = (0, 1), if 5 | (e/2), O_{a} belongs to the long G_{0}-orbit Ω_{2}. But if 5|(e/2), it follows from computations in [6] that O_{a} also belongs to the short orbit Ω_{1}.

So O_{(1,1)} and O_{(0,1)} are inequivalent if and only if 5 | (e/2). Below we give a more detailed picture of the induced groups for the ovals.

Theorem 179 ([100], 6.6). For e ≡ 2 (mod 4), d = ω and a = (1, 1), the complete group induced by G_{0} on P G(2, q) and stabilising

D_{q+1}(h_{(1,1)}(t)) = \{(1, t, \frac{ω^{2}(t^{4} + t)}{t^{4} + ω^{2}t^{2} + 1} + t^{1/2}) | t ∈ GF (q)\} ∪ \{(0, 1, 0)\} ,

has order 10e and is generated by the following three collineations:

(i) φ: (x, y, z) ↦ (y, x, z),

(ii) θ: (x, y, z) ↦ (x, y + ωx, z + ω^{2}x), and

(iii) ψ: (x, y, z) ↦ (x^{2}, x^{2} + ω^{2}y^{2}, x^{2} + ωz^{2}).

Proof. See pages 213–4 in [100]. We note that I_{∞} stabilises D_{q+1}(h_{a}(t)) for a = (1, 1) and acts on the oval as

I_{∞}: (x, y, z) ↦ (x, y + ωx, z + ω^{2}x).

Note. (i) φ leaves D_{q+1}(h_{a}(t))) invariant for a = (1, 1) but not for a = (0, 1). (ii) θ ∘ φ: (x, y, z) ↦ (y + ωx, x, z + ω^{2}x) has order 5.

Theorem 180 ([100], 6.7). For e ≡ 2 (mod 4), 5 | e̸, d = ω, and a = (0, 1), the complete group induced by G_{0} on P G(2, q) and stabilising

D_{q+1}(h_{(0,1)}(t)) = \{(1, t, \frac{t^{3} + t^{2} + ω^{2}t}{t^{4} + ω^{2}t^{2} + 1} + ωt^{1/2}) | t ∈ GF (q)\} ∪ \{(0, 1, 0)\} ,

has order 5e/2 and is generated by the following collineations:

(i) θ_{σ}: (x, y, z) ↦ (x^{σ}, y^{σ}, z^{σ}), σ ∈ Aut F with ω^{σ} = ω, (ii) θ ∘ φ: (x, y, z) ↦ (y + ωx, x, z + ωx).

14.4. Ovals for q = 4^{e}, e odd, 5|(e/2)

Now we want to prove that there are two inequivalent ovals in the herd of the Subiaco q–clan. We have shown this is true except for the case when 5|(e/2).

From the proof of Theorem 178, p. 111, we know that a (that is, G(a)) is in the long orbit Ω_{2} precisely if it is fixed by no involution I_{s}. So we may use Eq.(11) of [100] (determining the unique involution I_{s} fixing [A(s)]) to recognize members of Ω_{2}. Moreover, each oval D_{q+1}(h_{a}(t)) may be associated with an irreducible algebraic curve in P G(2, q) of degree 10 (see below).

There will be some manipulations made on the elements of the herd of the Subiaco q–clan (Payne version) so they correspond with the herd of the Subiaco q–clan in [30] (which corresponds to the rational function forms of the hyperovals H_{60} and H_{15} in P G(2, 64)).

We want to reindex the a ∈ P G(1, q) by c ∈ GF (q) ∪ {∞} in the following way. For c ∈ GF (q) ∪ \{∞\}, put a_{c} = (\frac{ω^{2}+ωc}{1+c^{1/2}+ωc}, \frac{ωc}{1+c^{1/2}+ωc}). Clearly 1 + c^{1/2} + ωc ≠ 0 for all c ∈ GF (q). Then a_{c} = (1, 1) if and only if c = ∞, and for c, d ∈ GF (q), a_{c} ≡ a_{d} if and only if c = d. For c = ω, a_{c} = (0, ω^{2}) ≡ (0, 1). For c ∈ GF (q) ∪ {∞} \ {∞, ω}, a_{c} ≡ (1, c/(c + ω)). Index the ovals D_{q+1}(h_{a}(t)) for a ∈ P G(1, q) so that O_{c} = D_{q+1}(h_{a_{c}}(t)), c ∈ GF (q) ∪ {∞}. Hence we have the following:

(5)  O_{∞} = {(1, t, f_{∞}(t)) | t ∈ F } ∪ {(0, 1, 0)},

where

f_{∞}(t) = \frac{ω^{2}t^{4}+ω^{2}t}{t^{4}+ω^{2}t^{2}+1} + t^{1/2}.

Then for c ∈ GF (q), using Payne’s version of the Subiaco q–clan, obtaining D_{q+1}(h_{a}(t)) from O_{a} and the Subiaco herd we obtain

O_{c} = {(1, t, f_{c}(t)) | t ∈ GF (q)} ∪ {(0, 1, 0)}

where

f_{c}(t) = (\frac{ω^{2}+ωc}{1+c^{1/2}+ωc})^{2} (\frac{ω^{2}t^{4}+t^{3}+t^{2}}{t^{4}+ω^{2}t^{2}+1} + ωt^{1/2})
+ (\frac{(ω^{2}+ωc)ωc}{1+c+ω^{2}c^{2}}) t^{1/2} + \frac{ω^{2}c^{2}}{1+c+ω^{2}c^{2}} (\frac{t^{3}+t^{2}+ω^{2}t}{t^{4}+ω^{2}t^{2}+1} + ωt^{1/2})
= \frac{1}{1+c+ω^{2}c^{2}} (\frac{(1+ωc^{2})t^{4}+ωt^{3}+ωt^{2}+ωc^{2}t}{t^{4}+ω^{2}t^{2}+1} + (ω^{2} + c + ω^{2}c^{2})t^{1/2}) .

Lemma 181 ([100], 6.8). Let g ∈ P GL(3, q) be defined by

g: (x, y, z) ↦ (y, x + ωy, ω^{2}y + z).

Then g is a collineation of order 5 stabilising O_{c} for every c ∈ GF (q) ∪ {∞}.

Proof. Using Eq.(4) we see that I_{1}∘I_{∞}: (a, c, b) ↦ (b, c+a∘b+g_{ω}(bP), a+ω^{2}b), and in particular, I_{1} ∘ I_{∞}: (aa, c, ba) ↦ (ba, c + g_{ω}(baP), (a + ω^{2}b)a). Hence I_{1} ∘ I_{∞} is a collineation of order 5 leaving G(a) invariant for each a ∈ P G(1, q). Use the semilinear map T: (x, y, z) ↦ (x^{2}, y^{2}, z^{2}) mapping O_{a} to D_{q+1}(h_{a}(t)) to transfer the action of I_{1} ∘ I_{∞} to the map

g_{a}: (x, y, z) ↦ (y, x + ωy, z + ya(P A_{ω}P^{T})a^{T})

which stabilises O_{a}. Note that P A_{ω}P^{T} ≡ A_{ω} = ( \begin{matrix}ω & ω^{2} \\ 0 & ω \end{matrix} ). For a = (1, 1),

aA_{ω}a^{T} = ω^{2}. Now put a = a_{c} and compute

aA_{ω}a^{T} = (\frac{ω+ω^{2}c^{2}}{1+c+ω^{2}c^{2}} ) ω + (\frac{(ω^{2}+ωc)ωc}{1+c+ω^{2}c^{2}}) ω^{2} + (\frac{ω^{2}c^{2}}{1+c+ω^{2}c^{2}} ) ω = ω^{2}.

Hence for every c ∈ GF (q) ∪ {∞}, g_{a} is the map g of the Lemma.

Lemma 182 ([100], 6.9). (i) O_{∞} is always in the short orbit Ω_{1}.

(ii) For c ∈ GF (q), O_{c} ∈ Ω_{2} if and only if

p_{c}(x) = x^{5} + (ω^{2}c^{2} + ω^{2})x^{4} + (ω^{2}c^{2} + 1)x + 1 = 0,

has no root in GF (q). (iii) There is a c ∈ GF (q) \ GF (4) for which both p_{c}(x) = 0 and

p_{(ω^{2}c^{2}+ω^{2})}(x) = x^{5} + (c^{4} + ω)x^{4} + c^{4}x + 1 = 0,

have no root in GF (q).

Define the conic C: xy = z^{2}, that is,

C = {(1, t, t^{1/2}) | t ∈ GF (q)} ∪ {(0, 1, 0)}.

Lemma 183 ([100], 6.10). (i) O_{∞} ∩ C = P G(2, 4) ∩ C,

(ii) For c satisfying condition (iii) of Lemma 182, we have

O_{c} ∈ Ω_{2} and O_{c} ∩ C = P G(2, 4) ∩ C.

Proof. For (i), (O_{∞} \ {(0, 1, 0)}) ∩ C = {(1, t, f_{∞}(t)) | f_{∞}(t) = t^{1/2}}. Clearly f_{∞}(t) = t^{1/2} if and only if t ∈ GF (4). Similarly, if c satisfies condition (iii) of Lemma 182, then O_{c} ∈ Ω_{2} and (O_{c} \ {(0, 1, 0)}) ∩ C = {(1, t, f_{c}(t)) | f_{c}(t) = t^{1/2}}. Using Eq. (5) we may compute that f_{c}(t) = t^{1/2} if and only if ω^{2}t(t^{3} + 1)(t^{5} + (c^{4} + ω)t^{4} + c^{4}t + 1) = 0, proving (ii).

For each c ∈ GF (q) ∪ {∞} we may define the hyperoval H_{c} = O_{c} ∪{(0, 0, 1)}, and with a natural abuse of language refer to H_{c} belonging to Ω_{1} or Ω_{2}. For example, H_{∞} ∈ Ω_{1}. And for c = c_{0} satisfying condition (iii) of Lemma 182, H_{c_{0}} ∈ Ω_{2}. From now on fix c = c_{0} to be such an element.

From Section 8.3, p. 57, the hyperovals H_{∞} and H_{c_{0}} are inequivalent for q = 64 and have full collineation stabilisers of order 60 and 15, respectively. So from now on we assume that q ≥ 1024.

Using the ideas of [87] we shall show that H_{∞} and H_{c_{0}} are the pointsets of two absolutely irreducible degree ten algebraic curves in P G(2, q). We remind the reader that in this section groups act on the right.

We refer the reader to Sections 2.6 and 10.1 in [55] for the algebraic geometric definitions used here.

Lemma 184 ([100], 6.11). For both c = ∞ and c = c_{0}, H_{c} is the pointset of an absolutely irreducible degree ten algebraic curve 𝒞 in P G(2, q) having (0, 0, 1) as its only singular point. The multiplicity of (0, 0, 1) for 𝒞 is 8, and in P G(2, q^{2}) there are two tangent lines for 𝒞 at (0, 0, 1). The tangent lines are Q = [η, 1, 0]^{T} : ηx + y = 0 and Q̄ = [η^{q}, 1, 0]^{T} : η^{q}x + y = 0, where η ∈ GF (q^{2}) \ GF (q) satisfies η^{q+1} = 1 and η^{q} + η = ω.

Proof. For c ∈ {∞, c_{0}}, write d = c/(c + ω), so d ∈ {1, c_{0}/(c_{0} + ω)}. In particular, we are avoiding the case c = ω with d = ∞. Put m_{c} = (\frac{ω^{2}+ωc}{1+c^{1/2}+ωc})^{2}. So m_{c} ≠ 0, m_{∞} = 1, and a_{c} = (m_{c})^{1/2}(1, d). Then H_{c} = {(1, t, f_{c}(t)) | t ∈ GF (q)} ∪ {(0, 1, 0), (0, 0, 1)} is projectively equivalent to H̄_{c} = {(1, t, f_{c}(t)/m_{c}) | GF (q) ∈ F } ∪ {(0, 1, 0), (0, 0, 1)}. Here

\frac{f_{c}(t)}{m_{c}} = f(t, d) = (\frac{ω^{2}t^{4}+t^{3}+t^{2}}{t^{4}+ω^{2}t^{2}+1} + ωt^{1/2}) + dt^{1/2} + d^{2} (\frac{t^{3}+t^{2}+ω^{2}t}{t^{4}+ω^{2}t^{2}+1} + ωt^{1/2})
= \frac{ω^{2}t^{4}+(1+d^{2})(t^{3}+t^{2})+ω^{2}d^{2}t}{t^{4}+ω^{2}t^{2}+1} + (ω + d + ωd^{2})t^{1/2}.

Define g_{c} ∈ P GL(3, q^{2}) by g_{c}: (x, y, z) ↦ (x, y, m_{c}z). So g_{c}: H̄_{c} ↦ H_{c}, and g_{c} fixes the two points (0, 1, 0) and (0, 0, 1). We show that H̄_{c} satisfies the conclusions of the Lemma. Since g_{c} also fixes the lines Q, Q̄, it will follow that H_{c} satisfies the same conclusions.

The point (1, t, f(t, d)) ≡ (1, y, z) satisfies

z = \frac{ω^{2}y^{4}+(1+d^{2})(y^{3}+y^{2})+d^{2}ω^{2}y}{y^{4}+ω^{2}y^{2}+1} + (ω + d + ωd^{2})y^{1/2}.

Multiply through by y^{4} + ω^{2}y^{2} + 1, square both sides, and then make the resulting equation homogeneous in x, y, z to get the algebraic curve 𝒞_{d} : h_{d}(x, y, z) = 0, where

h_{d}(x, y, z) = (z^{2} + (ω^{2} + d^{2} + ω^{2}d^{4})xy)(y^{2} + ωxy + x^{2})^{4}
+ ωx^{2}y^{8} + ωd^{4}x^{8}y^{2} + (1 + d^{4})(x^{4}y^{6} + x^{6}y^{4}).

By construction, for all t ∈ GF (q), (1, t, f(t, d)) ∈ 𝒞_{d}. Now h_{d}(0, y, z) = z^{2}y^{8} = 0 if and only if y = 0 or z = 0. Hence the pointset of 𝒞_{d} is exactly the hyperoval H̄_{c} (always d = c/(c + ω)). So |𝒞_{d} | = q + 2, and we now want to prove that h_{d}(x, y, z) is absolutely irreducible.

First compute \frac{∂h_{d}}{∂x} = (ω^{2} + d^{2} + ω^{2}d^{4})y(y^{2} + ωxy + x^{2})^{4}, \frac{∂h_{d}}{∂y} = (ω^{2} + d^{2} + ω^{2}d^{4})x(y^{2} + ωxy + x^{2})^{4}, and \frac{∂h_{d}}{∂z} = 0. Since ω^{2} + x + ω^{2}x^{2} is irreducible over GF (4), and hence over GF (q), ω^{2} + d^{2} + ω^{2}d^{4} ≠ 0 for all d ∈ GF (q). We have (0, 0, 1) as a singular point. The point (0, 0, 1) is incident with the lines l = [a, b, 0] for a, b ∈ GF (q) (recall the definition of multiplicity in [55], p. 52).

𝒞_{d} ∩ l = \{(x, y, z) | (a b 0) (\begin{matrix}x \\ y \\ z \end{matrix}) = 0 and h_{d}(x, y, z) = 0\}
= {(x, y, z) | ax + by = 0 and h_{d}(x, y, z) = 0}

now assume a ≠ 0, so a = 1,

= {(x, y, z) | x + by = 0 and h_{d}(x, y, z) = 0}
= {(by, y, z) | h_{d}(by, y, z) = 0}.

Let h′_{d} ∈ GF (q)[x, y] with h′_{d} homogeneous with h′_{d}(y, z) = h_{d}(by, y, z). Then

h′_{d}(y, z) = (z^{2} + (ω^{2} + d^{2} + ω^{2}d^{4})by^{2})(y^{2} + bωy^{2} + b^{2}y^{2})^{4} + ωb^{2}y^{10} + ωd^{4}b^{8}y^{10} + (1 + d^{4})(b^{4}y^{10} + b^{6}y^{10}).

Now (0, 1) is a zero of h′_{d} of multiplicity m if and only if 0 is a zero of h′_{d}(y, 1) of multiplicity m if and only if y^{m} divides h′_{d}(y, 1) but y^{m+1} does not divide h′_{d}(y, 1). So

h′_{d}(y, 1) = (1 + d^{4} + b^{8})y^{8} + (ω^{2}(1 + b^{2} + b^{8}) + (b^{4} + b^{6})(1 + d^{4}) + d^{2}b^{8}(1 + d^{2}) + d^{2} + d^{6}(1 + ωd^{2}))y^{10}.

The line [0, 1, 0] gives y = 0 and h_{d}(x, 0, z) = 0 if and only if z^{2}x^{8} = 0 if and only if h′_{d}(x, z) = 0 if and only if (0, 1) has multiplicity 8. The line [1, 0, 0] gives x = 0 and h_{d}(0, y, z) = 0 if and only if z^{2}y^{8} = 0 if and only if h′_{d}(y, z) = 0 if and only if (0, 1) has multiplicity 8. Hence (0, 0, 1) has multiplicity 8.

Let 𝔽 be the algebraic closure of GF (q). The lines [a, 1, 0], a ∈ 𝔽 and [1, 0, 0] are the lines on the point (0, 0, 1). Firstly, we notice that for the line l = [1, 0, 0], 𝒞_{d} ∩ l implies that h_{d}(x, y, z) = 0 and x = 0, so x = 0 and z^{2}y^{8} = 0. So the point (0, 0, 1) still has multiplicity 8, so [1, 0, 0] is not a tangent line to (0, 0, 1).

Let l = [a, 1, 0] for a ∈ 𝔽. Now h_{d}(x, y, z) = 0 and ax + y = 0 means ax = y and

0 = x^{10}(ωa^{8} + ωd^{4}a^{2} + (1 + d^{4})(a^{6} + a^{4}))
+ (z^{2} + (ω^{2} + d^{2} + ω^{2}d^{4})ax^{2})x^{8}(a^{2} + ωa + 1)^{4}

We have [a, 1, 0] being a tangent line to (0, 0, 1) if (a^{2} + ωa + 1)^{4} = 0 (with (0, 0, 1) having multiplicity 10 on 𝒞_{d} ∩l). So a^{2} +ωa+1 = 0 if and only if (x+a_{1})(x+a_{2}) = 0 for some a_{1}, a_{2} ∈ 𝔽 with a_{1}a_{2} = 1 and a_{1} + a_{2} = ω. Now a_{1} and a_{2} are conjugate in GF (q^{2}) so we let

x^{2} + ωxy + y^{2} = (x + ηy)(x + η^{q}y),

where η ∈ GF (q) \ GF (q^{2}), η^{q+1} = 1 and η + η^{q} = ω. So the tangent lines to (0, 0, 1) are

Q: [η, 1, 0] and Q̄: [η^{q}, 1, 0].

If one of Q, Q̄ is a component of 𝒞_{d}, so is the other, then y^{2} + ωxy + x^{2} must divide h_{d}(x, y, z). We now show that this is not the case. Put y^{2} = ωxy + x^{2}, and hence y^{4} = x^{3}(y + ωx), y^{6} = x^{5}y, y^{8} = ωx^{8} + ωyx^{7} into h_{d}(x, y, z). After a little simplification, h_{d}(x, y, z) = x^{9}(ω^{2}(1 + d^{4})y + x), which is not the zero polynomial. This proves that neither tangent to 𝒞_{d} at (0, 0, 1) is a component of 𝒞_{d}.

Now suppose that 𝒞_{d} has components C_{1}, …, C_{l}, l > 1, with deg C_{i} = k_{i}, and suppose (0, 0, 1) has multiplicity t_{i} for C_{i}. So ∑ k_{i} = 10 and ∑ t_{i} = 8. If some t_{i} = k_{i}, then t_{i} = k_{i} = 1 and C_{i} is a line. But in this case C_{i} must be tangent to C_{i} at (0, 0, 1), so C_{i} is tangent to 𝒞_{d} at (0, 0, 1), contradicting the previous paragraph. So the only possibility is that there are exactly two components C_{1} and C_{2} with t_{1} = k_{1} − 1 and t_{2} = k_{2} − 1.

First suppose C_{1} and C_{2} are defined over GF (q), and (0, 0, 1) is a (k_{i} − 1)-tuple point for C_{i}, i = 1, 2. Any tangent to C_{i} at (0, 0, 1) is tangent to 𝒞_{d} and not defined over GF (q). So any line (over GF (q)) through (0, 0, 1) meets C_{i} in a second point, which implies |C_{i}| ≥ q + 2. Since any point of C_{1} ∩ C_{2} is a singular point for 𝒞_{d} clearly |C_{1} ∪ C_{2}| ≥ 2(q + 2) − 1 > q + 2 = |𝒞_{d} |, an impossibility.

So we may suppose C_{1} and C_{2} are not defined over GF (q), but over some GF (q^{r}), r > 1. Let id ≠ σ ∈ Gal(GF (q^{r})/GF (q)). Then C_{1}^{σ} = C_{2}, so deg C_{1} = deg C_{2} = 5. By Lemma 10.1.1 of [55], |C_{i}| ≤ 5^{2}, so |𝒞_{d} | ≤ 49. Since q > 64 we have a contradiction.

For the remainder of this section write H_{1} = H_{∞} and H_{2} = H_{c_{0}}. Then for i = 1, 2, let G_{i} = P Γ L(3, q)_{H_{i}} be the full collineation stabiliser of the hyperoval H_{i}. Note that H_{1} and H_{2} are equivalent to the hyperovals similarly named in Chapter II, Section 8.3, p. 57.

Let g ∈ P Γ L(3, q) have companion automorphism σ ∈ Aut GF (q). Let h ∈ GF (q)[x, y, z] be a homogeneous polynomial. The image, h^{g}, of h under g is defined by:

h^{g} = g ∘ h ∘ σ^{−1},

and for d = 1 ∈ GF (q), write h_{1} = h_{d}(x, y, z) (since g_{∞} = id), and for d = c_{0}/(c_{0}+ω), let h_{2} be the image of h_{d}(x, y, z) under g_{c_{0}}^{−1}. Then we have H_{i} = {(x, y, z) ∈ P G(2, q) | h_{i}(x, y, z) = 0}, i = 1, 2.

Lemma 185 ([100], 6.12).

G_{i} = {g ∈ P Γ L(3, q) | h_{i}^{g} = λh_{i}, λ ∈ GF (q) \ {0}}, for i = 1, 2.

Proof. (⊇) Let g ∈ P Γ L(3, q) with h_{i}^{g} = λh_{i}, λ ∈ GF (q) \ {0} for i = 1, 2. Then g fixes the set of points (x, y, z) such that h_{i}(x, y, z) = 0. But these are just the points of the hyperoval H_{i} for i = 1, 2.

(⊆) Let g ∈ G_{i} and suppose h_{i}^{g} ≠ h_{i}. If 𝒞_{i} : h_{i}(x, y, z) = 0, i = 1, 2, then the pointsets of 𝒞_{i} and 𝒞_{i} ^{g} coincide. By Bezout’s Theorem (see [55] 10.1 VII), it follows that the number of points of 𝒞_{i} ∩ 𝒞_{i} ^{g}, counted according to multiplicity, is at most 100. This is contrary to our hypothesis that q ≥ 1024.

Since (0, 0, 1) is the unique singular point of 𝒞_{i}, i = 1, 2, G_{i} must fix (0, 0, 1) and the set {Q, Q̄} of tangents. Also, since (0, 0, 1) is on H_{i}, no nontrivial element of G_{i} can have (0, 0, 1) as its centre, i = 1, 2. Hence G_{1} and G_{2} fix (0, 0, 1) but not linewise, and so act faithfully on the quotient space P G(2, q)/(0, 0, 1).

We are now ready to prove the final theorem of this section.

Theorem 186. The hyperovals H_{1} and H_{2} in P G(2, q), q = 2^{e}, are inequivalent for e ≡ 2 (mod 4). The full collineation stabiliser of H_{1} in P Γ L(3, q) is isomorphic to C_{5} ⋊ C_{2e}, and the full collineation stabiliser of H_{2} in P Γ L(3, q) is isomorphic to C_{5} ⋊ C_{e/2}.

Proof. Let h ∈ P GL(2, q) with |h| = q+1. Then ⟨h⟩ is a Singer cycle of P G(1, q), so ⟨h⟩ = C_{q+1} acts regularly on P G(1, q). Since h is induced by an invertible 2 × 2 matrix over F with (conjugate) eigenvalues in GF (q^{2}) \ GF (q), we have ⟨h⟩ fixing a conjugate pair of points, say {P, P̄}. As N_{P Γ L(2,q)}(⟨h⟩) permutes the fixed points of ⟨h⟩, N_{P Γ L(2,q)}(⟨h⟩) stabilises the conjugate pair {P, P̄}. Hence N_{P Γ L(2,q)}(⟨h⟩) ≤ P Γ L(2, q)_{{P,P̄}}. And since P Γ L(2, q) acts transitively on conjugate pairs of points, then replacing h by a conjugate if necessary,

N_{P Γ L(2,q)}(⟨h⟩) ≤ P Γ L(2, q)_{{Q,Q̄}},

where {Q, Q̄} is the pair of conjugate tangent lines (which are conjugate points in the quotient space) to the hyperovals at (0, 0, 1) in P G(2, q). By Theorem 20, p. 7, we have

N_{P Γ L(2,q)}(⟨h⟩) = C_{q+1} ⋊ C_{2e}.

But the order of P Γ L(2, q)_{{Q,Q̄}} is 2(q + 1)e, so by comparing orders

C_{q+1} ⋊ C_{2e} = P Γ L(2, q)_{{Q,Q̄}}.

Hence both G_{1} and G_{2} are subgroups of C_{q+1} ⋊ C_{2e}. From this, for the g from Lemma 181, p. 113, we have that ⟨g⟩ is normal in P GL(2, q)_{{Q,Q̄}}, since ⟨g⟩ is characteristic in ⟨h⟩, which is normal in P GL(2, q)_{{Q,Q̄}}. Since q^{2} + q + 1 ≡ 1 (mod 5), g has a unique fixed line, namely x + y + ω^{2}z = 0. Since G_{1} and G_{2} normalise ⟨g⟩, they fix x + y + ω^{2}z = 0.

15. Tensor q–clan geometries  119

Let H_{i} = G_{i} ∩ C_{q+1}, for i = 1, 2. We have H_{1} and H_{2} fixing both tangent lines Q and Q̄, so fixing their product x^{2} + ωxy + y^{2} = 0. They also fix the line x + y + ω^{2}z = 0, so fix x^{2} + y^{2} + ωz^{2} = 0. Let P be the pencil generated by these two. The group induced on the set of conics of P by the stabiliser in P GL(3, q) of P is cyclic of order q − 1. Since (q + 1, q − 1) = 1 and H_{i} fixes the tangent lines then each H_{i} fixes every conic in P. So for i = 1, 2, H_{i} fixes the conic C: xy = z^{2}, so it fixes H_{i} ∩ C = C ∩ P G(2, 4), by Lemma 183, p. 114.

Now H_{i} acts semiregularly on P G(2, q)\{(0, 0, 1)} for i = 1, 2. All points (except the fixed point) have orbits of length |H_{i}|. But since H_{i} fixes C ∩ P G(2, 4) and acts semiregularly on C, we have |H_{i}| dividing |C ∩ P G(2, 4)| = 5. Hence |H_{i}| = 1 or 5. But ⟨g⟩ ≤ H_{i}, so H_{i} = ⟨g⟩ for i = 1, 2. Hence G_{1} and G_{2} are subgroups of ⟨g⟩ ⋊ C_{2e}.

From Theorem 179, p. 112 we have the induced group of H_{1} having order 10e so we have G_{1} = ⟨g⟩ ⋊ C_{2e}.

As q + 1 ≡ 0 (mod 5) with q = 2^{e}, then e = 2r where r is odd, so e/2 is odd. We now show that G_{2} ≤ ⟨g⟩ ⋊ C_{e/2}. The subgroup C_{5} ⋊ C_{4} contains all the involutions of ⟨g⟩ ⋊ C_{2e} (five in total) since e/2 is odd. These five involutions are conjugate under ⟨g⟩. If G_{2} contains an involution then it will contain the involution φ from Theorem 179, p. 112. Since

φ: {(1, t, f_{2}(t)) | t ∈ F } ↦ {(t, 1, f_{2}(t)) | t ∈ F } = {(1, u, uf_{2}(u^{−1})) | u ∈ F },

it follows that if G_{2} contains an involution, then f_{2}(t) = tf_{2}(t^{−1}) for all t ∈ F, but this is not so. Hence 2 does not divide |G_{2}|. That is, G_{2} ≤ ⟨g⟩ ⋊ C_{e/2}.

From the proof of Theorem 180, p. 112 we have the induced group of H_{2} has order 5e/2. Hence G_{2} = ⟨g⟩ ⋊ C_{e/2}.

This completes the proof of the theorem.

14.5. Remarks on the Subiaco hyperovals

A discussion of the Subiaco hyperovals and related structures can be found in Section 10, p. 158, Chapter IV.

15. Tensor q–clan geometries

While S. E. Payne visited The University of Western Australia in 1993 he was stimulated by a seemingly simple question. “Why is the c in the EGQ group G in the centre?” Payne’s answer to resolving this question was tensor q–clan geometry.

We will only talk briefly on this subject since much of the theory and applications are still in progress. The main references are [98] and [101].

Let q = 2^{e}. We will attempt to describe the automorphisms of the Fundamental Theorem in terms of tensor products of pairs of matrices in GL(2, q).

Let P = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) and define b ∘ a = bP a^{T} for a, b ∈ GF (q)^{2}. (Note this is

different from the definition used previously for b ∘ a.) Let

G = {(a, c, b) | a, b ∈ GF (q)^{2}, c ∈ GF (q)}

with binary operation

(a, c, b)(a′, c′, b′) = (a + a′, c + c′ + \sqrt{b ∘ a′}, b + b′).

Let

K = {((a, b), c) | a, b ∈ GF (q)^{2}, c ∈ GF (q)}

with binary operation

((a, b), c)((a′, b′), c′) = ((a + a′, b + b′), c + c′ + b ∘ a′).

The group K has centre Z = {((0, 0), c) | c ∈ GF (q)}. Note there is an isomorphism ψ from K to G defined by ((a, b), c) ↦ (a, c^{1/2}, b).

For each 0 ≠ a ∈ GF (q)^{2} we define

L_{a} = {(a ⊗ b, c) | b ∈ GF (q)^{2}, c ∈ GF (q)},

and for each 0 ≠ b ∈ GF (q)^{2} we define

R_{b} = {(a ⊗ b, c) | a ∈ GF (q)^{2}, c ∈ GF (q)}.

We have L_{a} and R_{b} being elementary abelian subgroups of K of order q^{3}. For each nonzero a, b ∈ GF (q)^{2}, L_{a} = L_{b} if and only if ⟨a⟩ = ⟨b⟩. So we can think of the L_{a} as indexed by the points of P G(1, q). The scalar multiplication

d(a ⊗ b, c) = (da ⊗ b, d^{2}c) = (a ⊗ db, d^{2}c)

makes each L_{a} into a 3-dimensional vector space over GF (q). So there is an associated projective plane L̄_{a} ≅ P G(2, q). Similarly, for R_{a}.

A vector v ∈ GF (q)^{4} is called pure, (or decomposable) if v = u ⊗ w for some vectors u, w ∈ GF (q)^{2}. (Generally, v = ∑ u ⊗ w. Indeed if V = U ⊗ W and B is a basis for U and C is a basis for W then {u ⊗ w | u ∈ B, w ∈ C} is a basis for V.) We have

(\begin{matrix}a \\ b \end{matrix}) ⊗ (c d) = (\begin{matrix}ac & ad \\ bc & bd \end{matrix}) ≅ (a, b) ⊗ (c, d).

The matrix (a, b) ⊗ (c, d) has rank one and so determinant zero and so is singular. We denote the set of all 2 × 2 matrices over GF (q) by M_{2}(q) and note that it is

isomorphic to GF (q)^{4}. Let A = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(q). Then A ∈ M_{2}(q) is pure if and

only if det(A) = 0 if and only if ad − bc = 0. Hence we can define a quadratic form, Q, from GF (q)^{4} to GF (q) by

Q(a, b, c, d) = ad − bc.

Hence all pure vectors of GF (q)^{4} are zeros of this form. (The associated bilinear form of Q is nondegenerate).

Define a map ϕ: K → K/Z where we identify K/Z with P G(3, q). Then ϕ(L_{a}) = L where L a totally singular line and ϕ(R_{a}) = R with R a totally singular line. The totally singular lines from L_{a} form a regulus of P G(3, q) with opposite regulus from the totally singular lines of R_{a}.

Now O^{+}(4, q) space has as its singular vectors the pure tensors in GF (q)^{4} = GF (q)^{2} ⊗ GF (q)^{2}. So K/Z and O^{+}(4, q) are isomorphic as spaces for q even. Let A, B ∈ GL(2, q). We have

A (\begin{matrix}a \\ b \end{matrix}) ⊗ (c d) B = A ⊗ B (\begin{matrix}ac \\ ad \\ bc \\ bd \end{matrix}) .

Hence A⊗B acts on GF (q)^{4} preserving the form Q. So GL(2, q)×GL(2, q) ≤ O^{+}(4, q).

In [98] Payne shows that by the Fundamental Theorem each collineation θ of GQ(C) of a q–clan C that fixes the two points (0, 0, 0) and (∞) is induced by an automorphism of the elation group of K and such a collineation is represented by θ(σ, H, S) where σ ∈ Aut GF (q) and H and S are 4 × 4 matrices over GF (q) with H being invertible. We will not go into the specifics of how θ acts on the 4-gonal family etc., instead we will say that in most circumstances we can define the map θ as θ(A⊗B) where H = A⊗B with A, B ∈ GL(2, q) and S is determined by A and B but in a slightly more complicated fashion. The matrix A acts on the totally singular lines L of O^{+}(4, q) and the B matrices act on the totally singular lines R, and so on the regulus R and the opposite regulus of R respectively.

Below is a result extensively used by Penttila and Royle.

Theorem 187 (T. Penttila, private communication). If O_{1} and O_{2} are equivalent ovals on {(0, 1, 0), (1, 0, 0), (1, 1, 1)} with nucleus (0, 0, 1) and O_{1} is in a herd then O_{2} is in a herd.

The above Theorem was used extensively by Penttila and Royle in [109]. The classification of flocks for small q via the trace equation T_{a}(f, g) (page 100) requires the choice of two o-polynomials f and g. The choice is narrowed down by the Theorem since we can pick our favourite o-polynomial for each inequivalent hyperoval. This allows a complete search for flocks for q small where all hyperovals have been classified.

15.1. Cyclic q–clans

After Payne invented the tensor q–clan approach Payne, Penttila and Royle noted that the map θ(M ⊗ M^{−5}) was a collineation of the Subiaco GQ of order q + 1. With this and other results it was decided that the collineations of GQ(C) of the form θ(M ⊗ M^{−p}) where M ∈ GL(2, q) and p is some rational number should be investigated.

The assumption is: There is a normalised q–clan C = {A_{t} | t ∈ GF (q)} with

A_{η} = (\begin{matrix}x_{η} & η^{1/2} \\ 0 & z_{η} \end{matrix}) given and an integer p modulo q + 1 for which θ(M ⊗ M^{−p}) is an

automorphism of K leaving ℱ invariant.

Basic goal is: Starting with A_{0} = (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}), A_{η} and θ as above we would like to

construct the entire q–clan C.

This can be done [101] by running a computer search where by

(i) fix η,

(ii) loop over values of p,

(iii) loop over values of x_{η},

(iv) loop over values of z_{η}.

By using the map θ the rest of the q–clan can be constructed given the right values of η, p etc. Royle found there is a large number of (x_{η}, z_{η}) pairs. Payne constructed an involution that chooses a ‘canonical’ representative pair.

The results are summarised below:

(i) For q = 64 we have p = 5 which always give the Subiaco q–clan. Also we have p = 21 giving a q–clan related to the hyperoval H_{12} of Penttila and Royle (see page 67).

(ii) For q = 256 we have p = 5 the Subiaco q–clan, and we have p = 85 giving a new q–clan related to the sporadic hyperoval in P G(2, 256) (see Section 9.1, Chapter II).

(iii) For q = 1024 we have the Subiaco q–clan and a new q–clan for p = 341.

The new q–clans seem to be related as the p values are equal to p = −2/3 modulo q + 1 for q = 64, 256, 1024 and new q–clans for q = 4096 and q = 16384.

16. Subquadrangles in GQs of order (s, s^{2})

Recall the definitions of subquadrangle in Section 6, p. 91, and the construction of subGQs from EGQs in Section 11, p. 104. An example of a subquadrangle is given on page 93. As another example consider the GQ T_{3}(Ω) where Ω is an ovoid of P G(3, q). Let π be a plane of P G(3, q) such that π ∩ Ω = O an oval of P G(2, q). Let Σ be a hyperplane of P G(4, q) that intersects P G(3, q) in π. Then we can construct T_{2}(O) in Σ as a subGQ of T_{3}(Ω).

The work below will use the results of rosettes of ovoids of a subGQ, induced by a GQ.

16. Subquadrangles in GQs of order (s, s^{2})  123

Theorem 188 ([130] 7.1). Let 𝒮 = (𝒫, ℬ, I) be a GQ of order (q, q^{2}), q even, having a subquadrangle 𝒮′ isomorphic to Q(4, q). If in 𝒮′ each ovoid 𝒪_{x} consisting of all points collinear with a given point x of 𝒮 \ 𝒮′ is an elliptic quadric, then 𝒮 is isomorphic to Q(5, q).

Note we are using the fact that Q(4, q) ≅ W (q) for q even and each ovoid of W (q) is an ovoid of P G(3, q) (Theorems 151, 152 and 153).

By the work of O’Keefe and Penttila [79] (see Section 11.3, p. 71, Chapter II) we have:

Corollary 189. If a GQ 𝒮 of order (16, 256) has a subquadrangle isomorphic to Q(4, 16), then 𝒮 ≅ Q(5, 16).

In [18] M. Brown extended this work.

Theorem 190 ([18], 5.5). If a GQ 𝒮 of order (q, q^{2}), q even, has a subquadrangle isomorphic to W (q) and each subtended ovoid is either an elliptic quadric or a Tits ovoid then 𝒮 is isomorphic to the classical GQ Q(5, q).

Corollary 191 ([18], 5.6). Suppose q is even and that the only ovoids of W (q) are Tits ovoids and the elliptic quadric ovoids. Then if 𝒮 is a GQ of order (q, q^{2}) containing a subquadrangle isomorphic to W (q), then 𝒮 is isomorphic to the classical GQ Q(5, q).

The below theorem shows (another!) possible way of trying to classify ovoids of P G(3, q).

Corollary 192 ([18], 5.7). If there exists a GQ 𝒮 of order (q, q^{2}) such that 𝒮 is not isomorphic to Q(5, q) and 𝒮 contains a subquadrangle isomorphic to W (q), then there is a new ovoid of P G(3, q).

Using the results from Section 11.3, p. 71, Chapter II, we have:

Corollary 193 ([18], 5.8). For q ≤ 32, q even, if 𝒮 is a GQ of order (q, q^{2}) containing a subquadrangle isomorphic to W (q) then 𝒮 is isomorphic to the classical GQ Q(5, q).

After a small result we shall look at the next possible cases for subGQs.

16.1. Flocks and subquadrangles

See Chapter IV for the definition of flocks. (Apology: This theorem seems more appropriate here than in the next chapter.)

Theorem 194. There is no flock which gives rise to a generalised quadrangle with a subquadrangle on (∞) isomorphic to a T_{2}(O) with O a pointed conic for q > 8.

Proof. Without loss of generality we let f(x) = x^{2} and

O = {(1, t, f(t)) | t ∈ GF (q)} ∪ {(0, 1, 0)}.

For T_{2}(O) to be a subquadrangle of a GQ(C) constructed from a q–clan, C, we require O to be in a herd. That is,

trace (\frac{(f(t) + f(u))(g(t) + g(u))}{(t + u)}) = trace ((t + u)(g(t) + g(u))) = 1 for all t ≠ u,

for some g. Now by Lemma 118, p. 72 and in its notation, this implies that we have a = t, b = u, d = g, and α = 1/2.

By the proof of Theorem 119, p. 72, it is shown that there is no such g unless α = 2 or α^{2} = 2, that is, unless q = 2, 4, or 8.

For q = 2, 4, 8 such flocks exist.

17. Generalised quadrangles of order (8, 64) with a subquadrangle of order 8

We show that only certain GQs of order (q, q^{2}) contain a given subquadrangle of order q via the ideas of rosettes introduced in Section 6, p. 91. Let 𝒮 be the GQ(q, q^{2}) and 𝒮′ be the GQ(q) that is a subGQ of 𝒮. By Theorem 156, p. 92, we know that every point in the 𝒮 \ 𝒮′ subtends an ovoid in 𝒮′, and that every external line (they are all tangent lines for a GQ(q) in a GQ(q, q^{2})) subtends a rosette of 𝒮′. Hence every ovoid in 𝒮′ gives rise to at least one point in 𝒮 \ 𝒮′ (but at most two by Theorem 134, p. 83) and every rosette in 𝒮′ gives rise to a line in 𝒮. These results allow a construction of a GQ(q, q^{2}) from a GQ(q).

What we now plan to do is do the above backwards. Given a 𝒮 = GQ(q) determine the ovoids in the GQ, and any ovoid will correspond to a point in 𝒮 \ 𝒮′ and each rosette of the GQ(q) will correspond to a (tangent) line of 𝒮. If fruitful this will give new GQs of order (q, q^{2}) with a subGQ of order q; if not it will give us back 𝒮.

To use the above results to construct a GQ(q, q^{2}) one must know all the ovoids of the GQ(q). For large q this is computationally and theoretically difficult. For small q (for instance, q = 8) it is possible. Later on, all the ovoids of T_{2}(O) will be constructed for q = 8. Another approach that can be used is to assume that only certain ovoids exist in the GQ(q) and construct rosettes from only these ovoids.

We continue the above line of thought: Which generalised quadrangles of order (q, q^{2}) contain a particular subGQ of order q? And can we characterise GQ(q, q^{2}) by their subGQs?

The first GQ of order q that is not classical is T_{2}(O) where O is a pointed conic (see chapter 12 in Payne and Thas), this GQ is self-dual, for q = 8.

17. GQ(8, 64) with subGQ(8)  125

From now on we denote the pointed conic of P G(2, 8) by O.

For q = 8 there are two known quadrangles of order (8, 64) containing T_{2}(O) as a subquadrangle. They are

(i) the dual of GQ(C_{2}), denoted FTWKB^{∧}, where C_{2} is the the Fisher-Thas-Walker-Kantor-Betten q–clan defined on page 147 and

(ii) the GQ T_{3}(Ω) where Ω is a Tits ovoid.

We first start with the latter approach. The former approach is explained at the end of this section, using the computations by G. F. Royle. Together these results will appear in [19]. We would like to warn the reader that some of the results in this section are still preliminary.

17.1. Preliminaries

We define G, ℱ and 𝒮 = 𝒮 (G, ℱ) = GQ(C) as in Section 10, p. 102.

We first look at the case when the ovoids of T_{2}(O) are subtended from the GQ(8, 64) of the dual Fisher-Thas-Walker-Kantor-Betten GQ (page 147) GQ. We take advantage of the fact that T_{2}(O) is self-dual by working in the FTWKB GQ of order (64, 8). Then every external line induces a spread of T_{2}(O), this spread dualises to an ovoid of T_{2}(O).

We will need to do some extensive calculations in a GQ constructed from a q–clan. Recall the calculations in Section 8.1, p. 98. Note the calculations in this section are for any subGQ 𝒮 (a) in a GQ(q^{2}, q) 𝒮.

It is easy to see that:

A(s)(c, c, d) = A(s)(0, c + g_{s}(c), s^{1/2}c + d), and
A(∞)(c, c, d) = A(∞)(c, c + d ∘ c, 0).

The line A(s)(c, c, d) is secant to 𝒮 (a) if and only if d + s^{1/2}c ∈ ⟨a⟩. But which s?

If c and a are GF (q)-independent then there exists a unique solution for s in the equation

d + s^{1/2}c ∈ ⟨a⟩.

So there is a unique line A(s)(c, c, d) through (c, c, d) secant to 𝒮 (a) which intersects 𝒮 (a) in the line

A_{a}(s)(0, c + g_{s}(c), s^{1/2}c + d).

If c = aa for some a ∈ GF (q) we have that the line A(∞)(c, c, d) being secant to 𝒮 (a) and intersects 𝒮 (a) in the line

A_{a}(∞)(c, c + d ∘ c, 0).

Every external line to 𝒮 (a) gives rise to a spread in 𝒮 (a). The only lines external to 𝒮 (a) are of type A(t)g and A(∞)g. Every point on these lines has a unique secant line to 𝒮 (a).

17.1.1. Lines of the form A(t)g

Let g = (g_{1}, g_{2}, g_{3}) with g_{1}, g_{3} ∈ GF (q)^{2} and g_{2} ∈ GF (q). Then

A(t)g = {(b, g_{t}(b), t^{1/2}b)(g_{1}, g_{2}, g_{3}) | b ∈ GF (q)^{2}}
= {(b + g_{1}, g_{t}(b) + g_{2} + (t^{1/2}b) ∘ g_{1}, t^{1/2}b + g_{3}) | b ∈ GF (q)^{2}}.

The line A(t)g is external to 𝒮 (a) if and only if for all b ∈ GF (q)^{2} we have b+g_{1} ∉ ⟨a⟩ or t^{1/2}b + g_{3} ∉ ⟨a⟩. (This implies that g_{1} and g_{3} are GF (q)-independent of a.)

Let g′ = (b + g_{1}, g_{t}(b) + g_{2} + (t^{1/2}b) ∘ g_{1}, t^{1/2}b + g_{3}) for some b ∈ GF (q)^{2}.

From above we have that if {b + g_{1}, a} are GF (q)-independent then the unique line secant to 𝒮 (a) on g′ is

A_{a}(s)(0, g_{t}(b) + g_{2} + (t^{1/2}b) ∘ g_{1} + g_{s}(b + g_{1}), t^{1/2}b + g_{3} + s^{1/2}(b + g_{1})), (6)

with s being the unique solution to

t^{1/2}b + g_{3} + s^{1/2}(b + g_{1}) ∈ ⟨a⟩.

If b + g_{1} = aa for some a ∈ GF (q) then the unique line secant to 𝒮 (a) on g′ in 𝒮 (a) is

A_{a}(∞)(aa, g_{t}(b) + g_{2} + (t^{1/2}b) ∘ g_{1} + (t^{1/2}b + g_{3}) ∘ (aa), 0). (7)

The unique line of 𝒮 (a) on the point A^{∗}(t)g on the line A(t)g is

[A_{a}(t)]. (8)

Given an external line A(t)g where g = (g_{1}, g_{2}, g_{3}) we have a spread in 𝒮 (a) being constructed from the q^{2} − q lines of Equation (6), q lines from (7) and the line (8).

17.1.2. Lines of the form A(∞)g

Let g = (g_{1}, g_{2}, g_{3}) with g_{1}, g_{3} ∈ GF (q)^{2} and g_{2} ∈ GF (q). Then

A(∞)g = {(0, 0, b)(g_{1}, g_{2}, g_{3}) | b ∈ GF (q)^{2}}
= {(g_{1}, g_{2} + b ∘ g_{1}, b + g_{3}) | b ∈ GF (q)^{2}}.

This line is external to 𝒮 (a) if and only if for all b ∈ GF (q)^{2} we have g_{1} ∉ ⟨a⟩ or b + g_{3} ∉ ⟨a⟩.

Let g′ = (g_{1}, g_{2} + b ∘ g_{1}, b + g_{3}) with b ∈ GF (q)^{2}.

From above we have that the unique secant line to 𝒮 (a) on g′ being

A_{a}(s)(0, g_{s}(g_{1}) + g_{2} + b ∘ g_{1}, s^{1/2}g_{1} + b + g_{3}),

with s being the unique solution to

s^{1/2}g_{1} + b + g_{3} ∈ ⟨a⟩.

The above q^{2} lines and the line [A_{a}(∞)] constructs the spread in 𝒮 (a) from the external line A(∞)g.

17.2. Fisher-Thas-Walker-Kantor-Betten q–clan

Let q = 8.

Let the A_{t} be the q–clan associated with the Fisher-Thas-Walker-Kantor-Betten (FTWKB) q–clan C_{2} (see page 147), that is, for q = 8 we have

A_{t} = (\begin{matrix}t^{2} & t^{4} \\ 0 & t^{6} \end{matrix}) .

Now A_{t} is a normalised q–clan since t^{4} = t^{1/2} when q = 8 and A_{0} = (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}).

We now investigate spreads in the subGQ, 𝒮 (a), of the GQ 𝒮 constructed from the FTWKB q–clan. Again any external line of 𝒮 (a) in 𝒮 creates a spread in 𝒮 (a). We need only look at the lines of type A(t)g and A(∞)g with appropriate g as constructed in Section 17.1.

Since there is only one orbit on herds of A_{t} we are transitive on the set of G(a), 0 ≠ a ∈ GF (q)^{2}, so we choose

a = (1, 0).
Now A_{a}(t) = {(aa, g_{t}(aa), t^{4}aa) | a ∈ GF (q)}, with

g_{t}(aa) = ((a 0) (\begin{matrix}t^{2} & t^{4} \\ 0 & t^{6} \end{matrix}) (\begin{matrix}a \\ 0 \end{matrix}))^{4} = at for q = 8.

As in Section 11, p. 104, in the basis (a, 0, 0), (0, 1, 0) and (0, 0, a) we have A_{a}(t) describing the oval

O_{a} = {(1, t, t^{4}) | t ∈ GF (q)} ∪ {(0, 0, 1)},

equivalent to the Tit’s oval for q = 8. Hence the spreads of 𝒮 (a) are in fact spreads of T_{2}(O_{a}).

17.2.1. Lines of type A(t)g

Let g = (g_{1}, g_{2}, g_{3}) where g_{1} = (g_{1x}, g_{1y}), g_{3} = (g_{3x}, g_{3y}) ∈ GF (q)^{2} and g_{2} ∈ GF (q). Let b = (b_{x}, b_{y}). The q^{2} − q b’s with b_{y} ≠ g_{1y} give q^{2} − q lines of the spread in 𝒮 (a) being

A_{a}(s)(0, tb_{x} + t^{2}b_{y}^{4}b_{x}^{4} + b_{y}t^{3} + g_{2} + t^{2}b_{x}^{4}g_{1y}^{4} + t^{2}b_{y}^{4}g_{1x}^{4} + s(b_{x} + g_{1x}) +
s^{2}(b_{x}^{4} + g_{1x}^{4})(b_{y}^{4} + g_{1y}^{4}) + (b_{y} + g_{1y})s^{3}, (t^{4}b_{x} + g_{3x} + s^{4}(b_{x} + g_{1x}))a),

with

s = \frac{tb_{y}^{2} + g_{3y}^{2}}{b_{y}^{2} + g_{1y}^{2}} .

When b_{y} = g_{1y} then b+g_{1} = aa for some a ∈ GF (q) we have q lines of the spread of 𝒮 (a) being

A_{a}(∞)((b_{x} + g_{1x})a, tb_{x} + t^{2}b_{y}^{4}b_{x}^{4} + b_{y}t^{3} + g_{2} + b_{y}^{4}g_{3x}^{4} + g_{3x}^{4}g_{1y}^{4} + g_{3y}^{4}b_{x}^{4} + g_{3y}^{4}g_{1x}^{4}, 0).

The final line of the spread in 𝒮 (a) containing (∞) is

[A_{a}(t)].

Example 1. Look at the external line A(0)(0, 0, (0, 1)), note a is still (1, 0). For any b = (b_{x}, b_{y}) ∈ GF (q)^{2} such that b_{y} ≠ 0 we have q^{2} − q lines of the spread being

A_{a}(b_{y}^{5})(0, b_{y}^{5}b_{x} + b_{x}^{4} + b_{y}^{2}, b_{y}^{6}b_{x}a),

where s = b_{y}^{−2} = b_{y}^{5}. The q lines of the spread in 𝒮 (a) being, for b_{x} ∈ GF (q), when b_{y} = 0,

A_{a}(∞)(b_{x}a, b_{x}^{4}, 0).

And the line of 𝒮 (a) containing (∞) is

[A_{a}(0)].

17.2.2. Lines of type A(∞)g

Let g = (g_{1}, g_{2}, g_{3}), g_{1} = (g_{1x}, g_{1y}), g_{3} = (g_{3x}, g_{3y}) ∈ GF (q)^{2}, and g_{2} ∈ GF (q). Let b = (b_{x}, b_{y}). Then for all b we have an element of the spread in 𝒮 (a) being

A_{a}(s)(0, sg_{1x} + s^{2}g_{1y}^{4}g_{1x}^{4} + g_{1y}s^{3} + g_{2} + b_{x}^{4}g_{1y}^{4} + b_{y}^{4}g_{1x}^{4}, (s^{4}g_{1x} + b_{x} + g_{3x})a),

with

s = \frac{b_{y}^{2} + g_{3y}^{2}}{g_{1y}^{2}} .

The line of the spread containing (∞) in 𝒮 (a) is

[A_{a}(∞)].

Example 2. Look at the external line A(∞)((0, 1), 0, 0). For any b = (b_{x}, b_{y}) ∈ GF (q)^{2} we have q^{2} lines of the spread of 𝒮 (a) being

A_{a}(b_{y}^{2})(0, b_{y}^{6} + b_{x}^{4}, b_{x}a),

where s = b_{y}^{2} and the line of the spread containing (∞) is

[A_{a}(∞)].

17.3. Constructing ovoids from spreads

The reference for this section is Chapter 12 of Payne and Thas and we recall that a = (1, 0).

Let O_{a} be the oval described by A_{a}(s) (in Section 17.2, p. 127) be in P G(2, 8). Embed P G(2, 8) as the plane x_{0} = 0 in P G(3, 8) and consider this as the GQ T_{2}(O_{a}). Then we have a geometric structure to the above spread constructions.

The line [A_{a}(∞)] of the GQ, 𝒮 (a), is associated with the point (0, 0, 0, 1) of P G(3, 8). The lines [A_{a}(s)], s ∈ GF (q), are associated with the points (0, 1, s, s^{4}). The lines A_{a}(∞)(ba, c, 0) of 𝒮 (a) are associated with the lines of P G(3, 8) tangent to the point (0, 0, 0, 1) on the oval O_{a} and containing the affine point (1, b, c, 0). The lines of type A_{a}(s)(0, c, ba) of 𝒮 (a) are associated with the lines of P G(3, 8) tangent to the point (0, 1, s, s^{4}) on the oval O_{a} and containing the affine point (1, 0, c, b).

In the notation of Chapter 12 of Payne and Thas we have:

[A_{a}(∞)] ↔ [∞],
[A_{a}(s)] ↔ [s],
A_{a}(∞)(ba, c, 0) ↔ [b, c], and
A_{a}(s)(0, c, ba) ↔ [s, b, c].

Also in Payne and Thas, p. 276, there is a duality defined on the lines of 𝒮 (a) as

[∞] ↦ (∞),
[s] ↦ (s^{4}),
[b, c] ↦ (b, c^{4}), and
[s, b, c] ↦ (s^{4}, b, c^{4}).

This takes a spread of 𝒮 (a) = T_{2}(O_{a}) to an ovoid of 𝒮 (a)^{∧} = T_{2}(O_{a})^{∧} = T_{2}(O), where O is the pointed conic O = {(1, t, t^{1/2}) | t ∈ GF (q)} ∪ {(0, 0, 1)}.

The points (s^{4}) of the GQ, 𝒮 (a)^{∧}, are associated with the planes of P G(3, 8) tangent to the point (0, 0, 0, 1) on O_{a} and containing the affine point (1, s^{4}, 0, 0). The points (b, c^{4}) are associated with the planes of P G(3, 8) tangent to the point (0, 1, b, b^{4}) on O_{a} and containing the affine point (1, 0, 0, c^{4}). The points (s^{4}, b, c^{4}) in 𝒮 (a)^{∧} are associated with the affine points (1, s^{4}, b, c^{4}) of P G(3, 8).

17.3.1. Ovoids from lines of type A(t)g

In the same notation as in Section 17.2.1 we have an external line A(t)g to 𝒮 (a) with g = (g_{1}, g_{2}, g_{3}), where g_{1} = (g_{1x}, g_{1y}), g_{3} = (g_{3x}, g_{3y}) and g_{2} ∈ GF (q) if and only if for all b = (b_{x}, b_{y}) ∈ GF (q) we have b + g_{1} ∉ ⟨a⟩ or t^{1/2}b + g_{3} ∉ ⟨a⟩.

The points of the ovoid in T_{2}(O) are:

(i) When b_{y} ≠ g_{1y} we have q^{2} − q affine points of P G(3, 8):

(1, \frac{t^{4}b_{y} + g_{3y}}{b_{y} + g_{1y}} , t^{4}b_{x} + g_{3x} + s^{4}(b_{x} + g_{1x}), t^{4}b_{x}^{4} + tb_{y}^{2}b_{x}^{2} + b_{y}^{4}t^{5} + g_{2}^{4} + tb_{x}^{2}g_{1y}^{2}

+ tb_{y}^{2}g_{1x}^{2} + s^{4}(b_{x}^{4} + g_{1x}^{4}) + s(b_{y}^{2} + g_{1y}^{2})(b_{x}^{2} + g_{1x}^{2}) + (b_{y}^{4} + g_{1y}^{4})s^{5}),

where s = (tb_{y}^{2} + g_{3y}^{2})/(b_{y}^{2} + g_{1y}^{2}). (ii) When b + g_{1} ∈ ⟨a⟩ we have q planes in P G(3, 8) tangent to O at the point

(0, 1, b_{x} + g_{1x}, b_{x}^{4} + g_{1x}^{4})

and containing the affine point

(1, 0, 0, t^{4}b_{x}^{4} + tb_{y}^{2}b_{x}^{2} + b_{y}^{4}t^{5} + g_{2}^{4} + b_{y}^{2}g_{3x}^{2} + g_{3x}^{2}g_{1y}^{2} + g_{3y}^{2}b_{x}^{2} + g_{3y}^{2}g_{1x}^{2}).

In plane co-ordinates of P G(3, 8) they are the planes

[t^{4}b_{x}^{4} + tb_{y}^{2}b_{x}^{2} + b_{y}^{4}t^{5} + g_{2}^{4} + b_{y}^{2}g_{3x}^{2} + g_{3x}^{2}g_{1y}^{2} + g_{3y}^{2}b_{x}^{2} + g_{3y}^{2}g_{1x}^{2}, b_{x}^{4} + g_{1x}^{4}, 0, 1].

(iii) The plane in P G(3, 8) tangent to O at the point

(0, 0, 0, 1)

and containing the affine point

(1, t^{4}, 0, 0) :

In plane co-ordinates of P G(3, 8) it is the plane

[t^{4}, 1, 0, 0].

Example (Continuation of Example 1, p. 128 ). The external line A(0)(0, 0, (0, 1)) describes an ovoid in T_{2}(O) as:

(1) The q^{2} − q affine points, with b_{y} ≠ 0,

(1, b_{y}^{6}, b_{y}^{6}b_{x}, b_{y}^{6}b_{x}^{4} + b_{y} + b_{x}^{2}).

(2) The q planes for b_{x} ∈ GF (q) tangent to O at the point (0, 1, b_{x}, b_{x}^{4}) and containing the affine point (1, 0, 0, b_{x}^{2}).

(3) The plane tangent to O at the point (0, 0, 0, 1) and containing the affine point (1, 0, 0, 0).

17.3.2. Ovoids from lines of type A(∞)g

In the same notation as in Section 17.2.2 we have an external line A(∞)g to 𝒮 (a) with g = (g_{1}, g_{2}, g_{3}), where g_{1} = (g_{1x}, g_{1y}), g_{3} = (g_{3x}, g_{3y}) and g_{2} ∈ GF (q) if and only if for all b = (b_{x}, b_{y}) ∈ GF (q) we have g_{1} ∉ ⟨a⟩ or b + g_{3} ∉ ⟨a⟩.

The points of the ovoid in T_{2}(O) are:

(i) The q^{2} affine points, b ∈ GF (q)^{2},

(1, \frac{b_{y} + g_{3y}}{g_{1y}} , s^{4}g_{1x} + b_{x} + g_{3x}, s^{4}g_{1x}^{4} + sg_{1y}^{2}g_{1x}^{2} + g_{1y}^{4}s^{5} + g_{2}^{4} + b_{x}^{2}g_{1y}^{2} + b_{y}^{2}g_{1x}^{2}),

where s = (b_{y}^{2} + g_{3y}^{2})/g_{1y}^{2}. (ii) The point (∞).

Example (Continuation of Example 2, p. 128). The external line A(∞)((0, 1), 0, 0) describes an ovoid in T_{2}(O) as:

(1) The q^{2} affine points, (b_{x}, b_{y}) ∈ GF (q)^{2}

(1, b_{y}, b_{x}, b_{y}^{3} + b_{x}^{2}).

(2) The point (∞).

17.4. Ovoids from the polarity

A polarity acts on the lines as such

[∞] ↦ (∞),
[s] ↦ (s^{2}),
[b, c] ↦ (b^{4}, c^{2}),
[s, b, c] ↦ (s^{2}, b^{4}, c^{2}).

of 𝒮 (a)^{∧} (see p. 276 of Payne and Thas):

An ovoid of 𝒮 (a)^{∧} can be constructed from the absolute points of this polarity. We have one absolute point via inspection: (∞). Points (s^{2}) are never incident with lines [s]. Points (b^{4}, c^{2}) are never incident with

lines [b, c].

Points P = (s^{2}, b^{4}, c^{2}) are incident with lines [s, b, c] if and only if P is one of the affine points of P G(3, 8),

(1, s^{2}, s^{3} + c, c^{2}), for s, c ∈ GF (q).

Together with the point (∞) these are the q^{2} + 1 points of the ovoid.

Hence the affine points defined by the polarity are:

(1, u, v, u^{3} + v^{2}) u, v ∈ GF (q). (9)

We can show that none of the ovoids in Section 17.3 come from polarities of T_{2}(O).

17.5. Constructing GQ(q, q^{2}) from T_{2}(O): FTWKB case

We now look at the special case of when T_{2}(O) can be a subGQ for a GQ(q, q^{2}) when q = 8, O is a pointed conic, and all the ovoids of T_{2}(O) are subtended from the dual FTWKB GQ. These ovoids where constructed in Section 17.1, p. 125. The GQ(q, q^{2}) is constructed from the ovoids by the correspondence that every ovoid of T_{2}(O) is a point of GQ(q, q^{2}) and every rosette of T_{2}(O) is a line of GQ(q, q^{2}). Now every point of T_{2}(O) is on at least q^{2} − q rosettes, since these are the number of external lines of the dual FTWKB GQ to T_{2}(O). Any point of T_{2}(O) with more than q^{2} −q rosettes is a candidate for constructing lines of a GQ(q, q^{2}) not isomorphic to the dual FTWKB GQ. Determining new line sets is straightforward. Let x be a point of T_{2}(O). Let 𝒮 be the GQ(q, q^{2}) we are trying to construct. Put

Y = {points of 𝒮 \ T_{2}(O) collinear with x }

= {ovoids on x}

(where ovoids on x really means ovoids from FTWKB construction on x). Now each partition of Y into lines corresponds to a set of q^{2} − q rosettes on x. If there is only one partitioning of Y into lines then we have only the original GQ(q, q^{2}) that gave the subtended ovoids.

To determine the partitioning we let Γ be a graph with vertices,

V (Γ) = {rosettes based at x}

and adjacency defined by

ℛ_{1} ∼ ℛ_{2} ⟺ ℛ_{1} and ℛ_{2} are rosettes not containing a common ovoid.

(Note the rosettes are only constructed from FTWKB ovoids.)

The number of cliques of size q^{2} − q (56 in our case), determine the number of partitions into lines of Y. An easy calculation shows that we only find one clique for points of T_{2}(O) when the ovoids are subtended from FTWKB GQ. Hence there are no new GQ(q, q^{2}) containing T_{2}(O) as a subGQ, when all the ovoids are subtended from the dual FTWKB GQ. Though we should stress that we are assuming that one ovoid only yields one point in 𝒮 \ T_{2}(O).

17.6. Subtended ovoids from T_{3}(Ω)

Recall the definition of T_{3}(Ω) in Section 2.5, p. 86. We make a distinction between ovoids subtended by a point of type (i) of T_{3}(Ω) and points of type (ii). We call the former projected ovoids and the latter planar ovoids. We call points of P G(4, q) − P G(3, q) affine points and hyperplanes of P G(4, q) meeting Ω in one point affine hyperplanes.

Also recall (page 122) how for a given secant plane π of Ω ⊆ P G(3, q) we can construct a T_{2}(O) with O = Ω ∩ π. We are interested when O is a pointed conic for q = 8, that is, a Tits oval. Hence Ω must be a Tits ovoid (see Section 11.3, p. 71, Chapter II). We label the projective three space that T_{2}(O) lives in as Σ = P G′(3, q), noting O ⊆ π.

We now describe the two types of ovoids constructed in T_{2}(O) by external points of T_{3}(Ω).

Result 195 (Planar Ovoids). Let x be a affine hyperplane. Then x intersects Ω at a point z and at a tangent plane π′. Let l be the line of intersection of π and π′. Let π′′ be the plane of intersection of P G′(3, q) and x (meeting at the line l). Then

Planar ovoid = {affine points of π′′} ∪ {(∞)}.

Result 196 (Projected ovoid). Let x be an affine point. The projection, x_{t}, of the q^{2} − q points of Ω \ O into P G′(3, q) gives a set of q^{2} − q points of T_{2}(O) that are not collinear in the subGQ. Let z be a point of O and let l be the unique tangent line on z in π. There is a unique tangent plane π′ on z, these generate the 3-space ⟨π′, x⟩. We label the plane of intersection of ⟨π′, x⟩ and P G′(3, q) as π_{z} (as there is a unique plane for every point of the pointed conic z).

Projected ovoid = {x_{t} | t ∈ Ω \ O} ∪ {π_{z} | z ∈ O}.

The number of external points to T_{2}(O) in T_{3}(Ω) is:

(1 + q)(1 + q^{3}) − (1 + q)(1 + q^{2}) = 4032 external points

= 4032 planar and projected ovoids.

These ovoids are divided into 3584 projected ovoids and 448 planar ovoids.

Let Ω = {(s, 1, t, s^{6} + st + t^{4}) | t ∈ GF (q)} ∪ {(0, 0, 0, 1)} be the Tits ovoid in P G(3, 8). Then the Tits ovoid defines a symplectic polarity with (alternating) form on P G(3, q)

f(x, y) = x_{2}y_{4} + x_{4}y_{2} + x_{3}y_{5} + x_{5}y_{3}

where points of P G(4, q) are of the form (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}). (This is derived from the fact that given the Tits oval as (s^{σ+2} + st + t^{σ}, 1, s, t) the polarity has the more usual form x_{1}y_{2} + x_{2}y_{1} + x_{3}y_{4} + x_{4}y_{3} in P G(3, q). We then permute the coordinates.)

Now the tangent plane to the point x of P G(3, q) is the perp of x under the polarity with form f. Let x = (0, s, 1, t, s^{6} + st + t^{4}), so

tangent plane at x = {y | f(x, y) = 0} ∩ [1, 0, 0, 0, 0]
= {y | sy_{4} + ty_{2} + y_{5} + (s^{6} + st + t^{4})y_{3} = 0} ∩ [1, 0, 0, 0, 0]

Hence the hyperplanes on x are [a, t, s^{6} + st + t^{4}, s, 1] with a ∈ GF (8). Though for the following computations we only need a to be in GF (8)^{∗} so that the hyperplanes are affine.

Let x = (0, 0, 0, 0, 1). Then we have f(x, y) = y_{3} = 0. Hence the planes on x are [a, 0, 1, 0, 0] with a as above.

17.6.1. Planar ovoids

We have

Ω \ O = {(0, s, 1, t, s^{6} + st + t^{4}) | t, s ∈ GF (8), s ≠ 0}.

An affine point is of the form (1, a, b, c, d) with a, b, c, d ∈ GF (8) a ≠ 0. An affine hyperplane is of the form [a, t, s^{6} + st + t^{4}, s, 1] with s ≠ 0 s, t, a ∈ GF (8) where the q affine hyperplanes meet at the point (0, s, 1, t, s^{6} + st + t^{4}).

It should be noted that planar ovoids have a curious feature; they are subtended by two external points of T_{3}(Ω). Recall that every ovoid of Q(4, q) is subtended by exactly two external points of Q(5, q). The proof is straightforward. We require:

Lemma 197 ([56], Chapter 16). For q even, every external line to an ovoid in P G(3, q) is on two tangent planes of the ovoid.

Using the notation of Result 195, p. 133, we have two affine hyperplanes on the line l intersecting P G′(3, q) at π′′. So each x gives exactly two planar ovoids in T_{2}(O).

It is easy to see that only one external point of T_{3}(Ω) subtends a projected ovoid.

17.7. Constructing GQ(q, q^{2}) from T_{2}(O): FTWKB and T_{2}(Ω) cases

This work is still in progress. There is the complication of ovoids being subtended by two points.

17.8. Ovoids of T_{2}(O), O pointed conic, q = 8

We summarise the results of G. F. Royle on his exhaustive search of ovoids of T_{2}(O). (As before O is a pointed conic and q = 8.) By using Brendan McKay’s nauty package [77] the automorphism group of T_{2}(O) can be determined and it has order 1204224 (though this can be easily worked out from the stabiliser of the pointed conic in P G(2, q) and the duality).

The orbits of the collineation group have sizes 1, 8, 64 and 512 on points (and the same on lines).

An ovoid of this GQ has size 65. An exhaustive search for ovoids yields the results that there are 5 ovoids, named Type I to Type V.

No. Name  |G| Orbits  (n_{0}, n_{1}, n_{2}) Plane intersections
I  FTWKB 168 {1, 8, 56} (0, 56, 9)  {0^{10}, 5^{168}, 7^{288}, 8^{63}, 13^{56}}
II Projected 168 {1, 8, 56} (0, 56, 9)  {0^{10}, 1^{56}, 7^{288}, 8^{63}, 9^{168}}
III Polarity 168 {1, 8, 56} (1, 64, 0)  {0^{1}, 4^{112}, 8^{360}, 12^{112}}
IV Planar 2688 {1, 64} (1, 64, 0) {0^{8}, 8^{576}, 64^{1}}
V FTWKB 1344 {1, 64} (1, 64, 0)  {0^{1}, 4^{112}, 8^{360}, 12^{112}}

The column Name gives the type of ovoid it is in the case of polarity, or if it is a planar or projected ovoid subtended from T_{3}(Ω) or subtended from the dual FTWKP GQs. The next column gives the size of the stabilisers of each ovoid and the column after that gives the orbits of the stabiliser on each type of ovoid. Let n_{0} be 1 if an ovoid contains (∞) or 0 if not. Let n_{1} be the number of affine points of P G(3, q) (points of type (i)) the ovoid has and n_{2} be the number of affine planes (points of type (ii)) the ovoid has. The last column shows how the 585 planes of P G(3, q) intersect the affine points of each ovoid type, where X^{Y} is read as there are Y planes intersecting the ovoid in X points. From the last column it is obvious that ovoid of type II is a projected ovoid and that the ovoids of type IV are planar. Ovoids of type I, III and V have been conjectured to be FTWKB type, from a polarity and FTWKB type, respectively, though we, as yet, can not prove this.

We can now determine whether the ovoids lie in rosettes or otherwise:

No. Point(orbit) Candidates  Rosettes
I  569(1)  28 I, 28 II  15
516(8)  14 I  2
13(56)  6 I, 1 III, 1 V 1
II 569(1)  28 I, 28 II  15
516(8)  7 II  1
11(56)  6 II, 3 III, 1 IV 1
III 0(1)  35 III  30
2(56)  3 II  0
29(8)  7 I  0
IV 0(1)  7 IV  1
2(64)  14 II  2
V 0(1)  35 V  30
2(64)  7 I  1

Observe that there are 3584 ovoids of type I, II and III, 224 of type IV and 448 of type V. Furthermore note that to form a GQ(8, 64) we need 4032 = 3584 + 448 ovoids.

We can conjecture, after looking at the above table:

Conjecture 198. For q = 8 the only generalised quadrangles of order (q, q^{2}) with T_{2}(O), O a pointed conic, as a subquadrangle of order q are T_{3}(Ω), Ω a Tits ovoid, and the dual of GQ(C_{2}) where C_{2} is the FTWKB q–clan.

18. End Note

Notes on Section 1, p. 81. The results in this Section are well known.

Notes on Section 2, p. 83. The results in this Section are well known.

Notes on Section 3, p. 87. A subspace W of P G(d, q) is absolute with respect to Δ if W ⊆ Δ(W). A polar space is the system of absolute subspaces where incidence is containment. Over a finite field, d > 2, this system is non-empty (in an infinite field we have counterexamples).

Properties of polar spaces: They satisfy GQ1 and ptGQ axioms and the axiom PS: Given an anti-flag (x, L) we have either GQ2 or that x is collinear with all

points of L Proof that axiom PS is satisfied in a GQ if the polar space has rank two: Suppose

x and y are absolute points. Then ⟨x, y⟩ is an absolute line if and only if x ⊆ Δ(y) if and only if y ⊆ π(x), since ⟨x, y⟩ ⊆ π(⟨x, y, ⟩) = π(x) ∩ π(y). Now let x be an absolute point and L an absolute line with x not incident with L. We have Δ(x) being a hyperplane with x ⊆ π(x). Either π(x) meets L in a point or contains L. As y ⊆ π(x) if and only if y ∼ x we have one point collinear with x or all points on L collinear with x. Notice axiom ptGQ requires the use of finite fields.

Suppose we have the anti-flag (x, L) with x being collinear with all the points of L. Then ⟨x, L⟩ is an absolute plane. We have L ⊆ Δ(x), x ⊆ π(L), x ⊆ π(x) and L ⊆ π(L), so

⟨x, L⟩ ⊆ Δ(x) ∩ π(L) = π(⟨x, L⟩).

If there are no absolute planes then the polar space is a generalised quadrangle (given that there are absolute lines). Hence the polar space has rank two.

Below is the theorem of Buekenhout and Lefévre:

Theorem 199 ([25]). A GQ 𝒮 = (𝒫, ℬ, I) whose incidence structure is a full substructure of a projective space P G(d, q) must be obtained in one of the following ways:

(i) There is a unitary or symplectic polarity π of P G(d, q), d = 3 or 4, such that 𝒫 is a set of absolute points of π and ℬ is the set of totally isotropic lines of π.

\underline{18. End Note 137}

(ii) There is a nonsingular quadric Q of projective index 1 in P G(d, q), d = 4, 5, such that 𝒫 is the set of points of Q and ℬ is the set of lines of Q.

Hence the classical GQs are W (q), Q(4, q), Q(5, q), H(3, q^{2}) and H(4, q^{2}).

The results in this Section are well known.

Notes on Section 4, p. 89. The results in this Section are well known. The proof of Theorem 151, p. 90, is different from the given reference and is due to Penttila.

Notes on Section 5, p. 91. The results in this Section are well known.

Notes on Section 6, p. 91. The results in this Section are well known. The model of Q(5, q) is somewhat new and is by Penttila.

Notes on Section 7, p. 94. Let 𝒮 = (𝒫, ℬ, I) be an EGQ(s, t) with two base points x_{1} ≠ x_{2} and two elation groups G_{1} and G_{2}. Then G_{1} has orbits {x_{1}}, x_{1}^{⊥} \ {x_{1}}, and 𝒫 \ x_{1}^{⊥}, and G_{2} has orbits {x_{2}}, x_{2}^{⊥} \ {x_{2}}, and 𝒫 \ x_{2}^{⊥} on 𝒫. So Aut 𝒮 has orbits being unions of both of the above orbits of G_{1} and G_{2}. Therefore Aut 𝒮 is transitive. Hence Aut 𝒮 has rank 3 on 𝒫. We have Aut 𝒮 being primitive since it has subdegrees 1, s(t+1) and s^{2}t, and none of 1+s(t+1), 1+s^{2}t, s(t+1)+s^{2}t divides |𝒫|. Primitive rank 3 groups have been classified by Liebeck [73]. It is well known that this forces Aut 𝒮 to be classical. (Though the reference is hard to find. Folklore possibly?)

In P G(2n + m − 1, q) consider a set O(n, m, q) of q^{m} + 1 (n − 1)–dimensional subspaces [n − 1]_{0}, …, [n − 1]_{q^{m}}, every three of which generate a (3n − 1)-dimension (projective) space, and such that each element of [n − 1]_{i} of O(n, m, q) is contained in a [n + m − 1]_{i} having no point in common with any other [n − 1]_{j} for i ≠ j. The [n + m − 1]_{i} is uniquely determined and is called the tangent space of O(n, m, q) at [n − 1]_{i}.

Now we embed P G(2n + m − 1, q) in a P G(2n + m, q) and construct a point-line geometry T (n, m, q) as follows. Points are of three types:

(i) The points of P G(2n + m, q) − P G(2n + m − 1, q).

(ii) The (n+m)–dimensional subspaces of P G(2n+m, q) which intersect P G(2n+ m − 1, q) in one of the [n + m − 1]_{i}.

(iii) The symbol (∞). Lines are of two types:

(a) The n–dimensional subspaces of P G(2n + m, q) which intersect P G(2n + m − 1, q) in a [n − 1]_{i}.

(b) The elements of O(n, m, q). Incidence: A type (i) point is incident only with type (a) lines (incidence of P G(2n + m, q)). A type (ii) point is incident with all type (a) lines contained in it and with the unique element of O(n, m, q) contained in it. The point (∞) is incident with no type (a) lines and all type (b) lines. There are no further incidences.

We have O(1, 1, q) = O is an oval of P G(2, q) and T (1, 1, q) is the GQ T_{2}(O). We have O(1, 2, q) = Ω is an ovoid of P G(3, q) and T (1, 2, q) is the GQ T_{3}(Ω).

The only known TGQs for q even are the T_{2}(O) and T_{3}(Ω) generalised quadrangles.

Let q be odd and Ω be an ovoid of P G(3, q). That is, a set of q^{2} + 1 1-spaces in GF (q)^{4}. Let q = q_{0}^{m}. Then these q^{2} + 1 1-spaces are q_{0}^{2m} + 1 m-spaces (over GF (q_{0})) in GF (q_{0})^{4m} with two distinct m-spaces generating a 2m space that is disjoint from the q_{0}^{2m} + 1 set. Any point of Ω is on a unique 3m-space Σ with Σ ∩ Ω = {x}. Hence this collection of q_{0}^{2m} + 1 m-spaces is a O(m, 2m, q_{0}), sometimes referred to as a bad egg, and so constructs a TGQ(q, q^{2}). Now the tangent spaces Σ form a bad egg, Ω^{∗}, in P G(4m − 1, q_{0}). The bad egg Ω^{∗} constructs a TGQ that is not necessarily isomorphic to the original TGQ. For q = 3^{e}, e odd, there is a bad egg, Ω, that can be constructed from the semifield planes of Ganley [41]. The TGQ constructed from Ω is not isomorphic to the TGQ constructed by Ω^{∗}. These TGQs from Ω^{∗} are the Roman generalised quadrangles constructed by Payne [94][95].

The results in this Section are well known.

Notes on Section 8, p. 97. Theorem 168, p. 97 is proved for the construction of an EGQ 𝒮 (G, ℱ) of order (q^{m}, q^{n}) for positive integers m and n. Since we are already quite comprehensive for EGQs of order (q^{2}, q) to generalise even further will be of little benefit to the reader.

The results in this Section are well known.

Notes on Section 9, p. 99. The Theorem 169, p. 100, first appeared in [30]; a collaborative effort between the author, Penttila, Royle and Cherowitzo.

Notes on Section 10, p. 102. The results in this Section are well known.

Notes on Section 11, p. 104. The results in this Section are well known.

Notes on Section 12, p. 105. The Subiaco q–clan first appeared in [30]; a collaborative effort between the author, Penttila, Royle and Cherowitzo.

Notes on Section 13, p. 105. This Section is from the work of Payne [102, 6, 96].

Notes on Section 14, p. 108. The work first appeared in the papers of Payne. Lemma 184, p. 115, Lemma 185 and Theorem 186, p. 118, first appeared in [100]. The results were a collaborative effort between the author and Penttila, with contributions by Payne. They were influenced by the ideas in [87].

Notes on Section 15, p. 119. The question (I think) was asked by T. Penttila. It was asked in the view that it will hopefully make certain GQ calculations easier, in particular those requiring the Fundamental Theorem. The answer to that question was yes!

This Section is from [98] and [101].

Notes on Section 16, p. 122. The results in this Section are well known though recently published.

Notes on Section 17, p. 124. The introductory work in this Section is well known.

Work on the FTWKB q–clan: Sections 17.1–17.7 is by the author. Section 17.8, p. 134 is by Royle.

CHAPTER IV Flocks

1. Introduction

Take an oval (or hyperoval, if q is even) O in P G(2, q). Take a point v in P G(3, q) not in P G(2, q). The union of the points on the lines joining the vertex v to the oval (or hyperoval) O is called a cone. The oval (hyperoval) O is sometimes referred to as the base of the cone and the lines from the vertex to the base are occasionally referred to as the generators of the cone. A quadratic cone is a cone in which the base is a conic. A translation oval cone is a cone whose base is a translation oval (see Section 6, p. 46, Chapter II). A flock F of a cone K with vertex v is a set of q disjoint ovals (hyperovals) partitioning K − {v}. A flock F is called linear if all the planes containing the elements of the flock meet in a (external) line.

Let Ω be an ovoid of P G(3, q). A flock of Ω is a partition of Ω into q − 1 ovals with the remaining two points of the ovoid x and y, the carriers of the flock. If the planes containing the ovals of the flock meet in a (external) line then we say the flock is linear.

Let H be the hyperbolic quadric of P G(3, q), (that is H is O^{+}(4, q)). A partition of H into q + 1 irreducible conics is called a flock of H. If the planes containing the elements of the flock meet in a (external) line then we say the flock is linear.

In all of the above cases the ovals of the flocks uniquely define the planes containing them, hence we just as easily define flocks as a set of planes intersecting a cone (minus the vertex), ovoid (minus the carriers) or hyperbolic quadric.

2. Translation Planes from Flocks of Quadrics

We now look at flocks of quadrics, that is, flocks arising from a quadratic cone, hyperbolic quadric, or a elliptic quadric.

We denote O^{+}(6, q), the Klein quadric in P G(5, q), by K.

We embed the flock of a quadratic cone, hyperbolic quadric, and elliptic quadric in K. There is a unique embedding of a quadratic cone (up to similarity), hyperbolic quadric, and of an elliptic quadric in K since all 3-spaces intersecting K in a quadric are isomorphic (see Theorem 22.6.6 in [57]).

So each flock, F, is a set of planes in P G(5, q) with each plane intersecting K in a conic. The polarity, ⊥, defined by K of P G(5, q) takes planes of this type to planes of this type. (That is because we have an orthogonal sum O^{+}(6, q) = O(3, q) ⊥ O(3, q), since a plane of this type is an O(3, q).)

Let Ω be the set of totally singular points x with x in π^{⊥}, for some π in F, a flock of a quadratic cone. That is,

Ω = ⋃_{π_{i}∈F}(π_{i}^{⊥} ∩ K).

We claim Ω is an ovoid.

Proof. Firstly we show |Ω| = q^{2} +1: There are q planes in a flock of a quadratic cone. The cone lives in a solid, Σ, whose perp is a tangent line L (since Σ meets K in a cone). The point v of tangency of L is the vertex of the cone. Thus all π^{⊥} are on L and so on v. Moreover, since the π’s in F are distinct the π^{⊥}’s are distinct, so they pairwise intersect exactly in L. Other than v, each π^{⊥} has q points of K on it; q^{2} points in all. Adding v we have the result.

Secondly, no two points of Ω are perpendicular (that is, no two lie on a line of K): If π, π′ in F, then π intersects π′ in an external line M to K (this is the first time we have used this fact). Thus M^{⊥} is an elliptic quadric solid, as

O^{+}(6, q) = O^{−}(2, q) ⊥ O^{−}(4, q).

But M^{⊥} is the span of π^{⊥} and π′^{⊥}. In an elliptic quadric solid, no two points are perp. Therefore, if x, y ∈ Ω and x ∈ π^{⊥}, y ∈ π′^{⊥} with π, π′ distinct, x is not perp to y. If π = π′, then x and y both lie in a conic, so again they are not perp.

For a hyperbolic quadric a flock has q + 1 planes, and the perp of the solid is a secant line, so we have (q + 1)(q − 1) + 2 = q^{2} + 1. The rest of the proof is identical. For an elliptic quadric, a flock has q − 1 planes, and two points not in the partition; so Ω must be taken to be the union of the π^{⊥}’s plus these two points; then we get (q − 1)(q + 1) + 2 as for this case the perp of the solid is an external line.

Now each π^{⊥} is a conic, and so under the Klein correspondence, which takes points of K to lines of P G(3, q), (see Section 15.4 in [56]) there corresponds a regulus of P G(3, q). Together, the q^{2} + 1 points of Ω Klein correspond to a set of q^{2} + 1 skew lines of P G(3, q), that is, a spread, S, of P G(3, q) (see Chapter I, 10). We use the Bruck-Bose construction, Theorem 22, p. 10, to construct a translation plane of order q^{2} with kernel containing GF (q). (Note: For each of the π^{⊥}’s being a conic this says that the resulting translation planes have q, q + 1 or q visible derivation sets [36, pp.

224–5] accordingly as the quadric is a cone, hyperbolic or elliptic. Of course, some of these may lead to the same derived plane up to isomorphism.)

Now the above translation plane is Desarguesian if and only if S is regular if and only if Ω is an elliptic quadric if and only if the π’s have a common line if and only if F is linear.

3. Classification of Flocks of Ovoids

Theorem 200. Any flock of an ovoid in P G(3, q) is linear.

Proof. For the q is odd case see [88], or [40] Theorem 3.2(b) for another proof. For the q even case see [40] Theorem 3.1. See also Sections 17.1 and 17.2 in [56].

Hence all translation planes from a flock of an elliptic quadric are Desarguesian.

4. Classification of Flocks of a Hyperbolic Quadric

Let H be the hyperbolic quadric O^{+}(4, q) in P G(3, q). The hyperbolic quadric can be thought of as a grid of type (q + 1, q + 1). This is a regulus and its opposite regulus.

Theorem 201. Any flock of H in P G(3, q), where q is even, is linear.

Proof. See [40] Theorem 3.1.

In [126] we have a description of a non-linear flock for H in P G(3, q) when q is odd: We let C be the set of irreducible conics of H, and define an equivalence relation on C by: C_{1} ∼ C_{2} if and only if there exists a C ∈ C that is tangent to C_{1} and C_{2}. This equivalence relation defines two equivalence classes which we denote by I and II. Let L be an exterior line to H. The polarity induced by H takes exterior lines to exterior lines, we let L′ be the polar line of L with respect to H. Now we define some sets:

V = {planes on L intersecting H in a conic of class I},
V′ = {planes on L intersecting H in a conic of class II},
W = {planes on L′ intersecting H in a conic of class II,

if q = 1 mod 4, or class I, if q = −1 (mod 4)},

W′ = {planes on L′ intersecting H in a conic of class I,

if q = 1 mod 4, or class II, if q = −1 (mod 4)}.

Then V ∪ W and V′ ∪ W′ describe a non-linear flock of H, called a Thas flock.

There are also the three exceptional flocks of H in P G(3, q) for q = 11, 23, 59 [4]. Independently found by Baker-Ebert [10] for q = 11 and 23, and by Johnson [63] for q = 59.

Theorem 202. Any flock of the hyperbolic quadric H = O^{+}(4, q) in P G(3, q), q is even or odd, is:

(i) a linear flock, or (ii) a Thas flock, or (iii) one of the three exceptional flocks for q = 11, 23, 59.

Proof. See [5]. This paper uses the result of Thas [128] where the same result is proven under the extra hypothesis that q ≡ 1 (mod 4).

The non-linear flocks give rise to translation planes over nearfields. The Thas flock has the translation plane over a regular nearfield and the three exceptional flocks have their translation planes over some of the exceptional nearfields (hence their name) [36].

5. Flocks of Cones

In this section we will look at some results on flocks of cones over small fields.

Theorem 203. Any flock of a cone in P G(3, 2) is linear.

Proof. A flock consists of two planes of P G(3, 2). Any two planes meet in a line.

Theorem 204. Let F and F′ be two flocks of the same cone in P G(3, q). If |F ∩ F′| > (q − 1)/2 then the flocks are equal.

Proof. See [127] First Theorem of Section 1.5.1.

Corollary 205. Any flock of a cone in P G(3, 3) is linear.

The base of a cone can either be a hyperoval or an oval for q even. Now for q even we would like to know what happens to a flock, F, of a cone, K, in P G(3, q) when its base changes from an oval to a hyperoval; call this hyperoval cone K^{∗} and the flock F^{∗}. The Theorem after Remark 1.5.2 in [127] shows that the flock F defines the flock F^{∗}, so if C ∈ F then C ⊂ C^{∗} with C^{∗} ∈ F^{∗}. The converse is straightforward; remove the generator of K^{∗} that contains the point to be removed from the hyperoval to make it into an oval. This removes a point from each hyperoval of the flock F^{∗} to make a flock F. In short, when q is even, we can obtain flocks of K from K^{∗} and flocks of K^{∗} from K.

We can also understand flocks via the dual setting when q is even. Take a flock, F^{∗}, of a cone, K^{∗} with a hyperoval as its base in P G(2, q). Denote the planes that contain the elements of the flock π_{i} for i = 1, …, q, and the generators of K^{∗} denoted by L_{i} for i = 1, …, q+2. The generators L_{i} of K^{∗} have no three lying on a plane, this dualises to the lines L̂_{i} no three concurrent. The L̂_{i}’s lie in P G(2, q) since the L_{i}’s intersect at the vertex of K^{∗}. Now the points of P G(2, q) not incident with the lines L̂_{i} form a maximal (0, q/2)-arc, M, of size q(q − 1)/2 (see [49] for discussions about maximal arcs). The planes π_{i} dualise to the points π̂_{i} in P G(3, q) − P G(2, q). Each line containing π̂_{i} and π̂_{j} with i ≠ j intersects P G(2, q) in a point of the maximal arc M. So the flock F^{∗} is linear if and only if the q points π̂_{i} are collinear.

Theorem 206. Any flock of a cone in P G(3, 4) is linear.

Proof. Use the results in the dual setting and use the remarks on flocks of hyperovals being flocks of ovals.

6. Flocks of QuadraticCones

In this section we look at flocks of quadratic and translation oval cones settings in a more algebraic setting. We will also comment on the links between q-clans (see Section 9, p. 99, Chapter III) and flocks of quadratic cones and later of translation oval cones. We take the quadratic cone, K, in P G(3, q) to be the one defined by

xy = z^{2}.

Now the q planes, π_{i} with i = 1, …, q, of a flock F which do not contain the vertex (0, 0, 0, 1) of K can be described by the set of equations

a_{i}x + b_{i}y + c_{i}z + w = 0, for i = 1, …, q.

Theorem 207. Let T be a set of q ordered triples (a_{i}, b_{i}, c_{i}) with a_{i}, b_{i}, c_{i} ∈ GF (q).

If q is odd then,

(c_{i} − c_{j})^{2} − 4(a_{i} − a_{j})(b_{i} − b_{j}) = □̸, for i ≠ j

if and only if T describes a flock of a quadratic cone.

If q is even, with c_{i} ≠ c_{j} whenever i ≠ j, then

trace (\frac{(a_{i} + a_{j})(b_{i} + b_{j})}{(c_{i} + c_{j})^{2}}) = 1 for all i ≠ j

if and only if T describes a flock of a quadratic cone.

Proof. See [127].

Hence, C = \{(\begin{matrix}a_{i} & b_{i} \\ 0 & c_{i} \end{matrix}) | i ∈ GF (q)\} is a q–clan (Section 9, p. 99), if and only

if {a_{i}x + c_{i}y + b_{i}z + w = 0 | i ∈ GF (q)} is a flock of a quadratic cone, denoted F(C). We denote the translation plane from the flock F(C) by π(C).

Two flocks F = {π_{t} | t ∈ GF (q)} and F = {π′_{t} | t ∈ GF (q)} of a quadratic cone K are equivalent, or projectively equivalent, if there is an element of P Γ L(4, q) fixing K (so an element of P Γ O(4, q)) and mapping the set of planes {π_{t} | t ∈ GF (q)} onto the set of planes {π′_{t} | t ∈ GF (q)}. More formally:

Theorem 208 ([55]). The flocks F and F′ of a quadratic cone y^{2} = xz are equivalent if and only if there exists the following

(i) σ ∈ Aut GF (q), (ii) 0 ≠ u ∈ GF (q),

(iii) B = (\begin{matrix}a & c \\ b & d \end{matrix}) ∈ GL(2, q), and

(iv) a permutation of GF (q) π: t ↦ t̄, such that for some x_{0}, y_{0}, z_{0} ∈ GF (q) we have

(\begin{matrix}x′ ; t̄ \\ y′ ; t̄ \\ z′ ; t̄ \\ 1 \end{matrix}) = (\begin{matrix}ua^{2} & uab & ub^{2} & x_{0} \\ 0 & u(ad + bc) & 0 & y_{0} \\ uc^{2} & ucd & ud^{2} & z_{0} \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix}x_{t}^{σ} \\ y_{t}^{σ} \\ z_{t}^{σ} \\ 1 \end{matrix}) for all t.

It is of interest to the reader to compare the Fundamental Theorem of q–clan Geometry (page 106) to the above theorem.

7. Derivation of flocks of quadratic cones in P G(3, q), q odd

A BLT-set is a set of q + 1 (skew) lines in W (q) such that all lines of W (q) meets at most two of the lines of the set [7].

It has been shown (see for example [96]) that new flocks can be obtained from old, the fundamental object of study is a BLT-set in W (q).

We understand the construction of new flocks from old via the Knarr construction [24] where BLT-sets correspond to a generalised quadrangle, 𝒮 (q^{2}, q).

For q odd we have W (q)^{∧} ≅ Q(4, q). A BLT-set in Q(4, q) is a set, X, of q + 1 (pairwise non-perpendicular) points of Q(4, q) such that all points of Q(4, q) are perpendicular to at most two of the points of the set. Let x ∈ X. Then x^{⊥} is a quadratic cone K; and y^{⊥} ∩ x^{⊥} (for y ∈ X − {x}) is a flock F_{x} of K. Let y, z ∈ X − {x}, y ≁ x ≁ z, so y^{⊥} ∩ K is a conic of K, z^{⊥} ∩ K is a conic of K. We must show that y^{⊥} ∩ z^{⊥} ∩ K is an external line. For simplicity we assume that it is a line; if w ∈ y^{⊥} ∩ z^{⊥} ∩ K is singular, then {x, y, z} ⊆ w^{⊥} ∩ X a contradiction.

8. Known q–clans for q Even  147

The flock F_{x} gives a q-clan C_{x} so hence a generalised quadrangle GQ(C_{x}) ≅ 𝒮. For any two points, x, y, of a BLT-set we have GQ(C_{x}) ≅ GQ(C_{y}). We say F_{x} is derived from F_{y}.

If F is a flock of a quadratic cone K (with vertex x); embed K in the quadric P in P G(4, q). Each plane π in F is mapped under the polarity of P (in P G(4, q)) to a secant line l on x. Let y_{π} be the other singular point on l and let

X = {x} ∪ {y_{π} | π ∈ F}

Then X is a BLT-set.

From the results above and in Chapter III given a q–clan we can obtain from GQ(C) a flock F(C), but if GQ(C_{1}) ≅ GQ(C_{2}) this does not imply that F(C_{1}) ≅ F(C_{2}). This is the source of much subtlety.

8. Known q–clans for q Even

Since we have defined the links between herds, flocks and q-clans we now list the known q-clans and hence flocks of quadratic cones for q even.

The q–clan associated with the linear flocks [127] for even q:

A_{t} = (\begin{matrix}t & t \\ 0 & at \end{matrix}) , C_{1} :

where a ∈ GF (q) and trace(a) = 1. The herd H(C_{1}) consists of q+1 (nondegenerate) conics. The elation generalised quadrangle associated with this q–clan is isomorphic to H(3, q^{2}) [103].

The q–clan of Fisher-Thas-Walker-Kantor-Betten [40] [135] [65] [14] for q = 2^{e}, e odd:

A_{t} = (\begin{matrix}t & t^{2} \\ 0 & t^{3} \end{matrix}) . C_{2} :

Theorem 209. We have C_{2} is a q-clan for q = 2^{e}, e odd.

Proof. We need to show that for all x ≠ y we have

trace (\frac{(x + y)(x^{3} + y^{3})}{(x^{2} + y^{2})^{2}}) = 1.

Now

\frac{(x + y)(x^{3} + y^{3})}{(x^{2} + y^{2})^{2}} = \frac{x^{4} + xy^{3} + x^{3}y + y^{4}}{x^{4} + y^{4}} = 1 + \frac{xy^{3} + x^{3}y}{(x^{2} + y^{2})^{2}}

= 1 + \frac{xy(x^{2} + y^{2})}{(x^{2} + y^{2})^{2}} = 1 + \frac{xy}{x^{2} + y^{2}}

= 1 + \frac{x}{x + y} + \frac{x^{2}}{x^{2} + y^{2}} .

Since trace(\frac{x}{x+y} + \frac{x^{2}}{x^{2}+y^{2}} ) = 0 we have that

trace (\frac{(x + y)(x^{3} + y^{3})}{(x^{2} + y^{2})^{2}}) = trace(1).

But trace(1) = 1 if and only if q = 2^{e} with e odd.

The flock associated with this q–clan is linear when q = 2. The herd H(C_{2}) consists of q + 1 non-conical translation ovals if q > 2. The translation plane is due to Walker [135] for a similar q–clan for q odd (q ≡ −1 (mod 3)) and Betten [14] for q even. The flock is due to Thas and Fisher [40] and the generalised quadrangle due to Kantor [65].

The q–clan of Payne [93] for q = 2^{e}, e odd:

A_{t} = (\begin{matrix}t & t^{3} \\ 0 & t^{5} \end{matrix}) . C_{3} :

See Section 103, p. 49, for a proof that C_{3} is a q–clan. The flock associated with this q–clan is linear when q = 2. The herd H(C_{3}) consists of 2 Segre-Bartocci ovals and q − 1 Payne ovals, for q > 8. When q = 8, C_{3} is equivalent to C_{2}. The translation planes were found by Cohen and Ganley [31] deriving from a spread constructed by O. Barriga [11]. Kantor constructed a similar q–clan for q odd [67].

The q–clan, C_{4}, associated with the flock of De Clerck and Herssens [35] for q = 16. The herd H(C_{4}) consists of 17 Lunelli-Sce ovals (see Section 10.1, p. 158).

Payne [96] has shown that given an elation generalised quadrangle, GQ(C), associated with a q-clan C, it is possible to construct new flocks from GQ(C). This is, in some sense, the characteristic two analogue of derivation in the sense of BLT sets (see End Notes). Each of the above q-clans give rise to a unique flock except for the q-clan C_{3}. This non-linear flock, denoted F(C_{5}), of Payne [96] for q = 2^{e}, e > 3 odd, is constructed by recoordinatising one of the lines on the point (∞) of GQ(C_{3}) to obtain C_{5}. (Of course, GQ(C_{5}) is isomorphic to GQ(C_{3})).

There are some q-clans for q = 64, 256, 1024, 4096 and 16384 that appear in [109][101].

In earlier versions of [129] Thas gave as an open problem the construction of q–clans associated with non-linear flocks for q even, q square. The first example, F(C_{4}), of such a flock was found for q = 16 by De Clerck and Herssens [35]. We shall construct an infinite family of q-clans for all q even, which includes C_{4} for q = 16.

9. The Subiaco q–clans  149

9. The Subiaco q–clans

Let us recall the definition of T_{a}(f, g). For two permutation functions, f, g, defined on GF (q) with f(0) = g(0) = 0 and f(1) = g(1) = 1 with trace(a) = 1 then

A_{t} = (\begin{matrix}f(t) & t^{1/2} \\ 0 & g(t) \end{matrix})

is a q–clan if and only if

T_{a}(f, g) : trace (\frac{a(f(s) + f(t))(g(s) + g(t))}{s + t}) = 1 for all s ≠ t.

9.1. The case q = 2^{e} where e is odd.

Theorem 210. Let q = 2^{e}, e odd. Let

f(x) = \frac{x^{2} + x}{(x^{2} + x + 1)^{2}} + x^{1/2} and g(x) = \frac{x^{4} + x^{3}}{(x^{2} + x + 1)^{2}} + x^{1/2}.

Then

S′′ = \{A_{x} = (\begin{matrix}f(x) & x^{1/2} \\ 0 & g(x) \end{matrix}) | x ∈ GF (q)\}

is a q–clan.

Proof. We show that the matrices (\begin{matrix}f(x) & x^{1/2} \\ 0 & g(x) \end{matrix}), x ∈ GF (q), form a q–clan by

showing that (f(x) + f(y))(g(x) + g(y))/(x + y) has trace one for all x ≠ y; noting that trace(1) = 1 whenever q = 2^{e} for e odd. From now on we assume x ≠ y:

\frac{(f(x) + f(y))(g(x) + g(y))}{x + y}

= \frac{1}{x + y} (\frac{x^{2} + x}{(x^{2} + x + 1)^{2}} + x^{1/2} + \frac{y^{2} + y}{(y^{2} + y + 1)^{2}} + y^{1/2})

× (\frac{x^{4} + x^{3}}{(x^{2} + x + 1)^{2}} + x^{1/2} + \frac{y^{4} + y^{3}}{(y^{2} + y + 1)^{2}} + y^{1/2})

= \frac{1}{x + y} (\frac{(x^{2} + x)(x^{4} + x^{3})}{(x^{2} + x + 1)^{4}} + \frac{(x^{2} + x)(y^{4} + y^{3})}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}} + (x + y)^{1/2} \frac{x^{2} + x}{(x^{2} + x + 1)^{2}}

+ \frac{(y^{2} + y)(y^{4} + y^{3})}{(y^{2} + y + 1)^{4}} + \frac{(x^{4} + x^{3})(y^{2} + y)}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}} + (x + y)^{1/2} \frac{y^{2} + y}{(y^{2} + y + 1)^{2}}

+ (x + y)^{1/2} \frac{x^{4} + x^{3}}{(x^{2} + x + 1)^{2}} + (x + y)^{1/2} \frac{y^{4} + y^{3}}{(y^{2} + y + 1)^{2}} + (x + y)) .

We can express this last line as

A + B + 1

where

A = \frac{1}{x + y} (\frac{(x^{2} + x)(x^{4} + x^{3})}{(x^{2} + x + 1)^{4}} + \frac{(x^{2} + x)(y^{4} + y^{3})}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}

+ \frac{(y^{2} + y)(y^{4} + y^{3})}{(y^{2} + y + 1)^{4}} + \frac{(x^{4} + x^{3})(y^{2} + y)}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}} ) , and

B = \frac{1}{(x + y)^{1/2}} (\frac{x^{2} + x}{(x^{2} + x + 1)^{2}} + \frac{y^{2} + y}{(y^{2} + y + 1)^{2}} + \frac{x^{4} + x^{3}}{(x^{2} + x + 1)^{2}} + \frac{y^{4} + y^{3}}{(y^{2} + y + 1)^{2}} ) .

Hence we have “reduced” the problem to showing that A + B has trace zero, as trace(1) = 1 for e odd. Since all elements of trace zero are of the form X + X^{2}, this is equivalent to showing that A + B = X + X^{2} for some expression X. Since trace is additive, we have trace(A + B) = trace(A + B) + trace(B + B^{2}) = trace(A + B^{2}). (By showing A+B^{2} has trace zero, instead of A+B, we can eliminate the x^{1/2} terms from the latter.)

Now

A + B^{2} = \frac{1}{x + y} (\frac{(x^{2} + x)(x^{4} + x^{3})}{(x^{2} + x + 1)^{4}} + \frac{(x^{2} + x)(y^{4} + y^{3})}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}

+ \frac{(y^{2} + y)(y^{4} + y^{3})}{(y^{2} + y + 1)^{4}} + \frac{(x^{4} + x^{3})(y^{2} + y)}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}

+ \frac{x^{4} + x^{2}}{(x^{2} + x + 1)^{4}} + \frac{y^{4} + y^{2}}{(y^{2} + y + 1)^{4}} + \frac{x^{8} + x^{6}}{(x^{2} + x + 1)^{4}} + \frac{y^{8} + y^{6}}{(y^{2} + y + 1)^{4}} )

= \frac{1}{x + y} (\frac{x^{8} + x^{2}}{(x^{2} + x + 1)^{4}} + \frac{y^{8} + y^{2}}{(y^{2} + y + 1)^{4}}

+ \frac{(x^{2} + x)(y^{4} + y^{3}) + (x^{4} + x^{3})(y^{2} + y)}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}) .

Since x^{8} + x^{2} = (x^{4} + x^{2})(x^{2} + x + 1)^{2} all the terms can be placed over a common denominator, giving:

\frac{1}{x + y} (\frac{(x^{4} + x^{2})(y^{2} + y + 1)^{2} + (y^{4} + y^{2})(x^{2} + x + 1)^{2}}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}

+ .\frac{(x^{2} + x)(y^{4} + y^{3}) + (x^{4} + x^{3})(y^{2} + y)}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}) .

We expand, group, noting that x^{3} + y^{3} = (x + y)(x^{2} + xy + y^{2}), and divide by x + y to obtain:

\frac{(x + y)^{3} + x + y + x^{2}y^{2}(x + y) + x^{2}y^{2} + xy(x^{2} + xy + y^{2}) + xy(x + y)}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}.

With some cancellation we continue with:

\frac{x^{3} + y^{3} + x + y + x^{3}y^{2} + x^{2}y^{3} + x^{3}y + xy^{3}}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}

= \frac{(x + y)(x^{2} + x + 1)(y^{2} + y + 1) + (x + y)^{2}}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}}

= \frac{x + y}{(x^{2} + x + 1)(y^{2} + y + 1)} + \frac{(x + y)^{2}}{(x^{2} + x + 1)^{2}(y^{2} + y + 1)^{2}} .

Which is of the form X + X^{2} where

X = \frac{x + y}{(x^{2} + x + 1)(y^{2} + y + 1)}.

9.2. The case q = 4^{e}, where e is odd

Theorem 211. Let q = 4^{e}, e odd, with ω ∈ GF (q) satisfying ω^{2} + ω + 1 = 0. Let

f(x) = \frac{x^{2}(x^{2} + ωx + ω)}{(x^{2} + ωx + 1)^{2}} + ω^{2}x^{1/2}, and g(x) = \frac{ωx(x^{2} + x + ω^{2})}{(x^{2} + ωx + 1)^{2}} + ω^{2}x^{1/2}.

Then

S′ = \{A_{x} = (\begin{matrix}f(x) & x^{1/2} \\ 0 & ωg(x) \end{matrix}) | x ∈ GF (q)\}

is a q–clan. Proof. The proof is similar to that of Theorem 210, p. 149. For brevity we denote f and g by

f(x) = \frac{N_{f}(x)}{D(x)^{2}} + ω^{2}x^{1/2} and g(x) = \frac{N_{g}(x)}{D(x)^{2}} + ω^{2}x^{1/2},

where D(x) = x^{2} + ωx + 1, N_{f}(x) = x^{2}(x^{2} + ωx + ω), and N_{g}(x) = ωx(x^{2} + x + ω^{2}). We show that trace(ω(f(x) + f(y))(g(x) + g(y))/(x + y)) = 1 for all x ≠ y. From

now on we assume x ≠ y, so:

\frac{ω}{x + y} (f(x) + f(y))(g(x) + g(y))

= \frac{ω}{x + y} (\frac{N_{f}(x)}{D(x)^{2}} + ω^{2}x^{1/2} + \frac{N_{f}(y)}{D(y)^{2}} + ω^{2}y^{1/2}) (\frac{N_{g}(x)}{D(x)^{2}} + ω^{2}x^{1/2} + \frac{N_{g}(y)}{D(y)^{2}} + ω^{2}y^{1/2})

= \frac{ω}{x + y} (\frac{N_{f}(x)N_{g}(x)}{D(x)^{4}} + \frac{N_{f}(x)N_{g}(y) + N_{f}(y)N_{g}(x)}{D(x)^{2}D(y)^{2}} + \frac{N_{f}(y)N_{g}(y)}{D(y)^{4}})

+ \frac{1}{(x + y)^{1/2}} (\frac{N_{f}(x)}{D(x)^{2}} + \frac{N_{f}(y)}{D(y)^{2}} + \frac{N_{g}(x)}{D(x)^{2}} + \frac{N_{g}(y)}{D(y)^{2}} ) + ω^{2}.

We can express the last line as

A + B + ω^{2}

where

A = \frac{ω}{x + y} (\frac{N_{f}(x)N_{g}(x)}{D(x)^{4}} + \frac{N_{f}(x)N_{g}(y) + N_{f}(y)N_{g}(x)}{D(x)^{2}D(y)^{2}} + \frac{N_{f}(y)N_{g}(y)}{D(y)^{4}}) , and
B = \frac{1}{(x + y)^{1/2}} (\frac{N_{f}(x)}{D(x)^{2}} + \frac{N_{f}(y)}{D(y)^{2}} + \frac{N_{g}(x)}{D(x)^{2}} + \frac{N_{g}(y)}{D(y)^{2}} ) .

As ω^{2} has trace one in GF (q), q = 4^{e}, for any e odd, we have reduced the problem to showing that A + B has trace zero. As trace(A + B) = trace(A + B^{2}) this is equivalent to showing that A + B^{2} has trace zero:

A + B^{2} = \frac{1}{x + y} (\frac{ωN_{f}(x)N_{g}(x) + N_{f}(x)^{2} + N_{g}(x)^{2}}{D(x)^{4}}

+ \frac{ωN_{f}(y)N_{g}(y) + N_{f}(y)^{2} + N_{g}(y)^{2}}{D(y)^{4}} + \frac{ωN_{f}(x)N_{g}(y) + ωN_{f}(y)N_{g}(x)}{D(x)^{2}D(y)^{2}}) .

Now ωN_{f}(x)N_{g}(x)+N_{f}(x)^{2}+N_{g}(x)^{2} simplifies to x^{2}(x^{2}+ω^{2}x+1)(x^{2}+ωx+1)^{2} = x^{2}(x^{2} + ω^{2}x + 1)D(x)^{2}. Then the expression, placed over a common denominator D(x)^{2}D(y)^{2}, becomes:

\frac{1}{x + y} (\frac{x^{2}(x^{2} + ω^{2}x + 1)(y^{2} + ωy + 1)^{2} + y^{2}(y^{2} + ω^{2}y + 1)(x^{2} + ωx + 1)^{2}}{D(x)^{2}D(y)^{2}}

+ \frac{ω^{2}x^{2}y(x^{2} + ωx + ω)(y^{2} + y + ω^{2}) + ω^{2}y^{2}x(y^{2} + ωy + ω)(x^{2} + x + ω^{2})}{D(x)^{2}D(y)^{2}}) .

By expanding the terms we get

\frac{1}{x + y} (\frac{ω^{2}x^{4}y^{3} + ω^{2}x^{4}y^{2} + ωx^{4}y + ω^{2}x^{3}y + ω^{2}x^{2}y}{D(x)^{2}D(y)^{2}}

+ \frac{ω^{2}x^{3}y^{4} + ω^{2}x^{2}y^{4} + ωxy^{4} + ω^{2}xy^{3} + ω^{2}xy^{2}}{D(x)^{2}D(y)^{2}} ) ,

and then grouping to obtain

\frac{1}{x + y} (\frac{ω^{2}x^{3}y^{3}(x + y) + ω^{2}x^{2}y^{2}(x^{2} + y^{2}) + ωxy(x^{3} + y^{3})}{D(x)^{2}D(y)^{2}}

+ \frac{ω^{2}xy(x^{2} + y^{2}) + ω^{2}xy(x + y)}{D(x)^{2}D(y)^{2}}) .

Divide by x + y and simplify to get:

\frac{x^{3}y^{2} + x^{2}y^{3} + ωx^{3}y + ωx^{2}y + ωxy^{3} + ωxy^{2} + x^{3} + ω^{2}x^{2} + x + y^{3} + ω^{2}y^{2} + y}{(x^{2} + ωx + 1)^{2}(y^{2} + ωy + 1)^{2}}

= \frac{(x + y)(x^{2} + ωx + 1)(y^{2} + ωy + 1) + x^{2} + y^{2}}{(x^{2} + ωx + 1)^{2}(y^{2} + ωy + 1)^{2}}

= \frac{x + y}{(x^{2} + ωx + 1)(y^{2} + ωy + 1)} + \frac{(x + y)^{2}}{(x^{2} + ωx + 1)^{2}(y^{2} + ωy + 1)^{2}} .

Which is of the form X + X^{2}.

9. The Subiaco q–clans  153

9.3. The case q = 16^{e} where e is odd

Theorem 212. Let q = 16^{e}, e odd, with η ∈ GF (q) satisfying η^{4} = η + 1. Let

f(x) = \frac{η^{13}x^{4} + x^{3} + η^{2} + η^{6}x}{(x^{2} + η^{3}x + 1)^{2}} + η^{12}x^{1/2}
g(x) = \frac{η^{3}x^{4} + η^{7}x^{3} + η^{4}x^{2} + η^{12}x}{(x^{2} + η^{3}x + 1)^{2}}.

Then

S′′′ = \{A_{x} = (\begin{matrix}f(x) & x^{1/2} \\ 0 & η^{14}g(x) \end{matrix}) | x ∈ GF (q)\}

is a q–clan.

Proof. The proof is similar to that of Theorem 210, p. 149 and Theorem 211, p. 151. For brevity we denote f and g by

f(x) = \frac{N_{f}(x)}{D(x)^{2}} + η^{12}x^{1/2} and g(x) = \frac{N_{g}(x)}{D(x)^{2}} + η^{13}x^{1/2},

where

D(x) = x^{2} + η^{3}x + 1,
N_{f}(x) = η^{13}x^{4} + x^{3} + η^{2}x^{2} + η^{6}x, and
N_{g}(x) = η^{3}x^{4} + η^{7}x^{3} + η^{4}x^{2} + η^{12}x.

We show that trace(η^{14}(f(x)+f(y))(g(x)+g(y))/(x+y)) = 1 for all x ≠ y. From now on we assume x ≠ y, so:

\frac{η^{14}}{x + y} (f(x) + f(y))(g(x) + g(y))

= \frac{η^{14}}{x + y} (\frac{N_{f}(x)}{D(x)^{2}} + η^{12}x^{1/2} + \frac{N_{f}(y)}{D(y)^{2}} + η^{12}y^{1/2}) (\frac{N_{g}(x)}{D(x)^{2}} + η^{13}x^{1/2} + \frac{N_{g}(y)}{D(y)^{2}} + η^{13}y^{1/2})

= \frac{η^{14}}{x + y} (\frac{N_{f}(x)N_{g}(x)}{D(x)^{4}} + \frac{N_{f}(x)N_{g}(y) + N_{f}(y)N_{g}(x)}{D(x)^{2}D(y)^{2}} + \frac{N_{f}(y)N_{g}(y)}{D(y)^{4}})

+ \frac{η^{12}}{(x + y)^{1/2}} (\frac{N_{f}(x)}{D(x)^{2}} + \frac{N_{f}(y)}{D(y)^{2}}) + \frac{η^{11}}{(x + y)^{1/2}} (\frac{N_{g}(x)}{D(x)^{2}} + \frac{N_{g}(y)}{D(y)^{2}} ) + η^{9}.

We can express the last line as

A + B + η^{9}

where

A = \frac{η^{14}}{x + y} (\frac{N_{f}(x)N_{g}(x)}{D(x)^{4}} + \frac{N_{f}(x)N_{g}(y) + N_{f}(y)N_{g}(x)}{D(x)^{2}D(y)^{2}} + \frac{N_{f}(y)N_{g}(y)}{D(y)^{4}}) , and
B = \frac{η^{12}}{(x + y)^{1/2}} (\frac{N_{f}(x)}{D(x)^{2}} + \frac{N_{f}(y)}{D(y)^{2}}) + \frac{η^{11}}{(x + y)^{1/2}} (\frac{N_{g}(x)}{D(x)^{2}} + \frac{N_{g}(y)}{D(y)^{2}} ) .

As η^{9} has trace one in GF (q), q = 16^{e}, for any e odd, we have reduced the problem to showing that A + B has trace zero. As trace(A + B) = trace(A + B^{2}) this is equivalent to showing that A + B^{2} has trace zero:

A + B^{2} = \frac{1}{x + y} (\frac{η^{14}N_{f}(x)N_{g}(x) + η^{9}N_{f}(x)^{2} + η^{7}N_{g}(x)^{2}}{D(x)^{4}}

+ \frac{η^{14}N_{f}(y)N_{g}(y) + η^{9}N_{f}(y)^{2} + η^{7}N_{g}(y)^{2}}{D(y)^{4}} + \frac{η^{14}N_{f}(x)N_{g}(y) + η^{14}N_{f}(y)N_{g}(x)}{D(x)^{2}D(y)^{2}}) .

Now η^{14}N_{f}(x)N_{g}(x)+η^{9}N_{f}(x)^{2}+η^{9}N_{g}(x)^{2} simplifies to (η^{9}x^{4}+η^{10}x^{3}+η^{9}x^{2})D(x)^{2}. Then the above expression, placed over a common denominator D(x)^{2}D(y)^{2}, becomes:

\frac{1}{x + y} (\frac{η^{9}x^{4}y^{2} + η^{12}x^{4}y + η^{12}x^{3}y + η^{9}x^{2}y^{4} + η^{12}x^{2}y^{3} + η^{10}x^{2}y + η^{12}xy^{4} + xy^{3}}{D(x)^{2}D(y)^{2}}

+ \frac{η^{10}xy^{2} + η^{9}x^{4} + ηx^{3} + η^{9}x^{2} + η^{9}y^{4} + η^{10}y^{3} + η^{9}y^{2}}{D(x)^{2}D(y)^{2}} ) .

We group terms together in such a way that we can divide by (x + y) to get:

\frac{η^{9}x^{3}y^{2} + η^{9}x^{2}y^{3} + η^{12}x^{3}y + η^{12}xy^{3} + η^{7}x^{2}y + η^{7}xy^{2}}{D(x)^{2}D(y)^{2}}

+ \frac{η^{9}x^{3} + η^{9}y^{3} + η^{10}x^{2} + η^{10}y^{2} + η^{9}x + η^{9}y}{D(x)^{2}D(y)^{2}}

= \frac{η^{9}x + η^{9}y}{D(x)D(y)} + \frac{η^{3}x^{2} + η^{3}y^{2}}{D(x)^{2}D(y)^{2}} .

Which is of the form X + X^{2} so has trace zero.

9.4. Existence of Subiaco q–clans, in general

Theorem 213. Let d ∈ GF (q), q even, such that d^{2} +d+1 ≠ 0 and trace(1/d) = 1. Let

a = \frac{d^{2} + d^{5} + d^{1/2}}{d(1 + d + d^{2})} ,
f(x) = \frac{d^{2}(x^{4} + x) + d^{2}(1 + d + d^{2})(x^{3} + x^{2})}{(x^{2} + dx + 1)^{2}} + x^{1/2}, and
g(x) = \frac{d^{4}x^{4} + d^{3}(1 + d^{2} + d^{4})x^{3} + d^{3}(1 + d^{2})x}{(d^{2} + d^{5} + d^{1/2})(x^{2} + dx + 1)^{2}} + \frac{d^{1/2}}{d^{2} + d^{5} + d^{1/2}} x^{1/2}.

Then

S = S_{d} = \{A_{x} = (\begin{matrix}f(x) & x^{1/2} \\ 0 & ag(x) \end{matrix}) | x ∈ GF (q)\}

is a q–clan.

Proof. This proof is similar to the proof of Theorem 210, p. 149, and the proof of Theorem 211, p. 151, and the proof of Theorem 212, p. 153.

We start by showing trace(a) = 1 for all q even:

trace(a) = trace(a^{2})

= trace (\frac{d^{9} + d^{3} + 1}{d(d^{2} + d + 1)^{2}} )

= trace (\frac{1}{d}) + trace (d + \frac{d^{4}}{d^{2} + d + 1}) + trace (d^{2} + \frac{d^{8}}{(d^{2} + d + 1)^{2}} ) .

So trace(a) = trace(1/d) = 1.

In this form the equations will become quite unwieldy, so initially we will simplify f and g to

f(x) = \frac{N_{f}(x)}{D(x)^{2}} + x^{1/2} and g(x) = \frac{1}{d^{2} + d^{5} + d^{1/2}} \frac{N_{g}(x)}{D(x)^{2}} + \frac{d^{1/2}}{d^{2} + d^{5} + d^{1/2}} x^{1/2}.

We now show for GF (q), q = 2^{e}, that the matrices (\begin{matrix}f(t) & t^{1/2} \\ 0 & ag(t) \end{matrix}), t ∈ GF (q),

form a q–clan. We do this by showing that T_{a}(f, g) is true. That is we show a(f(x) + f(y))(g(x) + g(y))/(x + y) has trace one for all x ≠ y. From now on we assume that x ≠ y, so:

\frac{(f(x) + f(y))(ag(x) + ag(y))}{x + y}

= \frac{1}{x + y} (\frac{N_{f}(x)}{D(x)^{2}} + x^{1/2} + \frac{N_{f}(y)}{D(y)^{2}} + y^{1/2}) (\frac{N_{g}(x)}{d(1 + d + d^{2})D(x)^{2}} + \frac{d^{1/2}}{d(1 + d + d^{2})}x^{1/2}

+ \frac{N_{g}(y)}{d(1 + d + d^{2})D(y)^{2}} + \frac{d^{1/2}}{d(1 + d + d^{2})}y^{1/2})

= \frac{1}{x + y} (\frac{N_{f}(x)N_{g}(x)}{d(1 + d + d^{2})D(x)^{4}} + \frac{N_{f}(x)N_{g}(y)}{d(1 + d + d^{2})D(x)^{2}D(y)^{2}} + \frac{(x + y)^{1/2}}{d^{1/2}(1 + d + d^{2})} \frac{N_{f}(x)}{D(x)^{2}}

+ \frac{N_{f}(y)N_{g}(y)}{d(1 + d + d^{2})D(y)^{4}} + \frac{N_{f}(y)N_{g}(x)}{d(1 + d + d^{2})D(x)^{2}D(y)^{2}} + \frac{(x + y)^{1/2}}{d^{1/2}(1 + d + d^{2})} \frac{N_{f}(y)}{D(y)^{2}}

+ \frac{(x + y)^{1/2}}{d(1 + d + d^{2})} \frac{N_{g}(x)}{D(x)^{2}} + \frac{(x + y)^{1/2}}{d(1 + d + d^{2})} \frac{N_{g}(y)}{D(y)^{2}} + \frac{x + y}{d^{1/2}(1 + d + d^{2})}) .

If we let E_{1}, …, E_{8} correspond to the first eight terms inside the brackets of the above expression, we can express the last line as

\frac{1}{x + y} (E_{1} + E_{2} + E_{3}

+ E_{4} + E_{5} + E_{6}

+ E_{7} + E_{8} + \frac{x + y}{d^{1/2}(1 + d + d^{2})}) .

We will do a further substitution on the above line to obtain

A + B + \frac{1}{d^{1/2}(1 + d + d^{2})}

where

A = \frac{1}{x + y} (E_{1} + E_{2} + E_{4} + E_{5}) , and
B = \frac{1}{x + y} (E_{3} + E_{6} + E_{7} + E_{8}) .

Now

trace (\frac{1}{d^{1/2}(1 + d + d^{2})}) = trace (\frac{1}{d(1 + d^{2} + d^{4})})

= trace (\frac{1}{d}) + trace (\frac{d + d^{3}}{1 + d^{2} + d^{4}} )

= trace (\frac{1}{d}) + trace (\frac{d}{1 + d + d^{2}} ) + trace (\frac{d^{2}}{1 + d^{2} + d^{4}} )

= trace (\frac{1}{d}) = 1.

So it is now left to show that A + B has trace zero. Since trace(A + B) = trace(A + B^{2}), this is equivalent to showing that A + B^{2} has trace zero.

A + B^{2} = \frac{1}{x + y} (E_{1} + E_{2} + E_{4} + E_{5} + \frac{N_{f}(x)^{2}}{d(1 + d^{2} + d^{4})D(x)^{4}}

+ \frac{N_{f}(y)^{2}}{d(1 + d^{2} + d^{4})D(y)^{4}} + \frac{N_{g}(x)^{2}}{d^{2}(1 + d^{2} + d^{4})D(x)^{4}} + \frac{N_{g}(y)^{2}}{d^{2}(1 + d^{2} + d^{4})D(y)^{4}} )

= \frac{1}{x + y} (E_{2} + E_{5} + \frac{d(1 + d + d^{2})N_{f}(x)N_{g}(x) + dN_{f}(x)^{2} + N_{g}(x)^{2}}{d^{2}(1 + d^{2} + d^{4})D(x)^{4}}

+ \frac{d(1 + d + d^{2})N_{f}(y)N_{g}(y) + dN_{f}(y)^{2} + N_{g}(y)^{2}}{d^{2}(1 + d^{2} + d^{4})D(y)^{4}} ) .

After much simplification we obtain:

\frac{1}{x + y} (E_{2} + E_{5} + \frac{d^{5}(1 + d + d^{2})x^{8} + d^{6}(1 + d^{6})x^{7} + d^{5}(1 + d^{6})x^{6} + d^{8}(1 + d^{6})x^{5}}{d^{2}(1 + d^{2} + d^{4})D(x)^{4}}

+ \frac{d^{5}(1 + d^{6})x^{4} + d^{6}(1 + d^{6})x^{3} + d^{5}(1 + d + d^{2})^{2}x^{2}}{d^{2}(1 + d^{2} + d^{4})D(x)^{4}}

+ \frac{d^{5}(1 + d + d^{2})y^{8} + d^{6}(1 + d^{6})y^{7} + d^{5}(1 + d^{6})y^{6} + d^{8}(1 + d^{6})y^{5}}{d^{2}(1 + d^{2} + d^{4})D(y)^{4}}

+ \frac{d^{5}(1 + d^{6})y^{4} + d^{6}(1 + d^{6})y^{3} + d^{5}(1 + d + d^{2})^{2}y^{2}}{d^{2}(1 + d^{2} + d^{4})D(y)^{4}} ) .

Using 1 + d^{6} = (1 + d^{2} + d^{4})(1 + d^{2}) with more simplification, dividing by D(x)^{2} (or D(y)^{2}), and then placing over a common denominator, we obtain:

\frac{1}{x + y} (E_{2} + E_{5} + \frac{d^{3}x^{8} + d^{4}(1 + d^{2})x^{7} + d^{3}(1 + d^{2})x^{6} + d^{6}(1 + d^{2})x^{5}}{D(x)^{4}}

+ \frac{d^{3}(1 + d^{2})x^{4} + d^{4}(1 + d^{2})x^{3} + d^{3}x^{2}}{D(x)^{4}}

+ \frac{d^{3}y^{8} + d^{4}(1 + d^{2})y^{7} + d^{3}(1 + d^{2})y^{6} + d^{6}(1 + d^{2})y^{5}}{D(y)^{4}}

+ \frac{d^{3}(1 + d^{2})y^{4} + d^{4}(1 + d^{2})y^{3} + d^{3}y^{2}}{D(y)^{4}} )

= \frac{1}{x + y} (E_{2} + E_{5} + \frac{d^{3}(x^{4} + d(1 + d^{2})x^{3} + x^{2})D(x)^{2}}{D(x)^{4}}

+ \frac{d^{3}(y^{4} + d(1 + d^{2})y^{3} + y^{2})D(y)^{2}}{D(y)^{4}} )

= \frac{1}{x + y} (\frac{N_{f}(x)N_{g}(y)}{d(1 + d + d^{2})D(x)^{2}D(y)^{2}} + \frac{N_{f}(y)N_{g}(x)}{d(1 + d + d^{2})D(x)^{2}D(y)^{2}}

+ \frac{d^{3}(x^{4} + d(1 + d^{2})x^{3} + x^{2})}{D(x)^{2}} + \frac{d^{3}(y^{4} + d(1 + d^{2})y^{3} + y^{2})}{D(y)^{2}})

= \frac{1}{x + y} (\frac{d^{5}(1 + d^{3})(1 + d)x^{4}y^{3} + d^{6}(1 + d + d^{2})x^{4}y^{2} + d^{5}(1 + d + d^{2})x^{4}y}{d(1 + d + d^{2})D(x)^{2}D(y)^{2}}

+ \frac{d^{5}(1 + d^{3})(1 + d)x^{3}y^{4} + d^{6}(1 + d + d^{2})x^{2}y^{4} + d^{5}(1 + d + d^{2})xy^{4}}{d(1 + d + d^{2})D(x)^{2}D(y)^{2}}

+ \frac{d^{5}(1 + d + d^{2})^{3}x^{3}y^{2} + d^{6}(1 + d + d^{2})x^{3}y + d^{5}(1 + d)(1 + d^{3})xy^{2}}{d(1 + d + d^{2})D(x)^{2}D(y)^{2}}

+ \frac{d^{5}(1 + d + d^{2})^{3}x^{2}y^{3} + d^{6}(1 + d + d^{2})xy^{3} + d^{5}(1 + d^{3})(1 + d)x^{2}y}{d(1 + d + d^{2})D(x)^{2}D(y)^{2}}

+ \frac{d^{3}(x^{4} + d(1 + d^{2})x^{3} + x^{2})D(y)^{2}}{D(x)^{2}D(y)^{2}} + \frac{d^{3}(y^{4} + d(1 + d^{2})y^{3} + y^{2})D(x)^{2}}{D(x)^{2}D(y)^{2}} ) .

After some substantial simplification, dividing by d(1 + d + d^{2}) where appropriate and grouping, we obtain:

\frac{1}{x + y} (\frac{d^{3}x^{2}y^{2}(x^{2} + y^{2}) + d^{4}x^{2}y^{2}(x + y) + d^{4}xy(x^{3} + y^{3}) + d^{5}xy(x^{2} + y^{2})}{D(x)^{2}D(y)^{2}}

+ \frac{d^{4}(1 + d^{2})xy(x + y) + d^{3}(x^{4} + y^{4}) + d^{3}(x^{2} + y^{2}) + d^{4}(1 + d^{2})(x^{3} + y^{3})}{D(x)^{2}D(y)^{2}}) .

We divide by x + y and collect terms to obtain:

\frac{d^{4}x^{3}y + d^{4}xy^{3} + (d^{5} + d^{3})x^{2}y + (d^{5} + d^{3})xy^{2} + d^{3}x^{3}y^{2} + d^{3}x^{2}y^{3}}{D(x)^{2}D(y)^{2}}

+ \frac{d^{3}x^{3} + d^{3}y^{3} + d^{4}(1 + d^{2})x^{2} + d^{4}(1 + d^{2})y^{2} + d^{3}x + d^{3}y}{D(x)^{2}D(y)^{2}}

= \frac{(d^{3}x + d^{3}y)(x^{2} + dx + 1)(y^{2} + dy + 1) + d^{6}x^{2} + d^{6}y^{2}}{(x^{2} + dx + 1)^{2}(y^{2} + dy + 1)^{2}}

= \frac{d^{3}x + d^{3}y}{(x^{2} + dx + 1)(y^{2} + dx + 1)} + \frac{d^{6}x^{2} + d^{6}y^{2}}{(x^{2} + dx + 1)^{2}(y^{2} + dy + 1)^{2}} .

Which is of the form X + X^{2}.

We call S the Subiaco q–clan. We call the ovals of H(S) the Subiaco ovals and the resulting hyperovals the Subiaco hyperovals. We also call the flocks F(S) the Subiaco flocks, GQ(S) the Subiaco elation generalised quadrangles, and π(S) the Subiaco translation planes.

The q–clan S′ is a Subiaco q–clan for q = 4^{e}, e odd. Hence the herd of ovals H(S′), the flocks of the quadratic cone F(S′), the elation generalised quadrangles GQ(S′), and the translation planes π(S′) from the q–clan S′ are all Subiaco, for q = 4^{e}, e odd. Similarly for S′′′. For q = 2^{e}, where e is odd, we can let d = 1, hence we find S′′ = S_{1}. This gives a family of o-polynomials of the herd H(S_{1}) over GF (2) for q = 2^{e}, e odd.

The construction of S′′ for q = 2^{e}, e odd was the first Subiaco q–clan to be found. This was followed by the constructions of S′ for q = 4^{e}, e odd, and S′′′ for q = 16^{e}, e odd. From these three constructions it was possible to generalise to construct S.

10. Concluding Remarks

10.1. The Subiaco q–clans

In Chapter III it is shown that if d and d′ are elements of GF (q) with trace(1/d) = trace(1/d′) = 1, for q = 2^{e}, then S_{d} is equivalent to S_{d′}. In Chapter III it is shown that S is equivalent to S′, for q = 2^{e}, e ≡ 2 (mod 4), e ≠ 2 and the automorphism group of GQ(S) is calculated. For q = 2, 4, GQ(S) ≅ H(3, q^{2}). For q = 8, GQ(S) ≅ GQ(C_{2}). For q = 16, GQ(S) ≅ GQ(C_{4}) by results of [35]. For q ≥ 32, GQ(S) is new, (although for q = 32, 64, 128, 256, they appear as computer results of Penttila and Royle [111]). This can be seen from the automorphism groups. Alternatively, no previously known q–clans C gave rise to a generalised quadrangle GQ(C) with subquadrangle on (∞) and (0, 0, 0) isomorphic to T_{2}(O), for O a Subiaco oval. This follows from the results on the Subiaco hyperovals that follow in Section 8.4 for q ≥ 64. For q = 32, note that while the Subiaco hyperovals are Payne hyperovals, the ovals of the Subiaco herd, H(S), are not equivalent to the ovals of the Payne herd, H(C_{3}).

As H(S) contains no ovals that give rise to regular hyperovals, by [47, 80] we have all the ovals being from Lunelli-Sce hyperovals for q = 16. Since the Subiaco 16–clan is equivalent to C_{4}, it follows that H(C_{4}) consists of 17 Lunelli-Sce ovals.

10.2. The Subiaco flocks

In Chapter III it is shown that each Subiaco q–clan, S, gives rise to exactly one Subiaco flock of a quadratic cone in P G(3, q), up to isomorphism, by showing that the automorphism group of GQ(S) is transitive on the lines on (∞). This also determines the stabiliser in P Γ L(4, q) of the Subiaco flock in P G(3, q).

10.3. The Subiaco planes

In Section 7 of [6] the automorphism groups of the Subiaco planes π(S) are studied. The planes appear to be new for q ≥ 16.

10.4. The Subiaco hyperovals

In Chapter III it is shown that all Subiaco hyperovals in P G(2, q) are equivalent for q = 2^{e}, e ≢ 2 (mod 4). Also it is shown that there are two orbits in P G(2, q), for q = 2^{e}, e ≡ 2 (mod 4).

For q = 2, 4, 8, the Subiaco hyperovals are regular. For q = 16, they are Lunelli-Sce hyperovals [75]. For q = 32, they are Payne hyperovals. For q = 64, they are the hyperovals discovered by Penttila and Pinneri [107], with groups of orders 15 and 60. For q = 128, 256, they are the hyperovals discovered by Penttila and Royle [111].

In [87], it is shown that the stabiliser in P Γ L(3, q) of a Subiaco hyperoval in P G(2, q) is cyclic of order 2e, for q = 2^{e}, e ≢ 2 (mod 4). In Chapter III the stabilisers in P Γ L(3, q) of the Subiaco hyperovals in P G(2, q) for q = 2^{e}, e ≡ 2 (mod 4) are computed.

It is important to see how elements of the Subiaco herd intersect particular subplanes:

(i) For q = 4^{e}, e odd, if f ∈ H(S′) then D(f) ∩ P G(2, 4) is a regular hyperoval. (ii) For q = 2^{e}, e odd, if f ∈ H(S′′) then D(f) ∩ P G(2, 2) is the fundamental

quadrangle. (iii) For q = 16^{e}, e odd, if f ∈ H(S′′′) then D(f) ∩ P G(2, 16) is a Lunelli-Sce

hyperoval.

11. History of the Subiaco Constructions

We end this work with a story!

In 1992 I wanted to transfer from a Masters to a PhD. In doing so I needed to expand my mathematical horizons. My supervisor, T. Penttila, thought that the connection between hyperovals and GQs should be investigated first; since it seemed to be a natural extension of my work on hyperovals. (Always in the back of our minds were the possibilities of generalising the irregular hyperovals H_{60} and H_{15} of P G(2, 64).) We decided that the first paper to understand was the paper [93] by S. Payne. It seemed to be a ground breaking paper and was virtually a text for my initial work in this new area. Of course, at the time we didn’t know how influential the paper would be. I should also mention that in 1992 J. A. Thas gave the newly formed Combinatorial Computing Research Group (CCRG) a series of lectures on flocks of quadratic cones. The trace equation entered our minds and stayed there, in the background, until it was needed in a few months time. We decided the ideas of [93] needed to be simplified. This simplification introducted of the concept of herds — hyperovals from EGQs of order (q^{2}, q) — and the initial connection between flocks of quadratic cones. The concept of herd was refined to the point that we now had a more efficient way of searching for flocks of quadratic cones in P G(3, q) for small q. By January 1993 these ideas allowed Penttila and Royle to verify that the only non-linear flock in P G(3, 16) was the one constructed by De Clerck and Herssens. Also early in 1993 Penttila and Royle constructed a new flock in P G(3, 32) (later to be shown to be part of the Subiaco family) and two new flocks in P G(3, 64); one related to the hyperovals H_{60} and H_{15} of P G(2, 64) and one related to the Penttila-Royle hyperoval H_{12} of P G(2, 64). The numerology of the flock related to H_{60} and H_{15} forced us to look in P G(2, 4^{e}), e odd for any possible generalisations of the hyperovals. (For the later part of 1991 and most of 1992, Penttila and myself were certain that any generalisations of H_{60} and H_{15} must be in P G(2, q^{3}); we (vainly) looked at the case for q = 512 = 8^{3} instead of q = 1024 = 4^{5}.) Penttila and Royle then showed that any new flocks of P G(3, 64) must arise from new hyperovals in P G(2, 64).

In Feburary of 1994 we studied the o-polynomials of H_{60} over GF (4) and this was when we first noticed the pseudo-periodicity of the coefficients of the o-polynomial. With this information about the coefficients of the o-polynomial Penttila and myself constructed a new hyperoval in P G(2, 1024) with group of order 100. Penttila and Royle also classified all flocks of P G(3, 32) (using the classification of all hyperovals in P G(2, 32)).

In March 1993 Penttila and Royle constructed a new hyperoval in P G(2, 128) stabilised by an element of order 14 (this hyperoval was later to be part of the Subiaco family). This hyperoval also had pseudo-periodicity. This lead to a new construction of a flock in P G(2, 128) with the related o-polynomial (the “g”) also exhibiting pseudo-periodicity.

Around the beginning of the year W. Cherowitzo arrived in Perth from Denver, Colorado. Cherowitzo introduced the link between the pseudo-periodic o-polynomials and rational functions. By the end of March W. Cherowitzo proved the existence of a infinite family of flocks, hyperovals, etc. for q even but not a square (later to be part of the Subiaco family). Earlier it was noted that some of the o-polynomials of Payne’s hyperovals were pseudo-periodic, and together with the new hyperovals of P G(2, 128) the conjecture for the right form of the q non-square o-polynomials was found by Cherowitzo.

In April, Penttila and Royle constructed two new hyperovals in P G(3, 256). Penttila conjectured that a new family of hyperovals, flocks, translation planes, etc. existed for all q even and parameterised by an element of trace one, and that x^{4} + d^{2}x^{2} + 1 would be the denominator for the rational o-polynomials with trace(1/d) = 1. Cherowitzo continued his tradition of naming configurations from local names. The term Subiaco (from Subiaco Oval, the football stadium in the suburb Subiaco near the University of Western Australia and having as a sister city Subiaco, Italy) came to name the infinite family of hyperovals etc, firstly for the q non square case and then for the (then conjectured) q even infinite family. This conjectured infinite family of hyperovals would include all the hyperovals found to date except H_{12}. Penttila and Royle constructed a new flock in P G(3, 512) believing it to be in the Subiaco family and two new flocks in P G(3, 256).

In May Penttila and I found the “correct” rational function forms for the opolynomials of H_{60} and H_{15}. This then lent to the proof of an infinite family of hyperovals, flocks, translation planes, etc for q = 4^{e}, e odd, so generalising the hyperovals that we found in 1991! Penttila then generalised the De Clerck-Herssens flock, so generalising the Lunelli-Sce hyperoval of discovered in 1958, for q = 16^{e}, e odd.

Meanwhile, Cherowitzo, using his work on hyperovals of P G(2, 16) found the general form of the Subiaco o-polynomials and in doing so allowed for the proof of an infinite family of hyperovals, flocks etc for all q even, to be produced. (The proof was pieced together a few months later in Perth.)

It must be noted that this work was a truly collaborative effort. A special mention should go to G. Royle for his computational work that allowed the conjectures to be tested, corrected or discarded as soon as we thought of them.

12. End Notes

Notes on Section 1, p. 141. The term flock was popularised by Dembowski [36] in relation to inversive planes.

The results in this Section are well known.

Notes on Section 2, p. 141. The results in this Section are well known.

Notes on Section 3, p. 143. The results in this Section are well known.

Notes on Section 4, p. 143. There are two isometry classes of conics in P G(2, q), q odd, (but only one class up to similarity). When extended to GF (q)^{4} there are still two isometry classes of conics in O^{+}(4, q) giving the two classes in Thas’ construction of the Thas flocks.

The results in this Section are well known.

Notes on Section 5, p. 144. The results in this Section are well known.

Notes on Section 7, p. 146. Let C = {A_{t} | t ∈ GF (q)} be a q–clan. Fix s and let B_{t} = A_{t} − A_{s}. Let d_{t} = det(B_{t} + B_{t}^{T}) for t ≠ 0 and let d_{0} = 1. Let C_{t} = B_{t}/d_{t} and C′ = {C_{t} | t ∈ GF (q)}. Then C′ is a q–clan and corresponds to a derivation of a flock of the original flock F(C) [104]. The q choices of s give the q derived flocks. The above derivation is true for q even or odd, though there is no BLT setting for q even. (These matrices C_{t} are formed by hitting C with the map φ on page 107, the flip map.)

Let X be a BLT-set. The orbits of P Γ O(5, q)_{X} on X correspond to the isomorphism classes of flocks. Also, for x ∈ X, P Γ O(5, q)_{X,x} is isomorphic to the group of the flock corresponding to x.

The results in this Section are well known.

Notes on Section 8, p. 147. The results in this Section are well known.

Notes on Section 9, p. 149. The theorems in this Section appear in [30]. Theorem 210, p. 149, is by Cherowitzo; Theorem 211, p. 151, is by the author and Penttila; Theorem 212, p. 153, is by Penttila; and Theorem 213, p. 154, is by the author with initial work by Cherowitzo.

Notes on Section 10, p. 158. The results in this Section appear in [30, 100, 6].

CHAPTER V Open Problems

Standard open problems for configurations in projective geometries are:

• Classify all ovoids of P G(3, q) for q even.

• Classify all ovals of P G(2, q) for q even.

• Classify all flocks of a quadratic cone of P G(3, q).

BLT sets give a geometric interpretation of a GQ(C) arising from a flock F(C) for q odd.

• Is there a geometric interpretation of the GQ(C) arising from a flock F(C) for q even?

• Is every TGQ of order q isomorphic to a T_{2}(O) for some oval O?

• Is every TGQ of order (q, q^{2}), q even, isomorphic to a T_{3}(Ω) for some ovoid Ω?

At the time of writing W. Cherowitzo proved that his hyperovals form an infinite family using flocks of a translation oval cone, hence extending the link between hyperovals and flocks of cones.

• Is there a link between flocks of translation oval cones and GQs?

• Is the O’Keefe-Penttila hyperoval related to flocks of a translation oval cone? Which leads us to more questions on the O’Keefe-Penttila hyperoval:

• Does the O’Keefe-Penttila hyperoval live in an infinite family?

• Does it have an associated algebraic curve?

• Is normalising the q–clan in a different way a possible way of generalising this

hyperoval? It should be noted that it took over 35 years to generalise the Lunelli-Sce hyperoval.

Let Ω be an ovoid of P G(3, q) and let O = Ω ∩ π where π is a secant plane to Ω. Then T_{2}(O) is a subquadrangle of T_{3}(Ω).

• Given a T_{2}(O) when is it a subGQ for a T_{3}(Ω), for some ovoid Ω?

• Could progress be made on classifying ovoids in P G(3, q) by investigating sufficient conditions for a GQ of order (q^{2}, q) to be an EGQ, especially a GQ admitting a 3-regular point.

Determining all types of the above embedding is equivalent to determining all secant plane sections of ovoids.

• Can a herd be characterised by some of the ovals it contains?

• Similarly, can a GQ of order (q^{2}, q), q even, be characterised by the subquadrangles of the form T_{2}(O) it contains?

• For each flock for q even there corresponds an analogous flock for q odd, except for the Subiaco flock. Is there one for Subiaco?

• What is the connection of the use of the trace equation in [108] and the trace condition for q–clans.

• Are the cyclic q–clans for q = 4^{e}, e = 3, 4, 5, 6, 7, part of an infinite family?

• Classify all cyclic q–clans for q even.

Bibliography

[1] B. A. Anderson, Hyperovals and Howell designs, Ars Combin. 9 (1980), 29–38.

[2] J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationgruppe, Math. Z. 60 (1954), 156–186.

[3] E. Assmus and J. Key, Designs and their codes, Cambridge University Press, 1992.

[4] L. Bader, Some new examples of flocks of Q^{+}(3, q), Geom. Ded. 27 (1989), 213–218.

[5] L. Bader and G. Lunardon, On the flocks of Q^{+}(3, q), Geom. Ded. 29 (1989), 177–183.

[6] L. Bader, G. Lunardon, and S. E. Payne, On q–clan geometry q = 2^{e}, Bull. Belgian Math. Soc. 1 (1994), 301–328.

[7] L. Bader, G. Lunardon, and J. A. Thas, Derivation of flocks of quadratic cones, Forum Math. 2 (1990), 163–194.

[8] R. Baer, Polarities in finite projective planes, Bull. Amer. Math. Soc. 52 (1946), 77–93.

[9] , Projectivities with fixed points on every line of the plane, Bull. Amer. Math. Soc. 52 (1946), 273–286.

[10] R. D. Baker and G. L. Ebert, A non-linear flock in the Minkowski plane of order 11, Congr. Numer. 58 (1989), 75–81.

[11] O. Barriga, On the planes of Narayana Rao and Satyanarayana, J. Combin. Theory (A) 45 (1987), 148–151.

[12] L. M. Batten and A. Beutelspacher, The theory of finite linear spaces: Combinatorics of points and lines, Cambridge University Press, 1993.

[13] C. T. Benson, Generalized Quadrangles and (B, N)-pairs, Ph.D. thesis, Cornell University, 1965.

[14] D. Betten, 4-dimensionale Translationsebenen mit 8-dimensionaler Kollineations gruppe, Geom. Ded. 2 (1973), 327–339.

[15] M. Biliotti and G. Korchmáros, Hyperovals with a transitive collineation group, Geom. Ded. 24 (1987), 269–281.

[16] R. C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā 8 (1947), 107– 166.

[17] R. C. Bose and S. S. Shrikhande, Geometric and pseudo-geometric graphs (q^{2} + 1, q + 1, 1), J. Geometry (2/1) (1972), 75–94.

[18] M. R. Brown, Generalised quadrangles of order (q, q^{2}), q even, containing W (q) as a subquadrangle, (to appear).

[19] M. R. Brown, I. Pinneri, and G. F. Royle, Generalised quadrangles of order (8, 64) with a subquadrangle of order 8, (in preparation).

[20] B. H. Bruck and R. C. Bose, The construction of translation planes from projective spaces, Journal of Algebra 1 (1964), 85–102.

[21] F. Buekenhout, Etude intrinsèque des ovales, Rend. Mat. (5) 25 (1966), 333–393.

[22] , Plans projectifs à ovoides pascaliens, Arch. Math. 17 (1966), 333–393.

[23] , On the orbits of collineation groups, Math. Z. 119 (1971), 273–275.

[24] F. Buekenhout (ed.), Handbook of incidence geometry, Elsevier Science B. V., 1995.

[25] F. Buekenhout and C. Lefèvre, Generalized quadrangles in projective spaces, Arch. Math. 25 (1974), 540–552.

[26] J. J. Cannon, An introduction to the group theory language CAYLEY, Computational Group Theory (M. Atkinson, ed.), London Mathematical Society, Academic Press, 1984, pp. 145–183.

[27] W. Cherowitzo, Alpha flocks and hyperovals, (preprint).

[28] , Hyperovals in Desarguesian planes of even order: an update, (preprint).

[29] , Hyperovals in Desarguesian planes of even order, Annals Disc. Math. 37 (1988), 87–94.

[30] W. Cherowitzo, T. Penttila, I. Pinneri, and G. F. Royle, Flocks and ovals, Geom. Ded. 60 (1996), 17–37.

[31] S. D. Cohen and M. J. Ganley, Commutative semifields, two-dimensional over their middle nuclei, J. Algebra 75 (1982), 373–385.

[32] J. A. Conklin and S. E. Payne, An unusual generalized quadrangle of order 16, J. Combin. Theory (A) 24 (1978), 50–74.

[33] C. W. Curtis and I. Reiner, Representation thoery of finite groups and associated algebras, Interscience, New York, 1962.

[34] N. G. de Bruijn and P. Erdös, On a combinatorial problem, Indagationes Math. 10 (1948), 421–423.

[35] F. De Clerck and C. Herssens, Flocks of the quadratic cone in P G(3, q) for q small, The CAGe Report 8, Computer Algebra Group, The University of Gent, Belgium, 1992.

[36] P. Dembowski, Finite geometries, Springer, 1968.

[37] L. E. Dickson, Linear groups with an exposition of the Galois field theory, Dover, New York, 1958.

[38] M. M. Eich and S. E. Payne, Nonisomorphic symmetric block designs derived from generalized quadrangles, Atti Accad. Naz. Lincei Rend. 52 (1972), 893–902.

[39] G. Faina, The B-ovals of order q ≤ 8, J. Combin. Theory (A) 36 (1984), 307–314.

[40] J. C. Fisher and J. A. Thas, Flocks in P G(3, q), Math. Z. 169 (1979), 1–11.

[41] M. J. Ganley, Central weak nucleus semifields, Europ. J. Combin. 2 (1981), 339–347.

[42] D. G. Glynn, Two sequences of ovals in finite Desarguesian planes of even order, Combinatorial mathematics X: Lecture notes in mathematics (L. R. A. Casse, ed.), vol. 1036, Springer, 1983, pp. 217–229.

[43] , The Hering classification for inversive planes of even order, Simon Stevin 58 (1984), 319–353.

[44] , A condition for the existence of ovals in P G(2, q), Geom. Ded. 32 (1989), 247–252.

[45] M. Hall, Jr., Mathematical reviews 14 (1953), 1008, AMS.

[46] , The theory of groups, Macmillan Company, 1959.

[47] , Ovals in Desarguesian plane of order 16, Ann. Mat. Pura Appl. 102 (1975), 159–176.

[48] , Combinatorial theory, 2 ed., Wiley, 1986.

[49] N. Hamilton, Maximal arcs in finite projective planes and associated structures in projective spaces, Ph.D. thesis, University of Western Australia, 1995.

[50] C. Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geom. Ded. 2 (1974), 425–460.

[51] , Transitive linear groups and linear groups which contain irreducible groups of prime order II, J. Algebra 93 (1985), 151–164.

[52] D. G. Higman, Partial geometries, generalized quadrangles and strongly regular graphs, Atti convegno di geometria e sue applicazioni (A. Barlotti, ed.), Perugia, 1971, pp. 263–293.

[53] , Invariant relations, coherent configurations and generalized polygons, Combinatorics, part 3 (M. Hall, Jr. and J. H. van Lint, eds.), vol. 57, Math. Centre Tracts, 1974, pp. 27–43.

[54] J. W. P. Hirschfeld, Ovals in Desarguesian planes of even order, Ann. Mat. Pura Appl. 102 (1975), 79–89.

[55] , Projective geometries over finite fields, Oxford University Press, 1979.

[56] , Finite projective spaces of three dimensions, Oxford University Press, 1985.

[57] J. W. P. Hirschfeld and J. A. Thas, General Galois geometries, Oxford University Press, 1991.

[58] D. R. Hughes, Generalized incidence matrices over group algebras, Illinois Journal of Mathematics 1 (1957), 545–551.

[59] D. R. Hughes and F. C. Piper, Projective planes, Graduate texts in mathematics, vol. 6, Springer-Verlag, New York, 1973.

[60] S. H. Y. Hung and N. S. Mendelsohn, On Howell designs, J. Combin. Theory (A) 16 (1974), 174–198.

[61] B. Huppert, Endliche Gruppen I, Springer-Verlag, 1967.

[62] G. Järnefelt and P. Kustaanheims, An observation of finite geometries, Den Ilte Skandinaviske Matematikerkongress, Trondheim, 1949 (Johan Grundt, ed.), Tanums Forlag, Oslo, 1952, pp. 166–182.

[63] N. L. Johnson, Flocks of hyperbolic quadrics and translation planes admitting affine homologies, J. Geometry 34 (50–73), 50–73.

[64] W. M. Kantor, Classical groups from a non-classical viewpoint, Oxford Lectures 1978, Mathematical Institute, Oxford, 1979.

[65] , Generalized quadrangles associated with G_{2}(q), J. Combin. Theory (A) 29 (1980), 212–219.

[66] , Ovoids and translation planes, Canad. J. Math. 34 (1982), 1195–1207.

[67] , Some generalized quadrangles with parameters (q^{2}, q), Math. Z. 192 (1986), 45–50.

[68] G. Korchmáros, Grappi di collineazioni transitivi sui punti di un ovale [(q + 1)-arco] di S_{2,q}, q pari, Atti. Sem. Mat. Fis. Univ. Modena 27 (1978), 89–105.

[69] C. W. H. Lam, L. Thiel, and S. Swiercz, The non-existence of finite projective planes of order 10, Canad. J. Math. 41 (1989), 1117–1123.

[70] C. W. H. Lam, L. Thiel, S. Swiercz, and J. McKay, The non-existence of ovals in a projective plane of order 10, Discrete Math. 45 (1983), 319–321.

[71] E. S. Lander, Symmetric designs: an algebraic approach, Cambridge Univeristy Press, 1983.

[72] S. Lang, Algebra, Addison-Wesley Publishing Company, 1965.

[73] M. W. Liebeck, The finite primitive permutation groups of rank three, Bull. London Math. Soc. 18 (1986), 165–172.

[74] H. Lüneburg, Über projektive Ebenen, in denen jede Fahne von einer nichttrivialen Elation invariant gelassen wird, Abh. Hamburg 29 (1965), 37–76.

[75] L. Lunelli and M. Sce, k-archi completi nei piani proiettivi Desarguesiani di rango 8 e 16, Tech. report, Centro di Calcoli Numerici, Politecnico di Milano, 1958.

[76] R. Mathon, The partial geometries pg(5, 7, 3), Congr. Numer. 31 (1981), 129–139.

[77] B. D. McKay, nauty user’s guide (version 1.5), Department of Computer Science, Australian national University, 1990.

[78] C. M. O’Keefe, Ovals in Desarguesian planes, Australas. J. Combin. 1 (1990), 149–159.

[79] C. M. O’Keefe and T. Penttila, Ovoids of P G(3, 16) are elliptic quadrics, J. Geometry 38 (1990), 95–106.

[80] , Hyperovals in P G(2, 16), Europ. J. Combin. 12 (1991), 51–59.

[81] , Polynomials for hyperovals of Desarguesian planes, J. Aust. Math. Soc. (A) 51 (1991), 436–447.

[82] , A new hyperoval in P G(2, 32), J. Geometry 44 (1992), 117–139.

[83] , Ovoids of P G(3, 16) are elliptic quadrics, II, J. Geometry 44 (1992), 140–159.

[84] , Symmetries of arcs, J. Combin. Theory (A) 66 (1994), 53–67.

[85] C. M. O’Keefe, T. Penttila, and C. E. Praeger, Stabilisers of hyperovals in P G(2, 32), Advances in finite geometries and designs (J. W. P. Hirschfeld, D. R. Hughes, and J. A. Thas, eds.), Oxford University Press, 1991, pp. 337–351.

[86] C. M. O’Keefe, T. Penttila, and G. F. Royle, Classification of ovoids in P G(2, 32), J. Geometry 50 (1994), 143–150.

[87] C. M. O’Keefe and J. A. Thas, Collineations of Subiaco and Cherowitzo hyperovals, (to appear).

[88] W. F. Orr, The Miquelian inversive plane IP (q) and the associated projective planes, Ph.D. thesis, Madison Wisc., 1973.

[89] T. G. Ostrom, Semi-translation planes, Trans. Amer. Math. Soc. 111 (1964), 1–48.

[90] S. E. Payne, The Subiaco notebook.

[91] , Quadrangles of order (s − 1, s + 1), J. Algebra 22 (1972), 97–119.

[92] , Generalized quadrangles as group coset geometries, Congr. Numer. 29 (1980), 717–734.

[93] , A new infinite family of generalized quadrangles, Congr. Numer. 49 (1985), 115–128.

[94] , q-clans e quadrangoli generalizzati, Sem. Geom. Combin. 79 (1988), 1–16.

[95] , An essay on skew translation generalized quadrangles, Geom. Ded. 32 (1989), 93–118.

[96] , Collineations of the generalized quadrangles associated with q–clans, Annals of Disc. Math. 52 (1992), 449–461.

[97] , Collineations of the Subiaco generalized quadrangles, Bull. Belgian Math. Soc. 1 (1994), 427–438.

[98] , A tensor product action on q–clan geometries with q = 2^{e}, Linear algebra and its applications 226-228 (1995), 115–137.

[99] S. E. Payne and C. C. Maneri, A family of skew-translation generalized quadrangles of even order, Congr. Numer. 36 (1982), 127–135.

[100] S. E. Payne, T. Penttila, and I. Pinneri, Isomorphisms between Subiaco q–clan geometries, Bull. Belgian Math. Soc. 2 (1994), 197–222.

[101] S. E. Payne, T. Penttila, and G. F. Royle, Cyclic q–clans, (in preparation).

[102] S. E. Payne and L. A. Rogers, Local group actions on generalized quadrangles, Simon Stevin 64 (1990), 249–284.

[103] S. E. Payne and J. A. Thas, Finite generalized quadrangles, Research notes in mathematics, no. 110, Pitman Publishing, 1984.

[104] , Conical flocks, partial flocks, derivation and generalized quadrangles, Geom. Ded. 38 (1991), 229–243.

[105] T. Penttila, The Plane Equivalence Theorem, (in preparation).

[106] T. Penttila and I. Pinneri, Hyperovals, (in preparation).

[107] , Irregular hyperovals in P G(2, 64), J. Geometry 51 (1994), 89–100.

[108] T. Penttila and C. E. Praeger, Ovoids and translation ovals, (in preparation).

[109] T. Penttila and G. F. Royle, Flocks of quadratic cones in P G(3, q), q small, (in preparation).

[110] , Classification of hyperovals in P G(2, 32), J. Geometry 50 (1994), 151–158.

[111] , On hyperovals in small projective planes, J. Geometry (1994), (to appear).

[112] T. Penttila, G. F. Royle, and M. K. Simpson, Hyperovals in the known projective planes of order 16, (preprint).

[113] F. C. Piper, Collineation groups containing elations, Math. Z. 92 (1966), 281–287.

[114] B. Qvist, Some remarks concerning curves of the second degree, Ann. Acad. Sci. Fenn. (A) 134 (1952), 4–27.

[115] R. Roth, Collineation groups of finite projective planes, Math. Z. 83 (1964), 409–421.

[116] B. Segre, Ovals in a finite projective plane, Canad. J. Math. 7 (1955), 414–416.

[117] , Sui k-archi nei piani finiti di caratteristica due, Rev. Math. Pures Appl. 2 (1957), 289–300.

[118] , Le geometrie di Galois, Ann. Mat. Pura Appl. 48 (1959), 1–97.

[119] , On complete caps and ovaloids in three dimensional Galois spaces of characteristic two, Acta Arithm. 5 (1959), 315–332.

[120] , Ovali e curve σ nei piani di Galois di caratteristica due, Atti Accad. Naz. Lincei Rend. (8) 32 (1962), 785–790.

[121] B. Segre and U. Bartocci, Ovali ed altre curve nei piani di Galois di caratteristica due, Acta Arith. 8 (1971), 423–449.

[122] J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385.

[123] J. J. Sylvester, Collected mathematical papers, Cambridge University Press, 1904.

[124] D. E. Taylor, The geometry of the classical groups, Sigma series in pure mathematics, no. 9, Heldermann Verlag, Berlin, 1992.

[125] J. A. Thas, Ovoidal translation planes, Arch. Math. 23 (1972), 110–112.

[126] , Flocks of nonsingular ruled quadrics in P G(3, q), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 59 (1975), 83–85.

[127] , Generalized quadrangles and flocks of cones, Europ. J. Combin. 8 (1987), 441–452.

[128] , Flocks, maximal exterior sets and inversive planes, Finite geometries and combinatorial designs (E. S. Kramer and S. S. Magliveras, eds.), AMS, 1990, pp. 187–218.

[129] , Projective geometry over a finite field, In Buekenhout [24], pp. 295–347.

[130] J. A. Thas and S. E. Payne, Spreads and ovoids in finite generalized quadrangles, Geom. Ded. 52 (1994), 227–253.

[131] J. A. Thas, S. E. Payne, and H. Gevaert, A family of ovals with few collineations, European J. Combin. 9 (1988), 353–362.

[132] J. Tits, Sur la trialité ét certains groupes qui s’en déduisent, Inst. Hautes Etudes Sci. Publ. Maths. 2 (1959), 14–60.

[133] , Ovoïdes et groupes de Suzuki, Arch. Math. 13 (1962), 187–198.

[134] O. Veblen and J. W. Young, Projective geometry, Ginn and Co (Blaisdell, 1965), 1916, (two volumes).

[135] M. Walker, A class of translation planes, Geom. Ded. 5 (1976), 133–146.

[136] H. Wielandt, Finite permutation groups, Academic Press, 1964.

[137] H. Zassenhaus, Über endliche Fastkörper, Abh. Math. Sem. Univ. Hamburg 11 (1936), 187–220.

[138] K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. für Math. u. Phys. 3 (1892), 265–284.

170

Index

{x, y}^{⊥⊥}, 88 ≁, 13 ∼, 13 {x, y}^{⊥}, 88 1/u, 11

A^{∗}(i), 101 A1, 4 A2, 4 A3, 4 [A(i)], 101 absolute, 142 absolute trace, 11 acentric, 89 adjacent, 13 admissible points, 61 affine

hyperplane, 138 plane, 4 point, 138

AG(2, q), 6 A(i), 101 A(∞), 103 antiflag, 37 antiregular, 89 arc, 20 A(t), 103 Aut P G(V), 5 automorphic collineation, 5 automorphism, 5 axis, 30

of symmetry, 101

ℬ, 3 bad egg, 144 Baer

collineations, 29 subplane, 28

base, 149

point, 101 B-conic, 80 blocks, 3 BLT-set, 154, 170 B-oval, 80, 84 Bruck-Bose, 12 ℬ_{θ}, 100 Buekenhout oval, 80 bundle, 78

C^{⊥}, 73

C_{1}, linear q–clan, 155 C_{2}, FTWKB q–clan, 155 C_{3}, Payne q–clan, 156 C_{4}, De Clerck-Herssens q–clan, 156 C_{5}, Payne derived q–clan, 156 carrier

bundle, 78 flock, 78 pencil, 78

carriers

flock, 149 Cauchy’s Theorem, 7 CAYLEY, 82 centre, 30

of a GQ, 89

of symmetry, 101 centric, 89 C_{G}(H), 7 change of coordinates, 42 Cherowitzo hyperovals, 55 circles, 78 class, 19 closed configuration, 28 code, 73 collinear, 13 collineation, 6, 27 common tangent, 78 compatible, 77 complement, 7 complete bipartite graph, 14 complete graph, 80 concurrent, 13 cone, 149 conic, 23 coregular, 89 correlation, 44, 82

decomposable, 126 degenerate, 22 degenerate conics, 23 delta function, 48 Dempwolff plane, 36 derivation, 156, 170

sets, 150 derived, 155 Desarguesian projective plane, 6 diagonal line, 58 disjoint, 13 d_{t}, 103 d(t, u), 103 dual, 3

of the extended code, 73 duality, 44, 82

projective plane, 4 dually, 4

E(Γ), 80 egglike, 79 EGQ, 101 elation, 30

generalised quadrangle, 101 group, 101 of a GQ, 101

elliptic quadric, 75 End_{G}(V), 9 equivalent, 45, 46, 105

representation, 9 equivalent flocks, 154 even type, 74 exceptional

flocks, 152

nearfield, 152 exponential notation, 11 extended code, 73 external, 58

line, 20, 98

point, 21, 98 External Lines Lemma, 76

ℱ, 101

factorisation, 80

factors, 80 fan, 78 fano configurations, 58 f(b, a), 103 F(C), 154 finite

affine plane, 4

generalised quadrangle, 15 projective plane, 3

firm generalised quadrangle, 13

firm GQ, 13, 90

Fisher-Thas-Walker-Kantor-Betten q–clan, 155 Fix(G), 32 fixed

configuration, 28

line, 28 linewise, 28 point, 28 pointwise, 28

flag, 37 flip, 113 flock, 78, 149, 169

hyperbolic quadric, 149 ovoid, 149 quadratic cone, 97, 149

4-gonal, 102

f̄_{Q}, 23 f_{Q}, 22 Frattini argument, 8 FTWKB, 155 fundamental quadrangle, 25 Fundamental Theorem of Geometry, 6

G-endomorphism, 9

G_{0}, 115 G(a), 110 generalised

elation (of order t), 35 homology (of order t), 35 generalised quadrangle, 14, 87 generators of a cone, 149 G-invariant, 9 GL(V), 5 Γ L(V), 5 Glynn

hyperovals, 54 partial ordering, 48

GQ(C), 109 GQ1*, 13 GQ1, 13 GQ2*, 13 GQ2, 13 GQ(s, t), 87 graphs, 80 grid, 14, 88, 90

H, 115, 149 H_{1}, 64, 68 H_{12}, 71 H_{15}, 70 H_{2}, 64, 69 H(3, q^{2}), 155 H_{60}, 70 Hall

hyperoval, 82 plane, 36

H(C), 108 H(d, q^{2}), 93 height, 58, 66 herd of ovals, 108 Hirschfeld-Eich-Payne hyperoval, 54 homogeneous coordinates, 5 homography, 5 homology, 30 Howell design, 81 H(S), 166 H(s, 2n), 81 hyperbolic line, 88 hyperoval, 20 hyperplane, 4

ℐ^{}ˆ , 3

incidence matrix, 29 incidence relation, 3 incidence structure, 3 (∞), 90, 92, 93, 101, 143 internal

line, 98 point, 21, 98

intersect, 13 intertwiner, 9 inversive plane, 78, 89, 169 involution, 31

K, 150 K1, 101 K2, 101 K_{2n}, 80 k-arc, 20 k-cap, 75 k-factor, 80 Klein

correspondence, 150 quadric, 150

Knarr construction, 154 k-regular, 80 k-set, 19

ℒ, 3 Lagrange interpolation, 48 Lemma of Tangents, 24 Lenz type greater than I, 52 linear complex, 96 linear flock, 149, 155

hyperbolic quadric, 149 ovoid, 149

linear space, 16, 82 live, 68 Lunelli-Sce hyperoval, 51, 82, 156, 167

M, 115 matches, 78 matching point-ovoid pairs, 78 maximal arcs, 153 minimum weight, 74 monomial functions, 11

nauty, 140

nearfields, 152 N_{G}(H), 7 nice, 67 no triangles axiom, 13 non-adjacent, 13 non-collinear, 13 non-concurrent, 13 nondegenerate, 22 nonsingular, 22 nth cyclotomic polynomial, 8 nucleus, 22, 23 O^{+}(6, q), 150 O’Keefe-Penttila hyperoval, 57, 83 o-polynomial, 46 odd type, 74 one-factorisation, 80 O(n, m, q), 143 opposite regulus, 12 order

generalised quadrangle, 15 inversive plane, 78 projective plane, 3

orthogonal, 80 oval, 20 ovoid, 75, 95, 150 Ω, 149

𝒫, 3 p-subgroup, 7 Pappian plane, 84 parallel, 4 parameters, 15 partial geometry, 79 Pascal’s Theorem, 84 Payne and Thas GQ, 15 Payne hyperoval, 54, 156 pencil, 77, 78 Penttila-Royle hyperoval

in P G(2, 256), 72

in P G(2, 64), 71 perfect matchings, 80 ∣ð, 79 P G(d, 𝔽), 5 P G(d, q), 5 P G(V), 4 P GL(d + 1, q), 5 P Γ L(d + 1, q), 6 π(C), 154 planar, 29, 138 Plane Equivalence Theorem, 77 planes, 4 point graph, 79, 87 pointed conic, 25 polarity, 44, 82 PP1, 3 PP2, 3 PP3, 3 prime at a time, 59 profile, 58 projected, 138 projective

dimension, 4 plane, 3 plane of order 10, 75 space, 4

proper subGQ, 97 pseudo-axis, 35 pseudo-centre, 35 pseudo-periodicity, 68 P SL(d, q), 6 P (𝒮 , x), 92 𝒫_{θ}, 100 ptGQ*, 13 ptGQ, 13 pure, 126

Q, 22 Q̄, 23 q-pd, 8 q-ppd, 8 q-primitive divisor, 8 q-primitive part of q^{d} − 1, 8 q-primitive prime divisor, 8 Q(3, 2), 88 Q(4, 2), 88 Q(4, q), 93 Q(5, 2), 88 Q(5, q), 93 q–clan, 105–109, 111, 112, 114–116, 125–128,

155–156 Q_{st}, 105 quadrangle, 20 quadratic cone, 149 form, 22 quadric, 22

Rad f, 22 radical, 22 reducible, 9 regular, 89 graph, 79 hyperoval, 25 nearfield, 152

regulus, 12

representation, 9 Roman GQs, 144 Room design, 81 rosette of ovoids, 99

S, 111, 162 S_{d}, 111, 162 S′, 159 S′′, 157 S′′′, 161 Samosa Lemma, 8, 9, 16, 83 scale, 113 secant, 58

line, 20

plane, 76 Segre’s Theorem, 24 Segre-Bartocci hyperoval, 53, 156 self-dual, 75 self-orthogonal, 75 semi-translation hyperoval, 31 𝒮 (G, ℱ), 102, 103 shift, 113 Singer’s Theorem, 31 singular, 22

radical, 22 SL(d, q), 6 span, 88 spanning graph, 80 spread, 12, 95 SRG, 87 𝒮 (s, t), 87 𝒮_{θ}, 100 stabiliser

regular hyperoval, 26 strongly regular graph, 80, 87 subGQ, 97 Subiaco

q–clan, 111, 157–166

flock, 167

hyperoval, 167

hyperovals in P G(2, 64), 58 plane, 167

suborthogonal, 80 subplane, 28 subquadrangle, 97 subspaces, 5 subtends, 130 supplement, 7 support, 74 Suzuki group, 76 Syl_{p}(G), 8 Sylow p-subgroup, 7 Sylow’s Theorem, 7 Sylvester’s syntheme-duad geometry, 89 symmetry, 101

T_{2}^{∗}(O), 90 T_{2}(O_{x}), 93 T_{2}(O), 91 T_{3}(Ω), 93 tangent

circle, 78 line, 20, 98 plane, 76 point, 98 space, 101, 143

TGQ, 101

Thas flock, 151, 170

thickGQ, 14 thickness, 14 3-regular, 89 Tits

hyperoval, 76

oval, 76 ovoid, 75, 97 T (n, m, q), 143 totally singular points, 22 trace, 88 trace map, 11 translation q-arc, 31 generalised quadrangle, 101 group, 101 hyperoval, 31, 51 line, 31 oval cone, 149 plane, 31 transversal, 12 triad, 89 triangle, 20 element, 41

type, 19

unicentric, 89 U(ℤ_{p}), 8

V (Γ), 80

weight, 73 whorl, 100 Witt index, 23 W (q), 93

x^{⊥}, 87

Z, 5, 103 ℤ_{q−1}, 11