Université Libre de Bruxelles Année Académique 1997-1998 Faculté des Sciences Département de Mathématique

Classification of Rwpri Geometries for the Suzuki Simple Groups

Dimitri Leemans

Thèse présentée en vue Promoteurs de Thèse: de l’obtention du grade de Francis Buekenhout Docteur en Sciences Michel Dehon

Acknowledgements

I would like to thank my supervisors, Francis Buekenhout and Michel Dehon, for their support, their kindness and their disponibility throughout the elaboration of this work.

During the making of this thesis, I had the chance to visit Professor John Cannon at the University of Sydney. I would like to thank him and all his team for having made my stay there as nice as it was.

I would also like to thank several people with whom I had interesting conversations while working on this thesis: Wieb Bosma, Philippe Cara, Frank De Clerck, Jean Doyen, Harald Gottschalk, Cécile Huybrechts, Antonio Pasini, Leo Storme, Gernot Stroth, Joseph Thas and Hendrik Van Maldeghem.

Thanks must go to all my colleagues for their support, and also for the very nice working atmosphere that reigns in the Mathematics Department of the Université Libre de Bruxelles.

I also thank my parents for the opportunity they always gave me to let me choose what I wanted to do.

And last but certainly not least, I thank my wife Murielle for having pushed me into doing what I really wanted to do: mathematics.

Contents

Introduction 5

1 Definitions and Notation 9 1.1 Incidence Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 The Suzuki simple groups . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Some useful notation . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The Dihedral Groups 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 The Automorphism group of a dihedral group . . . . . . . . . . . 18 2.3 On the subgroups of a dihedral group . . . . . . . . . . . . . . . . 18 2.4 Geometries of rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 G_{0} = Z_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.2 G_{0} = D_{2m} and G_{1} ≠ Z_{n} . . . . . . . . . . . . . . . . . . . 20

2.5 Geometries of rank at least 3 . . . . . . . . . . . . . . . . . . . . . 22 2.5.1 G_{0} = Z_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5.2 G_{0} = D_{2n_{p_{i}}} and G_{i} ≠ Z_{n} . . . . . . . . . . . . . . . . . . . 23

3 The rank 2 geometries of Sz(q) 27 2 CONTENTS

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Some useful properties . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 The maximal subgroups of Sz(q) . . . . . . . . . . . . . . . . . . . 30 3.4 The maximal subgroups of the maximal subgroups of Sz(q) …. 31 3.5 Rank 2 Rwpri geometries . . . . . . . . . . . . . . . . . . . . . . 33 3.6 Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Observations on the results . . . . . . . . . . . . . . . . . . . . . . 48

4 The rank 3 geometries of Sz(q) 49 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Preliminary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Restricting the maximal parabolic subgroups . . . . . . . . . . . . 52 4.4 The rank 3 geometries . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4.1 First case: q − 1 is a prime number . . . . . . . . . . . . . 55 4.4.2 Second case: q − 1 is not prime and 2e + 1 is prime …. 56 4.4.3 Third case: 2e + 1 is not prime . . . . . . . . . . . . . . . 57

4.5 About the flag-transitivity . . . . . . . . . . . . . . . . . . . . . . 69 4.6 About the (2T)_{1} property . . . . . . . . . . . . . . . . . . . . . . 70

5 The pre-geometries of rank ≥ 4 of Sz(q) 71 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 The rank ≥ 4 pre-geometries of Sz(q) . . . . . . . . . . . . . . . . 72

5.2.1 s − 1 is prime . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.2 s − 1 not prime and 2f + 1 prime . . . . . . . . . . . . . . 77 CONTENTS 3

5.2.3 2f + 1 not prime . . . . . . . . . . . . . . . . . . . . . . . 77 5.3 Constructions and classification theorem . . . . . . . . . . . . . . 87 5.4 Observations and future work . . . . . . . . . . . . . . . . . . . . 91

A The RWPRI geometries of Sz(8) 93 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A.2 The geometries of the Suzuki group Sz(8) . . . . . . . . . . . . . . 94

A.2.1 Rank 2 geometries . . . . . . . . . . . . . . . . . . . . . . 94 A.2.2 Rank 3 geometries . . . . . . . . . . . . . . . . . . . . . . 98 A.2.3 Rank ≥ 4 geometries . . . . . . . . . . . . . . . . . . . . . 101

B Thin geometries for Sz(8) 105 B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 B.2 The thin geometries of Sz(8) . . . . . . . . . . . . . . . . . . . . . 106 B.3 Some observations . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.4 Applying the Neumaier construction . . . . . . . . . . . . . . . . 118

Bibliography 121 4 CONTENTS

Introduction

In this thesis, we classify geometries satisfying a set of properties for the Suzuki simple groups.

The classification of the finite simple groups, summarized around 1981, is one of the most spectacular achievement in mathematics of all time. This is due to the efforts of hundreds of researchers, over a period of thirty years, in around five hundred publications, covering more than fiftheen thousand pages. A coherent and revised presentation and proof is presently in the course of publication. The importance of the subject is due, to a large extent to the fact that the groups are the best mathematical expression of symmetry and, as their name may suggest, the simple groups are the basis that constitutes the more general groups. The classification leads to eighteen infinite families of groups and twenty-six sporadic groups, the largest of them being the Monster. While some people could think that the classification finishes the study of these groups, there are still open questions in this field. One of them is the classification of all the maximal subgroups of the simple groups. Also, these families of groups are known relatively to other things that are supposed to be known, as for example the prime numbers. As we can see in the work we do here, certain prime numbers lead to results that are completely different from others.

One of the most important open questions on these groups concerns a unified geometric interpretation of them which should help understanding them better. This theory already exists for the majority of them, namely for the groups of Lie type, which means all of them except the alternating groups and the sporadic groups : it is the theory of Buildings, due to Jacques Tits, for which he received the Wolf prize in 1993 (see [5], chapter 11 for a historical survey). This theory has given one of the most used geometrico-combinatorial structures in other fields of mathematics, and in particular, in the classification itself. Its extension to the sporadic groups is far from being easy; many efforts have been made in that sense during the last twenty years (see [5], chapter 22 for a historical survey). One of the reasons of the difficulty is precisely the sporadic character of these groups, and the fact that geometrically speaking, some groups of Lie type might need to be considered as groups isomorphic to further sporadic groups.

In [5] (chapter 22), the approach which is developed is the following: given a diagram Δ, try to find a geometry Γ belonging to Δ and a group G which is an automorphism group of Γ.

Here we work with another approach: we start from a group and we try to determine all geometries, satisfying a set of axioms, that can be constructed from this group and some of its subgroups. This approach is strongly based on experimental work.

We really think that experimental work can play an important role in mathematics, helping to solve theoretical problems. Some mathematical journals are now entirely devoted to experimentation, as for example Experimental Mathematics, whose advisory board is composed of great mathematicians as Hendrik W. Lenstra Jr., Jean-Pierre Serre and William P. Thurston.

It seems appropriate to elaborate an exhaustive experimental search of geometries satisfying some properties and corresponding to the smallest simple groups, be them sporadic or not. Such a research concentrates on reasonable conditions to be able to determine the geometries of all simple groups.

Experimentation has already been achieved with two slightly different conditions denoted Pri and Rwpri, with help of the computer algebra packages Cayley [10, 11] and Magma [2, 12, 13]. For experimental results in the Pri case, we refer to [7, 8, 17]. This work covers the groups G with P SL(2, q) ≤ G ≤ Aut(P SL(2, q)), for 5 ≤ q ≤ 19, the Mathieu group M_{11}, and P SU(4, 2). For the Rwpri case, we refer to [9, 18, 19, 20, 21, 22, 23]. These references cover experimental results for the groups Alt(n) and Sym(n) with 5 ≤ n ≤ 7, the groups G with P SL(2, q) ≤ G ≤ Aut(P SL(2, q)) for q = 7, 8, the Mathieu group M_{11}, P SU(4, 2), P SL(3, 4) and the Janko group J_{1}.

If G is one of the studied groups, we would like to determine the class Γ(G) of geometries in the Buekenhout-Tits sense, in which G acts transitively on the maximal flags, and which satisfy additional conditions, as for example the maximality, in G, of the stabilizers of some elements. We have, at present, a set of conditions or axioms which apparently permits that the size of Γ(G) remains reasonable, as in the building case, and which accepts an efficient programming using the computer algebra packages Cayley or Magma. Also, it should now be possible to start a purely theoretical approach, in the same way than experimentation.

In its actual state, experimental work showed us that an interactive study of the geometries associated to a group G depends on the knowledge of the geometries corresponding to some subgroups of G. The next step might reasonably begin with a family of groups having a well known subgroup structure. Here the candidates that seem more appropriate are the P SL(2, q) groups and the Suzuki groups Sz(q). The latter appear to us as the easiest to start with for such a general study. Most of the subgroups of these groups are soluble groups. Also, we notice very early that the subgroups that play an important role are most of the time dihedral groups. In that spirit, we first study in general geometries associated to the dihedral groups.

Most of the definitions and notation that are used throughout this thesis are introduced in the first chapter. There are notions about incidence geometries and Suzuki groups, but also some notational shortcuts that are described in distinct sections.

Chapter 2 is devoted to the classification of the Rwpri geometries of dihedral groups. We start by giving an exhaustive list of the maximal subgroups of a dihedral group D_{2n} for n an integer. Then we classify the rank two geometries. And finally we determine the geometries of rank at least three.

In Chapter 3 we start the classification of the Rwpri geometries of the Suzuki groups. Again, we first recall what are the maximal subgroups of a Suzuki group Sz(q). Then we classify the rank two geometries. We also determine how many rank two geometries there are up to isomorphism for any Suzuki group Sz(q). Some very surprising results appear here. Let us give a few examples.

Let q = 2^{2e+1} and G ≅ Sz(q).

• If 2e + 1 is a prime number, that is, the corresponding Suzuki group does not have proper subgroups isomorphic to a Suzuki group, then it has either one or a huge number of geometries.

• By looking at these geometries, we see that if we add a condition called (2T)_{1}, that is two transitivity on rank 1 residues, then all geometries except one (or three for the case q = 8) are rejected. So this property which we tested the last few years seems to be a good candidate to enter our set of axioms.

In Chapter 4, we classify the Rwpri geometries of rank three for all Suzuki simple groups Sz(q) with q an odd power of 2. One big observation arises again from the results obtained there: if we impose the (2T)_{1} property, only Suzuki simple groups Sz(q) with q − 1 prime have geometries of rank greater than 2. All of these are thin geometries of rank 3.

In chapter 5, we determine all Rwpri pre-geometries of rank ≥ 4 for any Suzuki simple group Sz(q). To do that, we start from the classification theorems stated in chapter 4 for the rank 3 case. We get, as final result, eleven constructions leading to all Rwpri pre-geometries of rank ≥ 4 for any Sz(q) (see theorem 5.3.1).

This thesis ends with two appendices: the first one gives the classification of all firm, residually connected geometries on which the group Sz(8) acts flagtransitively and residually weakly primitively. It is the smallest Suzuki group, but also the only one that is currently possible to investigate fully with a computer. It helped us very well in guessing the way to follow for the general case. The second one gives all thin residually connected geometries on which Sz(8) acts flagtransitively. We hoped that this work would help us to determine, in the general case, how many thin geometries are appearing up to isomorphism. Unfortunately, we didn’t succeed. In this appendix, we also put more structure on the diagrams and we introduce a new property that seems very promising to us. It is the (SIP)_{2} property. It seems promising because it is a necessary condition to have the intersection property, and also, it reduces quite significantly the huge number of thin geometries obtained.

The classification we obtain in this thesis for the Suzuki groups and for the dihedral groups strenghten our belief that the Rwpri condition is a very efficient one to impose on the subgroups forming the geometries. Indeed, even if this condition gives many geometries for some groups, their number seems to stay under control.

Chapter 1

Definitions and Notation

In a first section, we recall the basics of incidence geometry and fix some notation. In a second section, we introduce the Suzuki simple groups as done in [33]. In a third section, we fix some notation that is used throughout this thesis.

1.1 Incidence Geometry

We recall the basic definitions about incidence geometry and group-geometry pairs. For a complete introduction on that subject, we refer to [5].

The basic concepts about geometries constructed from a group and some of its subgroups are due to Tits [41] (see also [5], chapter 3). Let I be a finite set and let G be a group together with a finite family of subgroups (G_{i})_{i∈I}. We define the pre-geometry Γ = Γ(G, (G_{i})_{i∈I}) as follows. The set X of elements of Γ consists of all cosets gG_{i}, g ∈ G, i ∈ I. We define an incidence relation * on X by :

g_{1}G_{i} * g_{2}G_{j} iff g_{1}G_{i} ∩ g_{2}G_{j} is non-empty in G.

The type function t on Γ is defined by t(gG_{i}) = i. The type of a subset Y of X is the set t(Y); its rank is the cardinality of t(Y) and we call | t(X) | the rank of Γ. The Borel subgroup of the pre-geometry is the subgroup B = ∩_{i∈I}G_{i}. A flag is a set of pairwise incident elements of X and a chamber of Γ is a flag of type I. An element of type i is also called an i-element. The group G acts on Γ as an automorphism group, by left translation, preserving the type of each element. As in [17], we call Γ a geometry provided that every flag of Γ is contained in some chamber and we call Γ flag-transitive (FT) provided that G acts transitively on all chambers of Γ, hence also on all flags of any type J, where J is a subset of I. It is obvious that any rank two pre-geometry Γ(G; G_{0}, G_{1}) is a flag-transitive geometry.

Lemma 1.1.1 Let Γ(G; G_{0}, G_{1}, G_{2}) be a rank 3 pre-geometry. Then Γ is a geometry (not necessarily flag-transitive).

Proof: We have to show that every flag is contained in a chamber. Suppose without loss of generality that we have a flag F = {g_{0}G_{0}, g_{1}G_{1}}. Then, there exists an element h ∈ G such that h ∈ g_{0}G_{0}∩g_{1}G_{1}. Thus the coset hG_{2} is incident to both g_{0}G_{0} and g_{1}G_{1} and F is contained in the chamber {g_{0}G_{0}, g_{1}G_{1}, hG_{2}}. □

This lemma permits us to talk about geometries instead of pre-geometries in the rank three case. Let Γ(G; G_{0}, …, G_{n−1}) be a rank n pre-geometry. We call C = {G_{0}, …, G_{n−1}} the maximal parabolic chamber of Γ. Assuming that F is a subset of C, the residue of F is the pre-geometry

Γ_{F} = Γ(∩_{j∈t(F)}G_{j}, (G_{i} ∩ (∩_{j∈t(F)}G_{j}))_{i∈I\t(F)})

If F = {G_{i}} for some i ∈ I = {0, …, n − 1} then Γ_{F} is also called the G_{i}-residue of Γ. In that case, we sometimes write Γ_{i} instead of Γ_{F}. If Γ is a flag-transitive geometry and if F is any flag of Γ, of type t(F), then the residue Γ_{F} of Γ is isomorphic to the residue of the flag {G_{i}, i ∈ t(F)} ⊆ C. Let J be a subset of I. The J-truncation of Γ is the pre-geometry consisting of the elements of type j ∈ J, together with the restricted type-function and induced incidence relation. In group-geometry terms, the J-truncation of Γ(G, (G_{i})_{i∈I}) is the pre-geometry Γ(G, (G_{j})_{j∈J}).

Assume Γ is a pre-geometry. We call Γ firm (F) (resp. thick, thin) provided that every flag of rank | I | −1 is contained in at least two (resp. three, exactly two) chambers. We call Γ residually connected (RC) provided that the incidence graph of each residue of rank ≥ 2 is a connected graph. Let Γ(G; G_{0}, …, G_{n−1}) be a pre-geometry and put I = {0, …, n − 1}. As in [23], for any nonempty subset J ⊂ I, we set G_{J} = ⋂_{j∈J} G_{j}, B = G_{I} and G_{∅} = G. Then we call L(Γ) = {G_{J} : J ⊆ I} the sublattice (on the subgroup lattice of G) spanned by the collection (G_{i})_{i∈I}. The elements of the lattice are called the parabolic subgroups and the subgroups G_{i}’s are the maximal parabolic subgroups. When the context is clear, we write ”sublattice” instead of ”sublattice spanned by …”. We call Γ primitive (Pri) provided that G acts primitively on the set of i-elements of Γ for each i ∈ I. We call Γ residually primitive (Rpri) provided that for any J ⊂ I and for all i ∈ I\J, the group G_{J∩{i}} is a maximal subgroup of G_{J}. Notice that this definition of Rpri differs slightly from the one given in [9]. Moreover, if Γ is a flag-transitive geometry, the definition given here is equivalent to the one given in [9]. We call the pre-geometry Γ residually weakly primitive (Rwpri) provided that for any J ⊂ I there exists at least one element i ∈ I\J such that G_{J∪{i}} is maximal in G_{J}. Again, this definition of Rwpri differs slighty from the one given in [9]. If the pre-geometry Γ is a flag-transitive geometry, then the definition of Rwpri given here is equivalent to the one given in [9]. If Γ is a Rwpri pre-geometry then, without loss of generality, we may assume that

• ∩_{j∈{0,...,i}}G_{j} is a maximal subgroup of ∩_{j∈{0,...,i−1}}G_{j} for all i = 1, …, n − 1;

• ∩_{j∈I}G_{j} is a maximal subgroup of ∩_{j∈I\{i}}G_{j} for all i ∈ I.

A subgroup H of a group G is called quasi-maximal if there exists a unique chain of intermediate subgroups from G to H. We call Γ quasi primitive (Qpri) provided that every subgroup G_{i} forming the geometry is a quasi-maximal subgroup of G. If Γ is a geometry of rank 2 with I = {0, 1} such that every of its 0-elements is incident with each of its 1-elements, we call Γ a generalized digon. We call the pre-geometry Γ locally 2-transitive and we write (2T)_{1} for this, provided that the stabilizer G_{F} of any flag F ⊂ C of rank | I | −1 acts 2-transitively on the residue Γ_{F}. Again here, if Γ is a flag-transitive geometry, then the (2T)_{1} property is the same as the one given in [9].

We denote by σ_{0}(F) the 0-shadow of a flag F, that is the set of elements of type 0 that are incident with the flag F. We say that a geometry Γ, over a set of types I, satisfies the intersection property (IP) when for every types 0, 1 ∈ I, for every 1-element x, and for every flag F, | σ_{0}(x) ∩ σ_{0}(F) |> 1 implies that there exists a flag F′ such that x and F are incident with F′ and σ_{0}(F′) = σ_{0}(x) ∩ σ_{0}(F). We say that Γ satisfies the rank two intersection property (IP)_{2} if in every rank 2 residue of Γ other than a generalized digon, any two elements of the same type are incident with at most one element of the other type.

Following [3] and [4], the diagram of a firm, RC, FT geometry Γ is a graph together with additional structure, whose vertices are the elements of I, which is further described as follows. To each vertex i ∈ I, we attach the order s_{i} which is | Γ_{F} | −1, where F is any flag of type I\{i}, the number n_{i} of varieties of type i, which is the index of G_{i} in G, and the subgroup G_{i}. Elements i, j of I are not joined by an edge of the diagram provided that a residue Γ_{F} of type {i, j} is a generalized digon. Otherwise, i and j are joined by an edge endowed with three positive integers d_{ij}, g_{ij}, d_{ji} where g_{ij} (the gonality) is equal to half the girth of the incidence graph of a residue Γ_{F} of type {i, j} and d_{ij} (resp. d_{ji}), the i-diameter (resp. j-diameter) is the greatest distance from some fixed i-element (resp. j-element) to any other element in the incidence graph of Γ_{F}. On a picture of the diagram, this structure will often be depicted as follows.

[figure: d_{ij}; g_{ij}; d_{ji}; s_{i}; s_{j}; n_{i}; n_{j}; G_{i}; G_{j}]

If g_{ij} = d_{ij} = d_{ji} = n, then Γ_{F} is called a generalized n-gon and on a picture, we do not write d_{ij} and d_{ji}. If (d_{ij}, g_{ij}, d_{ji}) = (5, 5, 6) we write P on the corresponding edge instead of the 3 parameters. This is because such a rank 2 residue is a Petersen graph. If d_{ij} = g_{ij} = 3, d_{ji} = 4 and s_{i} = 1, we write ⊂ on the corresponding edge. We sometimes call a rank 2 residue with parameters (d_{ij}, g_{ij}, d_{ji}) a d_{ij} − g_{ij} − d_{ji}- residue. A star-diagram is a diagram with one vertex joined to all other and no other edge.

The ordered pairs (Γ, G) and (Γ′, G) are isomorphic (resp. conjugate) if there exists an automorphism (resp. internal automorphism) of G mapping Γ onto Γ′.

As to notation for groups, we follow the conventions of the Atlas [14] up to slight variations. The symbol ”:” stands for split extensions, the ”hat” symbol ”ˆ.”stands for non split extensions and the symbol × stands for direct products. Sometimes it is not known whether an extension is split or not. In that case, we write ”·” or nothing to denote that it is an extension. We denote by Z_{n} (or just n when the context is not ambiguous) a cyclic group of order n.

Following ideas developped in [39], we say that a group G acts chirally on a geometry Γ if the chambers of Γ are divided in two orbits of the same length on which G acts transitively. We also say that the group G acts regularly on Γ if G is transitive on the set of chambers of Γ.

1.2 The Suzuki simple groups

We recall the definition of the Suzuki groups as given in [33]. Let K be a field of characteristic 2 with | K |> 2. Let σ be an automorphism of K such that x^{σ^{2}} = x^{2} for each x in K. Let B be the 3-dimensional projective space over K and let (x_{0}, x_{1}, x_{2}, x_{3}) be the coordinates of a point of B. Let E be the plane defined by the equation x_{0} = 0 and put U = (0, 1, 0, 0)K. We introduce coordinates in the affine space B_{E} by x = \frac{x_{2}}{x_{0}} , y = \frac{x_{3}}{x_{0}} and z = \frac{x_{1}}{x_{0}} . Finally, let D be the set of points of B consisting of U and all those points of B_{E} whose coordinates (x, y, z) satisfy the equation

z = xy + x^{σ+2} + y^{σ}

We denote by Sz(K, σ) the group of all projective collineations of B which leave D invariant.

Lemma 1.2.1 [33]

• If K is isomorphic to GF (2^{2e+1}) with e ≥ 1, then K admits exactly one automorphism σ with x^{σ^{2}} = x^{2} for all x in K.

• If K is isomorphic to GF (2^{2e}), then K does not possess an automorphism σ with x^{σ^{2}} = x^{2} for all x in K.

This theorem yields that we may write Sz(q) instead of Sz(K, σ) if K is isomorphic to GF(q) with q = 2^{2e+1}. The groups Sz(q) are the Suzuki groups named after Michio Suzuki who found them in 1960. The generalizations Sz(K, σ) are due to Rimhak Ree and Jacques Tits (see for example [42]). The set D is an ovoid, i.e. a non-empty point-set of a projective 3-space that satisfies the following three conditions.

1. No three points are collinear.

2. If p ∈ D, there exists a plane E of B with D ∩ E = {p}.

3. If p ∈ D and if E is a plane of B with D ∩ E = {p}, then all lines l through p which are not contained in E carry a point of D distinct from p.

For a, b ∈ K, we denote by τ(a, b) the mapping defined by

(x, y, z)^{τ(a,b)} = (x + a, y + b + a^{σ}x, z + ab + a^{σ+2} + b^{σ} + ay + a^{σ+1}x + bx)

We have τ(a, b)τ(c, d) = τ(a + c, ac^{σ} + b + d). For k ∈ K^{∗}, we define the collineation η(k) by

(x, y, z)^{η(k)} = (kx, k^{σ+1}y, k^{σ+2}z)

A straightforward computation shows that τ(a, b)η(k) = η(k)τ(ka, k^{σ+1}b). Let ω be the collineation of B defined by (x_{0}, x_{1}, x_{2}, x_{3})^{ω} = (x_{1}, x_{0}, x_{3}, x_{2}). The following lemma, due to Tits, will be used in a further section to study the structure of the stabilizer of a point of the ovoid.

Lemma 1.2.2 [42] Let K be a commutative field of characteristic 2 with | K |> 2 and assume that K admits an automorphism σ such that x^{σ^{2}} = x^{2} for all x ∈ K. If D is the point set defined above in the projective space of dimension 3 over K, then Sz(K, σ) acts doubly transitively on D. Moreover, if γ ∈ Sz(K, σ)_{U}, then there exists exactly one triple (k, a, b) ∈ K^{∗} × K × K with γ = η(k)τ(a, b), and if γ ∈ Sz(K, σ)\Sz(K, σ)_{U}, then there exists exactly one 5-tuple (k, a, b, c, d) ∈ K^{∗}×K×K×K×K with γ = η(k)τ(a, b)ωτ(c, d).

Let P G(3, q) be the projective space over the field GF (q) and let D be an ovoid of P G(3, q). If Π is a plane of P G(3, q) such that | Π∩D |> 1, then we call Π∩D a circle. The following lemma ensures that every circle has the same number of points, and moreover that these circles are ovals.

Lemma 1.2.3 [33] If D is an ovoid in P G(3, q), then | D |= q^{2} + 1. If E is a plane of P G(3, q), then E is either a tangent plane of D or E ∩ D consists of the q + 1 points of an oval of E.

1.3 Some useful notation

The symbol n_{p} denotes the number \frac{n}{p} when p is a divisor of n. The function ϕ is Euler’s function that associates to every positive integer n the number of positive integers i < n such that (i, n) = 1.

For some properties of this function, we refer to [37]. Since it is a well known function, almost every book on number theory is a good reference for this.

There are four numbers that play an important role in the subgroup structure of Sz(q). They are respectively q, q − 1, q + r + 1 and q − r + 1 where r = \sqrt{2q}. We note q + r + 1 = α_{q} and q − r + 1 = β_{q}.

When n is a proper divisor of m, we write n |_{P} m. When n is a maximal divisor of m, we write n |_{M} m.

16 CHAPTER 1. DEFINITIONS AND NOTATION

The Dihedral Groups

This chapter is taken from [25].

2.1 Introduction

The main part of this thesis is devoted to the classification of all firm residually connected geometries on which a group Sz(q), with q an odd power of 2, acts residually weakly primitively. In order to classify the geometries of rank greater than 2, we need to know the geometries of (some of) their maximal subgroups because of the Rwpri condition. Actually, every Suzuki simple group has maximal subgroups isomorphic to dihedral groups and other families of finite simple groups have some. So, this chapter may be seen as a necessary prelude to a classification process for some infinite families of finite simple groups.

The chapter is organised as follows. In section 2, we compute the automorphism group of a dihedral group. In section 3, we list the maximal subgroups of a dihedral group, as given in [36]. In section 4, we classify all the rank 2 geometries. Finally in section 5, we classify all the geometries of rank greater than 2.

2.2 The Automorphism group of a dihedral group

Let us recall that a dihedral group G = D_{2n}, with n ≥ 1 an integer, is a group generated by two elements a and b characterized by the relations

a^{2} = b^{2} = 1 = (ab)^{n}

as defined, for example, in [16]. The elements a and b are called generating reflections. The case n = 1 gives the cyclic group of order 2, and the case n = 2 gives the elementary abelian group 2^{2}. We find it convenient to represent G as the automorphism group of a regular n-gon denoted by P.

Observe that | Aut(G) |= 1 (resp. 6) for n = 1 (resp. 2).

Lemma 2.2.1 If n ≥ 3, then | Aut(G) |= ϕ(n) · n

Proof: The n-gon P has n symmetry axes. The elements a and b can be seen as two generating reflections, each of them stabilizing an axis of P. Remember that a rotation of P is the product of two reflections. The automorphism group of G is then the group that sends two generating reflections onto two other generating reflections. We can apply a onto one of the n reflections. Since the images of a and b must generate the cyclic group of rotations Z_{n} of D_{2n}, we may still apply b onto ϕ(n) different reflections. Thus | Aut(G) |= n · ϕ(n). □

2.3 On the subgroups of a dihedral group

The maximal subgroups of a dihedral group are given in [36]. Let G = D_{2n} be a dihedral group. Its maximal subgroups have the following structure.

• Z_{n}, which is the group of rotations of P.

• D_{2m} with m a maximal divisor of n. These subgroups are the stabilizers of the maximal subpolygons of P.

Remark that if n is even, there are two non-conjugate subgroups D_{n} in D_{2n}. One of them may be seen as the stabilizer of a \frac{n}{2}-subpolygon of the n-gon, and the other is the group generated by the rotations of the n-gon and the involution that exchanges the two \frac{n}{2}-subpolygons of the n-gon.

The following lemma is very useful in the next section while trying to construct Rwpri geometries.

Lemma 2.3.1 If \frac{n}{m} ≠ 2 and m ≠ 1, then D_{2m} is self-normalizing in D_{2n}. If \frac{n}{m} = 2 then D_{2m} is a normal subgroup of D_{2n}.

Proof: The second part of the lemma is obvious. Let us prove the first. Suppose \frac{n}{m} ≠ 2. We know that D_{2m} is a maximal subgroup of D_{2n} since its index is a prime number. The normalizer of D_{2m} is then either itself or D_{2n}. We know that there are \frac{n}{m} subgroups isomorphic to D_{2m} in D_{2n}. They are the stabilizers of the \frac{n}{m} different m-gons that we can construct in a n-gon by starting from the same edge. All of them are conjugate. □

2.4 Geometries of rank 2

To construct a Rwpri rank two geometry, we first choose a G_{0} which is maximal in G, then a G_{01} maximal in G_{0}. And finally, we look at all the G_{1} that contain G_{01} as a maximal subgroup.

2.4.1 G_{0} = Z_{n}

All maximal subgroups of G_{0} are cyclic subgroups of prime index. Thus G_{01} = Z_{m} with m a maximal divisor of n. The subgroups of G that contain G_{01} as a maximal subgroup are Z_{n} and D_{2m}. The first possibility leads to a contradiction since G_{0} must be distinct from G_{1}. Hence G_{1} = D_{2m}. Because [D_{2n} : Z_{n}] = [D_{2m} : Z_{m}], we can conclude that this geometry is a generalized digon. For given integers n and m, we now count how much geometries appear up to isomorphism. If n ≠ 2 and \frac{n}{m} ≠ 2, there is only one conjugacy class of D_{2m} in G. Then we may assume G_{1} is given and since there is only one Z_{n} in D_{2n}, we can construct only one such geometry up to isomorphism. If n ≠ 2 and \frac{n}{m} = 2, there are two classes of D_{2m} that are not fused under the action of Aut(D_{2n}). Thus is this case, we get two geometries that are nonisomorphic. If n = 2, there are three conjugacy classes of D_{2} = Z_{2} in D_{4} = 2^{2}. The automorphism group acts doubly transitively on the involutions of G. Thus there is only one such geometry up to isomorphism. Let n = p_{1}^{e_{1}} … p_{i}^{e_{i}} be the factorization of n. Then, there are i possible choices for G_{01} and thus i (resp. i + 1) for G_{1} if n is odd (resp. even).

2.4.2 G_{0} = D_{2m} and G_{1} ≠ Z_{n}

We have to distinguish two cases.

First case: n is prime. The case n = 2 has been analysed in section 2.4.1. So we now assume n > 2. Then G_{0} = Z_{2} and G_{1} = Z_{2} give a Rwpri geometry which is a regular n-gon. There are n subgroups of order 2. We may assume G_{0} is given. Then, there are n − 1 subgroups G_{1} that give a Rwpri geometry with G_{0}. These geometries are two by two conjugate, and they all are isomorphic since ϕ(n) = n − 1 when n is a prime number. Thus there is only one such geometry up to isomorphism.

Second case: n is not prime. The G_{01} candidates are Z_{m} and D_{2l} with l a maximal divisor of m. Suppose G_{01} = Z_{m}. The only subgroups containing it maximally have structure Z_{n} or D_{2m}. The first choice leads to geometries already obtained in section 2.4.1. Thus we may assume G_{1} = D_{2m}. We know that there are \frac{n}{m} = p subgroups isomorphic to G_{0} in G. Each of them contains only one Z_{m}. And Z_{m} is a normal subgroup of G. Thus there are p subgroups isomorphic to G_{0} which contain the same Z_{m}. We may assume G_{0} and G_{01} are given. The stabilizer of these two subgroups is G_{0}. There are p − 1 subgroups that can play the role of G_{1}. If p = 2, then p − 1 = 1 which means that there is only one such geometry up to isomorphism. Else, there are p−1 candidates for G_{1} and p−1 is an even number. The stabilizer of G_{0} and G_{01} being G_{0}, we have \frac{p−1}{2} non-conjugate geometries. It is easy to see that ϕ(n) ≥ p − 1. Thus the automorphism group | Aut(G) | fuses all the non-conjugate geometries inside a unique set and we have only one such geometry up to isomorphism.

Suppose G_{01} = D_{2l}. If l = 1, then m is a prime number and n is the product of two prime numbers. Let’s say n = m·p. Then G_{01} = Z_{2} and the G_{1} candidates are D_{2m} or D_{2p}. The first case gives a geometry which is a p-gon, while the second one gives a generalized digon. These two geometries are unique up to isomorphism. If l ≠ 1, then G_{1} must be a dihedral group. Suppose n = p_{1}^{e_{1}} … p_{i}^{e_{i}}. If n is odd, then we may assume m = n_{p_{a}}, l = m_{p_{b}} and G_{1} = D_{2m′} where m′ = n_{p_{b}} and a ≠ b thanks to lemma 2.3.1. This lemma implies also that such configurations are unique up to isomorphism. There are i possible choices for G_{0} and i − 1 choices for G_{1}. Thus there are \frac{i·(i−1)}{2} such geometries. If n is even, we must count twice the configurations with G_{0} = D_{n}. Thus we get \frac{i(i−1)}{2} + i − 1 geometries.

Now we can state a classification theorem for the rank 2 Rwpri geometries of a dihedral group.

Theorem 2.4.1 Let G = 2^{2}. Up to isomorphism, G has only one rank 2 geometry satisfying F, RC, FT, and Rwpri. It has two maximal parabolic subgroups isomorphic to cyclic groups of order 2 and it is a generalized digon. Let G = D_{2n}, with n ≥ 3 an integer, be a dihedral group and suppose n = p_{1}^{e_{1}} … p_{m}^{e_{m}}. If n is odd (resp. even), up to isomorphism, the group G has 2m + \frac{m(m−1)}{2} (resp. 3m + \frac{m(m−1)}{2}) geometries of rank 2 satisfying F, RC, FT, and Rwpri. They are given below.

• m (resp. m + 1) geometries with the following diagram:

[figure: 1; 2; Z_{n}; p_{i} − 1; p_{i}; D_{2n_{p_{i}}}; B = Z_{n_{p_{i}}}]

• m (resp. m) geometries with the following diagram:

[figure: p_{i}; 1; p_{i}; D_{2n_{p_{i}}}; 1; p_{i}; D_{2n_{p_{i}}}; B = Z_{n_{p_{i}}}]

• \frac{m(m−1)}{2} (resp. \frac{m(m−1)}{2} + m − 1) geometries with the following diagram:

[figure: p_{i} − 1; p_{i}; D_{2n_{p_{i}}}; p_{j} − 1; p_{j}; D_{2n_{p_{j}}}; B = D_{2n_{p_{i}p_{j}}}]

2.5 Geometries of rank at least 3

The following result implies that all the Rwpri geometries constructed in this section are residually connected and flag-transitive. As in [1], a string is a graph on a set I such that we can order I = {1, …, n} so that the edges of the graph are {i, i + 1}, 1 ≤ i < n.

Lemma 2.5.1 [1] Let G be a group, I a finite set, and Γ = (G_{i} : i ∈ I) a family of subgroups of G. Assume:

(i) for each subset J of I of corank at least 2, G_{J} =< G_{J∪{i}} : i ∈ I \ J >, and (ii) the connected components of the graph of the diagram of Γ are strings. Then (1) G is flag-transitive on Γ.

(2) Γ is residually connected.

We classify directly geometries for all ranks higher than 2. Let n = p_{1}^{e_{1}} … p_{m}^{e_{m}} and m ≥ 1 (the case m = 0 gives the cyclic group of order 2 whose only Rwpri geometry is of rank 1). We construct all Rwpri geometries of rank α = 3, …, m+ 1 which is the highest rank.

2.5.1 G_{0} = Z_{n}

The residue of G_{0} is of the following form.

[figure: …; p_{i_{1}} − 1; p_{i_{1}}; Z_{n_{p_{i_{1}}}}; p_{i_{2}} − 1; p_{i_{2}}; Z_{n_{p_{i_{2}}}}; p_{i_{α−1}} − 1; p_{i_{α−1}}; Z_{n_{p_{i_{α−1}}}}]

An easy exercise shows that, here, the highest rank of the geometries for G is m + 1. The G_{j} candidates cannot be cyclic groups. Thus, every G_{j} is a dihedral group having as subgroup a cyclic group Z_{n_{p_{j}}}. This implies G_{j} ≅ D_{2n_{p_{j}}}. And no two distinct G_{j}’s are isomorphic. Thus we can conclude by saying that the corresponding geometries have the following diagrams.

[figure: …; 1; 2; Z_{n}; p_{i_{1}} − 1; p_{i_{1}}; D_{2n_{p_{i_{1}}}}; p_{i_{α−1}} − 1; p_{i_{α−1}}; D_{2n_{p_{i_{α−1}}}}]

There are (_{α−1}^{m}) distinct diagrams. If n is odd, each one corresponds to a unique geometry up to isomorphism, for every rank α = 3, …, m + 1. If n is even, since there are two subgroups D_{n} which are non-conjugate under the action of Aut(G), there are (_{α−1}^{m−1})+2·(_{α−2}^{m−1}) geometries up to isomorphism. Thanks to lemma 2.5.1, we know that all these geometries are residually connected and flag-transitive.

2.5.2 G_{0} = D_{2n_{p_{i}}} and G_{i} ≠ Z_{n}

Let us first consider the case where all the G_{i}’s are pairwise non-isomorphic. Then the sublattice spanned by the maximal parabolic subgroups is composed of cyclic groups of prime order or dihedral groups. Thus we obtain the following geometries which are all unique up to isomorphism.

[figure: …; p_{i_{2}} − 1; p_{i_{2}}; D_{2n_{p_{i_{2}}}}; p_{i_{1}} − 1; p_{i_{1}}; D_{2n_{p_{i_{1}}}}; p_{i_{α}} − 1; p_{i_{α}}; D_{2n_{p_{iα}}}]

If n is odd, there are (_{α}^{m}) such geometries and the highest rank is m. If n is even, there are two D_{n} that are non-conjugate under the action of Aut(G) and so there are (_{α}^{m−1}) + 2 · (_{α−1}^{m−1}) geometries up to isomorphism. Thanks to lemma 2.5.1, we know that all these geometries are residually connected and flag-transitive.

Now suppose that exactly two of the G_{i}’s are isomorphic. While descending in the sublattice, we keep two isomorphic dihedral groups that intersect on a cyclic group at the bottom of the sublattice. This gives a rank 2 residue which is a polygon. All other rank 2 residues are generalized digons. Thus we have the following diagram.

[figure: p_{i_{1}}; …; 1; p_{i_{1}}; D_{2n_{p_{i_{1}}}}; 1; p_{i_{1}}; D_{2n_{p_{i_{1}}}}; p_{i_{2}} − 1; p_{i_{2}}; D_{2n_{p_{i_{2}}}}; p_{i_{α−1}} − 1; p_{i_{α−1}}; D_{2n_{p_{i_{α−1}}}}]

If n is odd, there are m · (_{α−2}^{m−1}) such geometries and the highest rank is m + 1. If n is even, since there are two D_{n} that are non-conjugate under Aut(G), there are (_{α−2}^{m−1}) + 2 · (m − 1) · (_{α−3}^{m−2}) + (m − 1) · (_{α−2}^{m−2}) geometries up to isomorphism. Thanks to lemma 2.5.1, we know that all these geometries are residually connected and flag-transitive.

It is easy to show that there cannot be three G_{i}’s that are isomorphic. Indeed, suppose it is possible. Then the lattice must have two subgroups that are equal, a contradiction.

It is also easy to show that there cannot be two pairs of isomorphic subgroups in the G_{i}’s. Again, in the lattice, this would imply that two subgroups are the same, a contradiction.

This ends the classification of the F, RC, FT, Rwpri geometries of a dihedral group. We now resume the preceding discussion in a theorem.

Theorem 2.5.1 Let G = D_{2n} be a dihedral group with n = p_{1}^{e_{1}} … p_{m}^{e_{m}} > 2. If n is odd (resp. even), up to isomorphism, the group G has (_{α−1}^{m}) + (_{α}^{m}) + m · (_{α−2}^{m−1}) (resp. (_{α}^{m−1}) + 3 · (_{α−1}^{m−1}) + 3 · (_{α−2}^{m−1}) + (m − 1) · (_{α−2}^{m−2}) + 2 · (m − 1) · (_{α−3}^{m−2})) + geometries of rank α, that satisfy F, RC, FT and Rwpri (α = 2 … m + 1). They are given below.

• (_{α−1}^{m}) (resp. (_{α−1}^{m−1}) + 2 · (_{α−2}^{m−1})) geometries with the following diagram:

[figure: …; 1; 2; Z_{n}; p_{i_{1}} − 1; p_{i_{1}}; D_{2n_{p_{i_{1}}}}; p_{i_{α−1}} − 1; p_{i_{α−1}}; D_{2n_{p_{i_{α−1}}}}; B = Z_{n_{p_{i_{1}} ...p_{i_{α−1}}}}]

• (_{α}^{m}) (resp. (_{α}^{m−1}) + 2 · (_{α−1}^{m−1})) geometries with the following diagram:

[figure: …; p_{i_{2}} − 1; p_{i_{2}}; D_{2n_{p_{i_{2}}}}; p_{i_{1}} − 1; p_{i_{1}}; D_{2n_{p_{i_{1}}}}; p_{i_{α}} − 1; p_{i_{α}}; D_{2n_{p_{iα}}}; B = D_{2n_{p_{i_{1}} ...p_{iα}}}]

• m · (_{α−2}^{m−1}) (resp. (_{α−2}^{m−1}) + 2 · (m − 1) · (_{α−3}^{m−2}) + (m − 1) · (_{α−2}^{m−2})) geometries with the following diagram:

[figure: p_{i_{1}}; …; 1; p_{i_{1}}; D_{2n_{p_{i_{1}}}}; 1; p_{i_{1}}; D_{2n_{p_{i_{1}}}}; p_{i_{2}} − 1; p_{i_{2}}; D_{2n_{p_{i_{2}}}}; p_{i_{α−1}} − 1; p_{i_{α−1}}; D_{2n_{p_{i_{α−1}}}}; B = Z_{n_{p_{i_{1}} ...p_{i_{α−1}}}}]

Remark that this theorem also gives the rank 2 classification as it was stated in theorem 2.4.1, except for G = 2^{2}. Two easy corollaries can be obtained from the preceding discussion.

Corollary 2.5.1 All the Rwpri geometries of a dihedral group are Rpri.

Corollary 2.5.2 All the Rwpri geometries of a dihedral group satisfy (IP)_{2}.

Theorem 2.5.1 shows that the number of non-isomorphic geometries obtained for a dihedral group D_{2n} depends only on the number of distinct prime divisors of n and on the fact that n is either odd or even. Table 2.1 gives, for every number n = p_{1}^{e_{1}} … p_{m}^{e_{m}} ≥ 3, with m ≤ 10, the number of non-isomorphic geometries of rank 2, …, m+1. For example, if we take D_{60}, we have n = 30 = 2 · 3 · 5. Thus m = 3. Since n is even, we can say thanks to table 2.1 that D_{60} has 12, (resp. 15,

7) geometries of rank 2 (resp. 3, 4) up to isomorphism. This table, which goes up to m = 10, covers all the dihedral groups D_{2n} for n < 2 · 3 · 5 · 7 · 11 · 13 · 17 · 19 · 23 · 29 · 31 = 200, 560, 490, 130 and, if n is odd, it covers all dihedral groups D_{2n} up to n < 3 · 5 · 7 · 11 · 13 · 17 · 19 · 23 · 29 · 31 · 37 = 3, 710, 369, 067, 405. It is given to show that the number of geometries obtained remains reasonable.

m n odd  n even
1 2  3
2 5, 3  7, 5
3 9, 10, 4  12, 15, 7
4 14, 22, 17, 5  18, 31, 27, 9
5 20, 40, 45, 26, 6  25, 54, 67, 43, 11
6 27, 65, 95, 81, 37, 7  33, 85, 135, 126, 63, 13
7 35, 98, 175, 196, 133, 50, 8 42, 125, 240, 291, 214, 87, 15
8 44, 140, 294, 406, 364, 204, 52, 175, 392, 581, 560, 337,
65,9  115,17
9 54, 192, 462, 756, 840, 624, 63, 236, 602, 1050, 1246, 988,
297,82,10  501,147,19
10 65, 255, 690, 1302, 1722,  75, 309, 882, 1764, 2478,
1590,1005,415,101,11  2430,1629,712,183,21

Table 2.1: The number of non-isomorphic geometries for a dihedral group D_{2n}, with n ≥ 3 an integer.

The rank 2 geometries of Sz(q)

This chapter is taken from [26].

3.1 Introduction

We begin the classification of all firm residually connected geometries on which a Suzuki simple group Sz(q), with q an odd power of 2, acts residually weakly primitively, by determining all the rank 2 geometries. In section 2, we give a list of useful properties that are used throughout the chapter to achieve this classification. In section 3, we recall the structure of maximal subgroups of a Suzuki simple group. In section 4, we give the structure of the maximal subgroups of each maximal subgroup of a Suzuki group. In section 5, we determine all rank 2 geometries that are obtained. The results of the quite long discussion and case analysis of section 5 leads to a classification theorem that we state in section 6. Finally, we close this chapter with section 7, in which we make some observations on the results obtained in the preceding sections.

In this chapter, we consider geometries of rank 2, i.e. geometries Γ(G; G_{0}, G_{1}). We call the quadruple (G, G_{0}, G_{1}, G_{01}) an Rwpri configuration (of rank 2) provided that G_{0} is maximal in G and G_{01} = G_{0} ∩ G_{1} is maximal in both G_{0} and G_{1}.

Lemma 3.1.1 If Γ is a Rwpri configuration, then Γ is firm, residually connected and flag-transitive.

Proof: This is a direct consequence of the results obtained in [17]. □

This lemma implies that all geometries of rank 2 which are Rwpri are also F, RC, and FT. Thus the search of the rank 2 geometries of the Suzuki group Sz(q) that are F, RC, FT and Rwpri is reduced to the search of all Rwpri configurations.

3.2 Some useful properties

In this section, we give some properties concerning q − 1, α_{q}, and β_{q}.

Lemma 3.2.1 Cataldi, 1603 If 2^{2e+1} − 1 is a prime number, then 2e + 1 is a prime number.

The following lemma implies that if q − 1 is a prime number then the Suzuki group Sz(q) does not have any subgroup isomorphic to a Suzuki group of lower order. Let us remark that the converse is not true.

Lemma 3.2.2 If there exist s = 2^{2f+1} and an integer a such that q = s^{a}, then q − 1 is not a prime number.

Proof: s = 2^{2f+1} and q = 2^{2e+1} = 2^{a·(2f+1)} implies that 2e + 1 is not a prime number. Suppose q − 1 is a prime number. Then lemma 3.2.1 says that 2e + 1 must be a prime number. Thus we have a contradiction. □

Lemma 3.2.3 The numbers q - 1, α_{q} and β_{q} are pairwise coprime.

Proof: (q - 1, α_{q}) = p implies that p divides α_{q}- (q - 1) and α_{q}+ (q - 1). Thus p divides 2^{e+1} + 1 and 2^{e+1} + 2. This implies p = 1. By a similar argument, we see that (q - 1, β_{q}) = p implies that p divides both 2^{e+1} − 2 and 2^{e+1} − 1. Thus p = 1. At last, suppose (α_{q}, β_{q}) = p ≠ 1. Then p divides α_{q}- β_{q} which is a power of 2. Thus p is a even number dividing two odd numbers, a contradiction. □

Lemma 3.2.4 Let s = 2^{2e+1} and q = s^{2f+1}. Then s^{2} + 1 divides q^{2} + 1.

Proof: s^{2} + 1 = 2^{4e+2} + 1 and q^{2} + 1 = 2^{(4e+2)(2f+1)} + 1. Thus we have

\frac{q^{2} + 1}{s^{2} + 1} = ∑_{i=1}^{2f+1}(−1)^{i+1} · 2^{(4e+2)(2f+1−i)}

□

Lemma 3.2.5 Let s = 2^{2e+1} and q = s^{2f+1}. Then, s^{2} + 1 does not divide α_{q} nor β_{q}.

Proof: The following argument is due to J.A. Thas. Assume first that s^{2} + 1 | α_{q}. Then

2^{2(2e+1)} + 1 | 2^{(2e+1)(2f+1)} + 2^{2ef+e+f+1} + 1

Suppose first that f is odd. Then

\begin{aligned}2^{2(2e+1)} + 1 & | & (2^{2(2e+1)f} + 1 − 1)2^{2e+1} + 2^{2ef+e+f+1} + 1 \\ & | & −2^{2e+1} + 2^{2ef+e+f+1} + 1 \end{aligned}

Because f is odd, we can write f = 2r + 1 with r an integer. Then

\begin{aligned}2^{2(2e+1)} + 1 & | & −2^{2e+1} + 2^{4er+2e+2r+1+e+1} + 1 \\ & | & −2^{2e+1} + 2^{2(2e+1)r+3e+2} + 1 \end{aligned}

Suppose that r is odd. Then

2^{2(2e+1)} + 1 | −2^{2e+1} + (2^{2(2e+1)r} + 1 − 1)2^{3e+2} + 1
| −2^{2e+1} − 2^{3e+2} + 1
| 2^{2e+1} + 2^{3e+2} − 1

But 2^{4e+2} + 1 > 2^{2e+1} + 2^{3e+2} − 1 thus we have a contradiction. Suppose that r is even. Then r = t + 1, with t odd and we have the following.

2^{2(2e+1)} + 1 | −2^{2e+1} + (2^{2(2e+1)t} + 1 − 1)2^{7e+4} + 1
| −2^{2e+1} − 2^{7e+4} + 1
| (2^{2(2e+1} + 1 − 1)2^{3e+2} + 2^{2e+1} + 1
| 2^{3e+2} − 2^{2e+1} − 1

But 2^{4e+2} + 1 > 2^{3e+2} − 2^{2e+1} − 1 thus we have a contradiction. Hence we have proved that f cannot be odd. The same reasoning can be made for f even. And the same arguments can be used to show that 2^{2(2e+1)} + 1 cannot divide β_{q}. □

Lemma 3.2.6 Let s = 2^{2e+1} and q = s^{2f+1}, with e and f some positive integers. Then α_{q} and β_{q} are not prime.

Proof: Easy to see thanks to lemmas 3.2.4, 3.2.5. □

Structure  Order  Index Description
(qˆ .q):(q-1)  q^{2} · (q − 1)  q^{2} + 1 Normalizer of a 2-Sylow,

stabilizer of a point of the ovoid.

D_{2(q−1)} 2 · (q - 1) \frac{(q^{2}+1)·q^{2}}{2} Stabilizer of a pair of

points of the ovoid.

α_{q}:4  α_{q}·4  \frac{q^{2}(q−1)}{4β_{q}} Normalizer of a Z_{α_{q}}.
β_{q}:4  β_{q}·4  \frac{q^{2}(q−1)}{4α_{q}} Normalizer of a Z_{β_{q}}.
Sz(2^{2f+1})  (s^{2} + 1) · s^{2} · (s − 1)

with 2f + 1 |_{M} 2e + 1

Table 3.1: The maximal subgroups of Sz(q)

Lemma 3.2.7 Fermat, 1640 If p is a prime number then p divides 2^{p} − 2.

3.3 The maximal subgroups of Sz(q)

Table 3.1 gives a description of the conjugacy classes of maximal subgroups of Sz(q). Those results have been obtained by Suzuki [40]. It is from these subgroups that we start to classify all Rwpri configurations in the subgroup lattice of Sz(q). For a Suzuki simple group Sz(q), with q = 2^{2e+1}, if 2e + 1 is a prime number, there are no proper subgroups isomorphic to a Suzuki simple group. This implies that if 2e + 1 is a prime number, the group Sz(q) has only four conjugacy classes of maximal subgroups. Some Suzuki groups may have subgroups of Suzuki type which are not isomorphic. For example, Sz(32768) has maximal subgroups that are respectively isomorphic to Sz(8) and Sz(32). This means we don’t have five classes of maximal subgroups there but six. In fact, the number of non-isomorphic classes of Sz(s) is equal to the number of prime divisors of 2e + 1. Dividing 2e + 1 by every of the latter primes gives us a number 2f + 1 for which there exists a Sz(2^{2f+1}) which is a maximal subgroup of Sz(q).

3.4. THE MAX. SUBGROUPS OF THE MAX. SUBGROUPS OF SZ(Q) 31

3.4 The maximal subgroups of the maximal subgroups of Sz(q)

Lemma 3.4.1 The maximal subgroups of (qˆ .q) : (q-1) have one of the following structures.

• (qˆ .q):n where n is a maximal divisor of q - 1.

• q:(q - 1)

Proof: The subgroup G_{0} is the stabilizer of a point. Let us take U as the fixed point. Then, a complete description of this group is given in [33]. Lüneburg also shows that the stabilizer of a circle is of order q(q−1). As we defined it in chapter 1, a circle is obtained by intersecting the ovoid with a non-tangent plane. If we take the plane Π defined by a cartesian equation x = 0, we see that the τ(0, b) and the η(k) stabilize this plane for b ∈ GF (q) and k ∈ GF (q)^{∗} = GF (q) − {0}. Looking at the subgroup generated by these collineations, we have a subgroup of order q(q−1) which is the stabilizer of a circle. This group has a normal subgroup formed by the τ(0, b), of order q. And its structure is q : (q − 1). We denote it by G_{C}. We still have to show that it is a maximal subgroup of G_{0}. Thanks to lemma 1.2.2, every element of G_{0} can be written in a unique form as η(k)τ(a, b) with a, b ∈ GF (q) and k ∈ GF (q)^{∗}. Thus we see that the only elements we could add are of the form τ(a, 0) with a ∈ GF (q), a ≠ 0. If we add any of these elements to G_{C}, the η(k)’s act so that every τ(a, 0) is generated, and thus G_{0} is generated. We can then conclude that G_{C} is maximal in G_{0}.

Taking all τ(a, b), we obtain a subgroup of structure qˆ .q which is normal in G_{0}. The subgroup formed by the η(k)’s, with k ∈ GF (q)^{∗} is a cyclic group of order q − 1. Its maximal subgroups are the subgroups of order n, where n is a maximal divisor of q − 1. If we take one generator of such a subgroup, namely η(k) and we look at the subgroup generated by the τ(a, b) and this η(k), we obtain a subgroup of order q^{2} · n. Again, qˆ .q is normal in it and thus this subgroup has structure (qˆ .q):n. The only elements that are missing use η(l) with l not appearing in the multiplicative group generated by k. This multiplicative group is a maximal subgroup of GF (q)^{∗}. We can conclude that (qˆ .q):n is maximal in G_{0}.

The group generated by all the τ(a, 0) and the η(k), with a ∈ GF (q) and k ∈ GF (q)^{∗} is G_{0}. Thus no other maximal subgroups can be found. □

Lemma 3.4.2 The maximal subgroups of n : 4 have the following structures.

• m:4 where m is a maximal divisor of n.

• n:2

Moreover, if n is odd, every maximal divisor m of n gives only one conjugacy class of subgroups isomorphic to m : 4.

Proof: Obviously, n : 4 has a maximal subgroup whose structure is n : 2. In n : 4, there exists a normal subgroup isomorphic to m for each divisor m of n. Extending this cyclic group by 4, we obtain a group of order 4m which has structure m:4 because (m, 4) = 1. If m is a maximal divisor of n, then [n:4 : m:4] is a prime number, hence m:4 is a maximal subgroup of n:4. There remains to show that, if n is odd, for a given maximal divisor m of n, all subgroups m : 4 are conjugate in n : 4. Suppose we have two subgroups A and B isomorphic to m : 4. Then, A =< Z_{m}, Z_{4}^{A} > and B =< Z_{m}, Z_{4}^{B} > for some subgroups Z_{4}^{A} and Z_{4}^{B} of n : 4. If n is odd, the subgroups Z_{4}^{A} and Z_{4}^{B} are 2-Sylow subgroups. Hence they are conjugate in n : 4. Thus there exists an element g ∈ n : 4 such that (Z_{4}^{A})^{g} = Z_{4}^{B}. Since Z_{m} is a normal subgroup of n : 4, it is then obvious that A^{g} = B. □

Lemma 3.4.3 The maximal subgroups of D_{2(q−1)} have the following structure.

• the cyclic group of order q − 1

• D_{2n} where n is a maximal divisor of q − 1. If q − 1 is a prime number then D_{2} is the cyclic group of order 2.

Moreover, if q−1 is odd, every maximal divisor n of q−1 gives only one conjugacy class of subgroups isomorphic to D_{2n}.

Proof: Straightforward □

As to the maximal subgroups that are isomorphic to some Suzuki simple group, we refer to table 3.1.

3.5 Rank 2 Rwpri geometries

Let G be a group. Let G_{0} and G_{1} be subgroups of G and let G_{01} = G_{0} ∩ G_{1}. The Rwpri condition in rank 2 asks that either G_{0} or G_{1} must be a maximal subgroup of G and that G_{01} must be a maximal subgroup of G_{0} and G_{1}. We use the following method to determine all the Rwpri geometries of the Suzuki group Sz(q). We choose a subgroup G_{0} and we impose that it must be a maximal subgroup of Sz(q). Then we determine all the G_{01} possible groups. They are the maximal subgroups of G_{0}. Finally, we look at all the G_{1} candidates which must contain G_{01} as a maximal subgroup and which must be different from G_{0}.

Lemma 3.5.1 Let G ≅ Sz(q). There are no Rwpri geometries of rank > 1 with G_{0} ≅ (qˆ .q) : (q - 1).

Proof: We start from the maximal subgroups of G_{0}. They have been determined in lemma 3.4.1. We see that the only subgroups of Sz(q) that contain one of the G_{01} candidates are thus isomorphic to G_{0}. Because the normalizer of each G_{01} candidate must be a subgroup of G_{0}, we obtain that G_{1} must be equal to G_{0} which is a contradiction with the fact that G_{01} is their intersection. □

Lemma 3.5.2 Let e > 0 be a natural number. Let q = 2^{2e+1} and let G ≅ Sz(q) If 2e + 1 and q − 1 are prime numbers, the Rwpri rank 2 geometries of G with G_{0} ≅ D_{2(q−1)} are isomorphic to one of the geometries appearing in table 3.2. If 2e + 1 is prime and q − 1 is not prime, the only Rwpri rank 2 geometry of G with G_{0} ≅ D_{2(q−1)} is obtained with G_{1} ≅ q : (q-1). If 2e + 1 is not prime, the Rwpri rank 2 geometries of G with G_{0} ≅ D_{2(q−1)} are isomorphic to one of the geometries appearing in table 3.3.

Proof: By lemma 3.4.3, G_{01} can be isomorphic to either q − 1 of D_{2n} where n is a maximal divisor of q − 1.

Suppose first that G_{01} = q − 1.

The G_{1} candidates are neither subgroups of α_{q}:4, nor of β_{q}:4 because of lemma 3.2.3. The normalizer of a q − 1 is G_{0}. Thus two dihedral groups D_{2(q−1)} never intersect on a cyclic group of order q − 1 in Sz(q), and G_{1} cannot be a D_{2(q−1)}.

G_{1}  G_{01} #Γ
q : (q − 1)  q − 1 1
D_{2(q−1)} 2 \frac{(q−2)(q+4)}{4(2e+1)} + 1
D_{2n} where n is a prime divisor of α_{q} or β_{q} 2  \frac{(q−2)(q+2)}{4(2e+1)} + 1
2^{2}  2  \frac{q−2}{2(2e+1)}
4  2  \frac{q−2}{2(2e+1)} + 1
Table 3.2: G_{0} = D_{2(q−1)}, and 2e + 1 and q − 1 are prime numbers

G_{1}  G_{01}  #Γ
q : (q − 1)  q − 1 1
Sz(s) iff s-1 is a maximal divisor of q-1 D_{2(s−1)} 1
Table 3.3: G_{0} = D_{2(q−1)} and 2e + 1 is not prime

So the only candidates must be subgroups of (qˆ .q):(q-1) or Sz(s) with q = s^{a} for some integer a. Suppose G_{1} < (qˆ .q):(q-1). Then, in order to have G_{01} maximal in G_{1} the only candidate G_{1} is q : (q − 1). This gives rise to a Rwpri configuration. If we fix G_{1} and G_{01} (this can be done because all subgroups isomorphic to G_{01} are conjugate in G_{1}), there is only one D_{2(q−1)} intersecting G_{1} in G_{01}. Thus we have only one such configuration up to isomorphism. Suppose G_{1} < Sz(s). As Sz(s) exists, it implies that q − 1 is not a prime number. No maximal subgroup of Sz(s) contains a cyclic group of order q − 1. Thus G_{1} must be Sz(s). But q − 1 is not a maximal subgroup of Sz(s).

Now suppose that G_{01} = D_{2n} where n is a maximal divisor of q − 1.

Let us first examine the case where q − 1 is a prime number. Then G_{01} is a cyclic group of order 2. This gives rise to the following candidates for G_{1}.

• D_{2(q−1)}

• D_{2m} where m is a prime divisor of α_{q} or β_{q}.

• 2^{2}

• 4

All of these subgroups can be taken as G_{1} to form a Rwpri configuration. Suppose G_{1} = D_{2(q−1)}. We can fix G_{0} and G_{01} because all subgroups isomorphic to G_{01} are conjugate in G_{0}. We know that there are \frac{q^{2}·(q^{2}+1)}{2} subgroups isomorphic to G_{0} in Sz(q). We also know that G_{0} contains q − 1 subgroups of order 2. And the number of involutions in Sz(q) is equal to (q^{2} + 1)(q − 1). Thus we can conclude that there are \frac{q^{2}}{2} subgroups isomorphic to G_{0} intersecting in the same G_{01}. If we want to fix the types of our geometries, G_{01} cannot transform one of these geometries into another. If we do not fix the types, then we have to count how many of our geometries are self-dual by conjugacy. Suppose we have a geometry Γ(Sz(q); D_{2(q−1)} = G_{0}, D_{2(q−1)} = G_{1}). We know that the normalizer of G_{01} is a group of order q^{2} isomorphic to qˆ .q By looking at the action of H = qˆ .q/G_{01} on the \frac{q^{2}}{2} subgroups isomorphic to D_{2(q−1)} containing G_{01}, we see than it is transitive and hence striclty transitive because its order is equal to its degree. Because H contains q−1 involutions, there are q−1 geometries that are self-dual, and the \frac{q^{2}−2q}{2} elements of order 4 spread the \frac{q^{2}}{2} subgroups into squares. Every square appears two times, that is one time in each way. And no two disjoint squares can have a common vertex. This means that through every of the \frac{q^{2}}{2} points, there are \frac{q^{2}−2q}{4} disjoint squares that fuse \frac{q^{2}−2q}{2} geometries inside sets of 2 geometries. So finally, we have q − 1 + \frac{q^{2}−2q}{4} = 1 + \frac{(q−2)(q+4)}{4} non-conjugate geometries in this case. In Aut(Sz(q)), the stabilizer of G_{0} and G_{01} is a cyclic group of order 2(2e + 1). Its normalizer is two times bigger. The normalizer of G_{1} is G_{1} : (2e + 1). And each subgroup isomorphic to G_{1} : (2e + 1) contains q −1 subgroups isomorphic to Z_{2(2e+1)}. This implies that there are two subgroups isomorphic to G_{1} : (2e + 1) containing the same Z_{2(2e+1)}. Thus one geometry is fixed under the action of Aut(Sz(q)) and all the others are fused inside sets of 2e + 1. That is, we have 1 + \frac{(q−2)(q+4)}{4(2e+1)} non-isomorphic Rwpri configurations and we observe that this number is indeed an integer thanks to lemma 3.2.7.

Suppose G_{1} = D_{2m} where m is a prime divisor of α_{q} or β_{q}. We can decide to fix it without loss of generality. All its subgroups isomorphic to G_{01} are conjugate in it. This implies that we can also fix G_{01}. The normalizer of G_{1} in G is m:4. So the stabilizer of G_{1} and G_{01} is a cyclic group of order 4. And there are \frac{q^{2}}{2} subgroups isomorphic to G_{0} that intersect G_{1} in G_{01}. The cyclic group of order 4 fuses them inside sets of two. So there are \frac{q^{2}}{4} non-conjugate geometries. The subgroup which stabilizes G_{0} and G_{01} in Aut(G) is isomorphic to Z_{2(2e+1)}. The subgroup stabilizing G_{1} in Aut(G) is isomorphic to G_{1} : (2e + 1). Again, a simple counting argument permits us to conclude that there is only one subgroup isomorphic to G_{1} which is fixed under the action of Z_{2(2e+1)}. Thus, from the \frac{q^{2}}{4} non-conjugate geometries, one is fixed under the action of Aut(G) and all other are fused inside sets of 2e+1 geometries. This means we have 1+ \frac{(q−2)(q+2)}{4(2e+1)} non-isomorphic Rwpri configurations of that kind. And, again, thanks to lemma 3.2.7, we observe that this number is indeed an integer.

Suppose G_{1} = 2^{2}. Then G_{1} contains three subgroups of order 2. Also, G_{1} is contained in the stabilizer of some point. In each point-stabilizer, there are q − 1 Z_{2}. We must combine two of them to obtain subgroups isomorphic to 2^{2}. We first choose one Z_{2}. Then, the second one can be chosen in \frac{q−2}{2} ways. But there we have taken each possibility 3 times. So the number of 2^{2} is \frac{(q^{2}+1)(q−1)(q−2)}{6}. From this, we can count the number of 2^{2} containing the same Z_{2}. There are \frac{q−2}{2} such subgroups. Thus, by fixing G_{0} and G_{01}, we have \frac{q−2}{2} possible G_{1}. These give \frac{q−2}{2} non-conjugate geometries. The normalizer of G_{1} in Aut(G) is G_{1} : (2e + 1) and this subgroup does not contain the stabilizer of G_{0} and G_{01} in Aut(G). Thus no configuration of that kind is fixed under the action of Aut(G). This means that all of them are fused in sets of 2e + 1 geometries. Hence we have \frac{q−2}{2(2e+1)} nonisomorphic Rwpri configurations like that. Let us observe that in the smallest case (i.e. for q = 8), there is only one geometry of that kind up to isomorphism.

Suppose G_{1} = 4. Let us count the number of cyclic groups of order 4 contained in a Sz(q). Each of them is contained in the stabilizer of a point (there are q^{2} + 1 such stabilizers). In each stabilizer, the τ(a, b) with a ≠ 0 are elements of order 4. There are (q − 1)q such elements. In a cyclic group of order 4, there are 2 elements of order 4. Thus the number of cyclic groups of order 4 in Sz(q) equals (q^{2}+1)(q−1)\frac{q}{2}. We already know that the number of Z_{2} in Sz(q) is (q^{2}+1)(q−1). And it is well known that every Z_{4} contains only one Z_{2}. Hence the number of Z_{4} containing the same Z_{2} is \frac{q}{2}. Now we fix G_{0} and we can also fix G_{01} because all the subgroups that are isomorphic to G_{01} are conjugate in G_{0}. The stabilizer of G_{0} and G_{01} is G_{01}. Thus it fixes G_{1} also. This means we have \frac{q}{2} non-conjugate geometries. The normalizer of G_{1} in Aut(G) being a subgroup isomorphic to 4 : (2e + 1), it contains the stabilizer of G_{0} and G_{01} in Aut(G). This means that there is one configuration which is fixed under the action of Aut(G) and all the others are fused under Aut(G) inside sets of 2e + 1 geometries. Thus we have 1 + \frac{q−2}{2(2e+1)} non-isomorphic Rwpri configurations of that kind.

G_{1}  G_{01} #Γ
α_{q}: 4 4 \frac{q−2)}{2(2e+1)}
n : 4 where n is a prime divisor of β_{q} 4  \frac{q−2}{2(2e+1)} + 1
4 × 2  4  \frac{q−2}{2(2e+1)}
Table 3.4: The case G_{0} = α_{q}: 4 and α_{q} prime.

G_{1}  G_{01}  #Γ
Sz(s) iff α_{s} or β_{s} is a maximal divisor of α_{q} α_{s} : 4 or β_{s} : 4 1
Table 3.5: The case G_{0} = α_{q}: 4 and α_{q} not prime.

Suppose now that q − 1 is not a prime number. Then G_{01} is a dihedral group D_{2n}. This means that the G_{1} candidates will neither be subgroups of α_{q}:4, nor of β_{q}:4. Here we have two candidates for G_{1}. They are respectively D_{2(q−1)} and Sz(s) if s − 1 = n. The case G_{1} = D_{2(q−1)} is not possible. This is due to the fact that N_{Sz(q)}(D_{2n}) = D_{2n}. While fixing G_{0}, we can also fix G_{01} because all the subgroups isomorphic to G_{01} in G_{0} are conjugate. It is easy to see that there is only one subgroup isomorphic to Sz(s) which contains G_{01}. So the G_{1} is uniquely determined. This gives us only one geometry up to conjugacy and thus up to isomorphism. □

An example of a configuration with G_{1} = Sz(s) can be obtained in Sz(512) by taking as G_{0} (resp. G_{1}, G_{01}) a subgroup isomorphic to D_{2(511)} (resp. Sz(8), D_{14}).

Lemma 3.5.3 Let e > 0 be a natural number. Let q = 2^{2e+1} and let G ≅ Sz(q) If α_{q} is a prime number, the Rwpri rank 2 geometries of G with G_{0} ≅ α_{q}: 4 are isomorphic to one of the geometries appearing in table 3.4. If α_{q} is not a prime number, the Rwpri rank 2 geometries of G with G_{0} ≅ α_{q}: 4 are isomorphic to one of the geometries appearing in table 3.5.

Proof: By lemma 3.4.2, G_{01} can be isomorphic to either α_{q}: 2 or n : 4 where n is a maximal divisor of α_{q}.

Suppose first that G_{01} = n : 4.

Let us first consider the case where n = 1 (i.e. α_{q} is a prime number). In this case, several candidates appear for G_{1}.

• 4 × 2

• m:4 where m | β_{q}, and m is a prime number.

• α_{q}:4

All these cases give rise to Rwpri candidates. Suppose G_{1} = 4 × 2. Let us count the number of subgroups isomorphic to 4 × 2 in Sz(q). Each of them is contained in the stabilizer of a point. In each point-stabilizer, there are (q − 1)\frac{q}{2} cyclic groups of order 4, as we saw before. The 4 × 2 are constructed by adding to these Z_{4} an element of order 2. This element must be different from the one in the Z_{4}. Thus we have q − 2 such possible elements. But a 4 × 2 contains 3 subgroups of order 2. This implies that there are \frac{q−2}{2} subgroups isomorphic to 4 × 2 which contain the same Z_{4}. And every 4 × 2 contains 2 subgroups of order 4. Thus the number of 4 × 2 is (q^{2} + 1)(q − 1)\frac{q}{2} \frac{q−2}{2} \frac{1}{2} = \frac{(q^{2}+1)q(q−1)(q−2)}{8}. Now let us count the number of nonconjugate and non-isomorphic geometries. We can fix G_{0}. All the subgroups isomorphic to Z_{4} are conjugate in G_{0}. This implies that we can also fix G_{01}. The stabilizer of G_{0} and G_{01} being G_{01}, we have \frac{q−2}{2} non-conjugate geometries. The (2e + 1) classes of subgroups isomorphic to 4 × 2 in G are fused under the action of Aut(G). This means that the normalizer of a subgroup isomorphic to 4×2 will not contain elements of order (2e + 1). The stabilizer of G_{0} and G_{01} in Aut(G) is a subgroup isomorphic to 4 : (2e + 1). Thus, there is no configuration which is fixed under the action of Aut(G). Then, all the non-conjugate configurations are fused under the action of Aut(G) inside sets of 2e + 1 geometries. Hence we have \frac{q−2}{2(2e+1)} non-isomorphic Rwpri configurations like that. Let us observe that in the smallest case (i.e. for q = 8), there is only one such geometry up to isomorphism.

Suppose G_{1} = m:4. The subgroup G_{1} has its normalizer in β_{q}:4. Thus it is selfnormalizing. This gives us the number of subgroups isomorphic to m:4 in Sz(q). There are \frac{(q^{2}+1)q^{2}(q−1)}{4m} such subgroups. Each of them contains m cyclic subgroups of order 4. And we already know that there are \frac{(q^{2}+1)q(q−1)}{2} cyclic groups of order 4 in Sz(q). Thus every Z_{4} is contained in \frac{q}{2} subgroups m:4. We can fix G_{0}. All the subgroups isomorphic to Z_{4} are conjugate in G_{0}. Thus we can also fix G_{01}. The stabilizer of G_{0} and G_{01} is G_{01}. This implies that there are \frac{q}{2} non-conjugate geometries here. The stabilizer of G_{1} and G_{01} in Aut(G) being isomorphic to 4 : (2e + 1), there is only one configuration which is fixed under the action of Aut(G) and all the others are fused inside sets of 2e + 1 geometries. Thus we have 1 + \frac{q−2}{2(2e+1)} non-isomorphic Rwpri configurations in that case.

Suppose G_{1} = α_{q}:4. Then there are \frac{q}{2} subgroups isomorphic to G_{1} that contain the same Z_{4}. One of them is G_{0} so we have here \frac{q}{2} − 1 non-conjugate geometries if we decide to fix the types. It is easy to see that N_{G}(Z_{4}) is a group which is of order 2q and which is isomorphic to q ˆ. 2. By looking at the action of (q ˆ.2)/4 on the set of \frac{q}{2} subgroups isomorphic to α_{q}:4, we see that this group is transitive and hence strictly transitive. It is also easy to see that all its elements (except the identity) are involutions. This means that all the \frac{q}{2} − 1 geometries we found are self-dual. Thus there are \frac{q−2}{2} non-conjugate geometries. The stabilizer of G_{0} and G_{01} in Aut(G) is isomorphic to 4 : (2e + 1). Thus there is no geometry fixed under the action of Aut(G) and all of them are fused inside sets of 2e + 1 elements. This means we have \frac{q−2}{2(2e+1)} non-isomorphic Rwpri configurations of that kind. Let us observe that in the smallest case (i.e. for q = 8), there is only one geometry up to isomorphism.

Now suppose n ≠ 1, which means that α_{q} is not a prime number. Then G_{1} is not a subgroup of β_{q}:4 (resp. D_{2(q−1)}) because (α_{q}, β_{q}) (resp. (α_{q}, q − 1)) = 1. Two candidates arise for G_{1}. They are α_{q}:4 and Sz(s) if and only if α_{s} or β_{s} is a maximal divisor of α_{q}. The normalizer N_{Sz(q)}(n:4) is n:4. Thus the case G_{1} = α_{q}:4 leads to a contradiction. We then have just one candidate left, that is the one with G_{1} = Sz(s). We can fix G_{0} and G_{01} because G_{0} conjugates all its subgroups isomorphic to G_{01}, the stabilizer being G_{01}. And there is only one Sz(s) containing G_{01}. Thus there is only one such geometry up to conjugacy, and hence up to isomorphism.

Suppose that G_{01} = α_{q}:2.

This implies that G_{1} must be a subgroup of Sz(s). So q − 1 cannot be a prime number. The order of G_{01} must divide the order of G_{1}. Suppose G_{1} is a subgroup of (sˆ .s):(s-1) or D_{2(s−1)}. Then α_{q} divides s − 1 which is a contradiction with the fact that (q − 1, α_{q}) = 1 and s − 1 | q − 1. Suppose G_{1} is a subgroup of α_{s}:4 (resp. β_{s}:4). Then the order of G_{1} is smaller than the order of G_{01} which is also a contradiction. Finally, suppose G_{1} = Sz(s). Then G_{01} must be a maximal subgroup of Sz(s). This yields s − 1 = α_{q} and s − 1 | q − 1, a contradiction. □

G_{1}  G_{01} #Γ
β_{q}: 4 4 \frac{q−2)}{2(2e+1)}
n : 4 where n is a prime divisor of β_{q} 4  \frac{q−2}{2(2e+1)} + 1
4 × 2  4  \frac{q−2}{2(2e+1)}
Table 3.6: The case G_{0} = β_{q}: 4 and β_{q} prime.

G_{1}  G_{01}  #Γ
Sz(s) iff α_{s} or β_{s} is a maximal divisor of β_{q} α_{s} : 4 or β_{s} : 4 1
Table 3.7: The case G_{0} = β_{q}: 4 and β_{q} not prime.

Lemma 3.5.4 Let e > 0 be a natural number. Let q = 2^{2e+1} and let G ≅ Sz(q) If β_{q} is a prime number, the Rwpri rank 2 geometries of G with G_{0} ≅ β_{q}: 4 are isomorphic to one of the geometries appearing in table 3.6. If β_{q} is not a prime number, the only Rwpri rank 2 geometries of G with G_{0} ≅ β_{q}: 4 are isomorphic to one of the geometries appearing in table 3.7.

Proof: Same as proof of lemma 3.5.3 □

At this stage, we have determined the Rwpri configurations for the rank 2 geometries of the Suzuki groups Sz(q) where 2e + 1 is a prime number. If 2e + 1 is not a prime number, Sz(q) may have other maximal subgroups which are isomorphic to Sz(s). Before starting the discussion about this case, we demonstrate a few lemmas that are used in the lemma giving the Rwpri rank 2 geometries of Sz(q) with G_{0} ≅ Sz(s).

Lemma 3.5.5 For every subgroup isomorphic to (sˆ .s):(s-1), there is only one subgroup isomorphic to Sz(s) containing it.

Proof: Let P be a subgroup isomorphic to (sˆ .s):(s-1). All subgroups isomorphic to Sz(s) are conjugate in Sz(q). We only have to show that N_{Sz(q)}(P) ≤ Sz(s) in order to prove that there exists only one Sz(s) containing P. Sz(s) has an orbit of length s^{2} + 1. In this orbit, P fixes a point x and exchanges all of the other s^{2} points. We know that N_{Sz(q)}(P) fixes x. So, it is a subgroup of the stabilizer G_{x} of x which has structure (qˆ .q):(q-1). In G_{x}, there is a cyclic subgroup Z_{q−1} which has exactly two fixed points. And so does Z_{s−1} ≤ Z_{q−1}. Thus the second fixed point is in the orbit of s^{2} points. We also know that P has one orbit of length 1, one of length s^{2} and all others of length ≥ s^{2}. It can be easily seen that the other orbits have length s^{2}(s − 1). Since s ≠ 2, the normalizer N_{Sz(q)}(P) must stabilize the orbit of 1 point and the orbit of s^{2} points. Thus it must be a subgroup of Sz(s). □

Lemma 3.5.6 For every subgroup isomorphic to D_{2(s−1)}, there is only one subgroup isomorphic to Sz(s) containing it.

Proof: The normalizer N_{Sz(q)}(D_{2(s−1)}) must be a subgroup of D_{2(q−1)}. Because q − 1 and s − 1 are odd numbers, D_{2(s−1)} is self-normalizing in D_{2(q−1)}. □

Lemma 3.5.7 [40] Let Π be the partition consisting of four classes of conjugate subgroups of Sz(q):

• the Sylow 2-subgroups of Sz(q),

• one class of cyclic groups of order q-1,

• one class of cyclic groups of order q+r+1,

• one class of cyclic groups of order q-r+1.

If W is a subgroup of Sz(q), then there exists X ∈ Π with W ⊆ N_{Sz(q)}(X) or W is isomorphic to Sz(s) with s ≤ q.

Lemma 3.5.8 For every subgroup isomorphic to α_{s}:4, there is only one subgroup isomorphic to Sz(s) that contains it.

Proof: The number of Sz(s)’s in Sz(q) is equal to [Sz(q) : Sz(s)] because N_{Sz(q)}(Sz(s)) = Sz(s). By lemma 3.5.7, the normalizer N_{Sz(q)}(α_{s}:4) is either Sz(s) or a subgroup of α_{q}:4 or β_{q}:4. Suppose it is Sz(s). Then Sz(s) is not simple, a contradiction. So it must be a subgroup of either α_{q}:4 or β_{q}:4. Anyway, α_{s}:4 is self-normalizing in the Sz(s) that contains it. Thus N_{Sz(q)}(α_{s}:4) = α_{s}:4 and the

G_{1}  G_{01}  #Γ
(qˆ .s):(s-1)  (sˆ .s):(s-1) 1
Sz(s’) when it is a maximal subgroup and s’ ≠ s Sz(t)  1
D_{2n} where s-1 |_{M} n |_{P} q-1  D_{2(s−1)}  1
n:4 where α_{s} |_{M} n |_{P} α_{q} or β_{q}  α_{s}:4  1
n:4 where β_{s} |_{M} n |_{P} α_{q} or β_{q}  β_{s}:4  1

Table 3.8: The case G_{0} = Sz(s).

number of α_{s}:4’s in Sz(q) is equal to [Sz(q) : α_{s}:4]. Every Sz(s) contains exactly [Sz(s) : α_{s}:4] subgroups isomorphic to α_{s}:4. Thus the number of Sz(s) containing the same α_{s}:4 is equal to 1. □

Lemma 3.5.9 Let e > 0 be a natural number. Let q = 2^{2e+1} and let G ≅ Sz(q) Let G_{0} ≅ Sz(s) be a maximal subgroup of Sz(q) and suppose G_{1} is not one of D_{2(q−1)}, α_{q}: 4, or β_{q}: 4 (these cases have been analysed in lemmas 3.5.2,3.5.3 and 3.5.4). Then the Rwpri rank 2 geometries of G with G_{0} ≅ Sz(s) are isomorphic to one of the geometries appearing in table 3.8.

Proof: Recall that the maximal subgroups of Sz(s) have one of the following structures.

• (sˆ .s):(s-1)

• D_{2(s−1)}

• α_{s}:4

• β_{s}:4

• Sz(t) if and only if there exists a natural number b such that s = t^{b}.

Suppose G_{01} = (sˆ .s):(s-1).

Since G_{01} is the stabilizer of a point of the subovoid, G_{1} must be either a subgroup of Sz(s) or (qˆ .q):(q-1). Lemma 3.5.5 implies that G_{1} cannot be a subgroup isomorphic to Sz(s). Thus we have to look at (qˆ .q):(q-1) and its subgroups to find some G_{1} candidate. All elements of G_{01} are of the form η(k)τ(a, b) where a, b ∈ GF (s) and k ∈ GF (s)^{∗}. The subgroup < G_{01}, τ(0, c) > with c ∈ GF (q) − GF (s) has order q · s · (s − 1) because it gives elements of the form η(z)τ(x, y) with x ∈ GF (s), y ∈ GF (q) and z ∈ GF (s)^{∗}. It contains G_{01} as a maximal subgroup because GF (s) is a maximal subfield of GF (q). It is easy to see that it has structure (qˆ .s):(s-1). The subgroup < G_{01}, τ(c, 0) > with c ∈ GF (q) − GF (s) has order q^{2}(s − 1) because it generates all elements of the form η(z)τ(x, y) with x, y ∈ GF (q), z ∈ GF (s)^{∗}. It contains (qˆ .s):(s-1) and thus G_{01} is not maximal in it. The subgroup < G_{01}, η(l) > with l ∈ GF (q) − GF (s) is the group (qˆ .q):(q-1) which does not contain G_{01} as a maximal subgroup. We can fix G_{1} and G_{01} because G_{1} conjugates all its subgroups isomorphic to G_{01}. Since G_{01} is self-normalizing as shown in the previous lemma, the stabilizer of G_{1} and G_{01} is G_{01}. And there is only one subgroup isomorphic to Sz(s) which contains G_{01}. This means we have here only one geometry up to conjugacy, and hence up to isomorphism.

Suppose G_{01} = D_{2(s−1)}.

Since s − 1 divides q − 1 the only G_{1} candidates are Sz(s) and D_{2n} where n is a divisor of q − 1 and s − 1 is a maximal divisor of n. In fact, we can suppose that n is a proper divisor of q −1 because the case n = q −1 has been dealt with while determining the configurations with G_{0} = D_{2(q−1)}. Lemma 3.5.6 shows that two subgroups isomorphic to Sz(s) never intersect in a D_{2(s−1)}. Thus the only Rwpri candidates here are obtained by taking Sz(s) as G_{0}, and D_{2n} as G_{1} where n is a proper divisor of q − 1 and s − 1 is a maximal divisor of n. It is easy to see that there is only one such geometry up to conjugacy and hence up to isomorphism, for every such n.

Suppose G_{01} = α_{s} : 4.

This implies that G_{1} will certainly not be a subgroup of D_{2(q−1)} or (qˆ .q):(q-1). The candidates are then another Sz(s) or n:4 where n is a proper divisor either of α_{q} or β_{q} and α_{s} is a maximal divisor of n. Lemma 3.5.8 implies that G_{1} must be a n:4. We can fix G_{1} because all subgroups isomorphic to G_{1} in Sz(q) are conjugate. We can also fix G_{01} because all the subgroups isomorphic to G_{01} are conjugate in G_{1}. The stabilizer of G_{1} and G_{01} is then G_{01}. And there is only one Sz(s) containing G_{01}. Thus there is only one such geometry up to conjugacy and hence up to isomorphism.

Suppose G_{01} = β_{s} : 4.

Almost the same discussion as in the case G_{01} = β_{s} : 4 leads us to the conclusion that G_{1} must be n:4 where n is a proper divisor either of α_{q} or of β_{q} and β_{s} is a maximal divisor of n, and that there is only one such geometry up to conjugacy and hence up to isomorphism.

Finally, suppose G_{01} = Sz(t).

The only candidates are subgroups isomorphic to Sz(s’) for some s′ thanks to the order of Sz(t). The case s = s′ leads to a contradiction because, as Sz(q) is a simple group, and as the only subgroup between Sz(q) and Sz(t) is Sz(s) which is also simple, the normalizer of Sz(t) in Sz(q) is Sz(t). Thus we have here G_{1} = Sz(s’) with s ≠ s′ that gives a geometry. We can fix G_{0} and G_{01} here and there is only one subgroup isomorphic to G_{1} that contains G_{01}. Thus there is only one such geometry up to conjugacy and hence up to isomorphism. □

3.6 Classification Theorem

The results of the preceding section provide a complete classification theorem giving all the firm, residually connected, flag-transitive and residually weakly primitive geometries of rank 2 for a Suzuki group Sz(q) up to isomorphism. The symbol #Γ in tables 3.9, 3.10 and 3.11 is the number of non-isomorphic geometries that arise from a Rwpri configuration. The last column of the tables tells whether or not the corresponding geometry satisfies (2T)_{1}. This condition is so easy to test in rank 2 that we do not give details about the results obtained.

Theorem 3.6.1 Let e > 0 be a natural number. Let q = 2^{2e+1} and let Sz(q) be the Suzuki simple group over GF(q). If the numbers 2e + 1 and q − 1 are prime, every F, RC, FT, Rwpri rank 2 geometries of Sz(q) is isomorphic to one of the geometries appearing in table 3.9. If 2e + 1 is prime and q − 1 is not prime, every F, RC, FT, Rwpri rank 2 geometries of Sz(q) is isomorphic to one of the geometries appearing in table 3.10. If 2e + 1 is not a prime number, every F, RC, FT, Rwpri rank 2 geometries of

G_{0}  G_{1}  G_{01} #Γ  (2T)_{1}
D_{2(q−1)} D_{2(q−1)}  2  \frac{(q−2)(q+4)}{4(2e+1)} + 1 NO
D_{2n} where n is a prime divisor of α_{q} or β_{q} 2  \frac{(q−2)(q+2)}{4(2e+1)} + 1 NO
2^{2}  2  \frac{q−2}{2(2e+1)}  NO
4  2  \frac{q−2}{2(2e+1)} + 1  NO
q : (q − 1)  q − 1 1  YES

\underline{α_{q}prime}

α_{q}:4  α_{q}:4  4  \frac{q−2}{2(2e+1)}  NO
n:4 where n is a prime divisor of β_{q}  4  \frac{q−2}{2(2e+1)} + 1  NO
4 × 2  4  \frac{q−2}{2(2e+1)}  NO

\underline{β_{q}prime}

β_{q}:4  β_{q}:4  4  \frac{q−2}{2(2e+1)}  YES iff q=8
n:4 where n is a prime divisor of α_{q}  4  \frac{q−2}{2(2e+1)} + 1  NO
4 × 2  4  \frac{q−2}{2(2e+1)}  YES iff q=8

Table 3.9: The case 2e + 1 prime and q − 1 prime

Sz(q) is isomorphic to one of the geometries appearing in table 3.11.

Proof: This theorem is a compilation of the results obtained in lemmas 3.5.1, 3.5.2, 3.5.3, 3.5.4 and 3.5.9. Thanks to lemma 3.2.6, in the case 2e + 1 is not a prime number, we are sure that α_{q} and β_{q} are not prime numbers. □

Corollary 3.6.1 If 2e + 1 is a prime number and q-1, α_{q} and β_{q} are not prime, then the group Sz(q) has only one geometry of rank 2.

Proof: This is a direct consequence of theorem 3.6.1. □

Table 3.12 contains the number of prime divisors of q − 1 (#(q − 1)), α_{q} (#α_{q}), β_{q} (#β_{q}) and the number #Γ of non-isomorphic geometries obtained for every Suzuki group up to e = 60. Thanks to corollary 3.6.1, we know that there is only one geometry for e = 11, 18, 20, 21, 26, 29, 33, 35, 41, 48, 50, 51 and 54.

The smallest case, that is Sz(8), has been studied completely in [30]. It is included in this thesis as Appendix A.

G_{0}  G_{1}  G_{01} #Γ  (2T)_{1}
D_{2(q−1)} q : (q − 1)  q-1 1  YES

\underline{α_{q}prime}

α_{q}:4 α_{q}:4 4 \frac{q−2}{2(2e+1)} NO
n:4 where n is a prime divisor of β_{q} 4  \frac{q−2}{2(2e+1)} + 1 NO
4 × 2 4 \frac{q−2}{2(2e+1)} NO
\underline{β_{q}prime}  NO
β_{q}:4 β_{q}:4 4 \frac{q−2}{2(2e+1)} NO
n:4 where n is a prime divisor of α_{q} 4  \frac{q−2}{2(2e+1)} + 1 NO
4 × 2  4  \frac{q−2}{2(2e+1)}  NO

Table 3.10: The case 2e + 1 prime and q − 1 not prime

G_{0}  G_{1}  G_{01}  #Γ (2T)_{1}
D_{2(q−1)} q : (q − 1)  q-1  1 YES
Sz(s) if s-1 is a maximal divisor of q-1  D_{2(s−1)}  1 NO
α_{q}:4  Sz(s) if α_{s} or β_{s} is a maximal divisor of α_{q}  α_{s}:4 or β_{s}:4 1 NO
β_{q}:4  Sz(s) if α_{s} or β_{s} is a maximal divisor of β_{q}  α_{s}:4 or β_{s}:4 1 NO
Sz(s) (qˆ.s):(s-1)  (sˆ.s):(s-1) 1 NO
Sz(s’) when it is a maximal subgroup and s’ ≠ s Sz(t)  1 NO
D_{2n} where s-1 |_{M} n |_{P} q-1  D_{2(s−1)}  1 NO
n:4 where α_{s} |_{M} n |_{P} α_{q} or β_{q}  α_{s}:4  1 NO
n:4 where β_{s} |_{M} n |_{P} α_{q} or β_{q}  β_{s}:4  1 NO

Table 3.11: The case 2e + 1 not prime (and thus q − 1 not prime)

\begin{matrix}e & #(q-1) & #α_{q} & #β_{q} & #Γ & e & #(q-1) & #α_{q} & #β_{q} & #Γ \\ 1 & 1 & 1 & 1 & 29 & 2 & 1 & 1 & 1 & 177 \\ 3 & 1 & 2 & 1 & 2411 & 4 & 2 & 2 & 2 & 5 \\ 5 & 2 & 1 & 2 & 375 & 6 & 1 & 2 & 2 & ≥ 6.10^{6} \\ 7 & 3 & 2 & 3 & 14 & 8 & 1 & 2 & 2 & ≥ 1.10^{9} \\ 9 & 1 & 1 & 3 & ≥ 1.10^{10} & 10 & 3 & 3 & 3 & 15 \\ 11 & 2 & 3 & 2 & 1 & 12 & 3 & 2 & 3 & 8 \\ 13 & 3 & 3 & 3 & 5 & 14 & 3 & 1 & 2 & ≥ 3.10^{7} \\ 15 & 1 & 3 & 2 & ≥ 2.10^{17} & 16 & 4 & 3 & 3 & 15 \\ 17 & 4 & 3 & 3 & 17 & 18 & 2 & 2 & 3 & 1 \\ 19 & 4 & 4 & 5 & 22 & 20 & 2 & 3 & 2 & 1 \\ 21 & 3 & 3 & 2 & 1 & 22 & 6 & 5 & 5 & 23 \\ 23 & 3 & 3 & 1 & ≥ 7.10^{12} & 24 & 2 & 4 & 2 & 6 \\ 25 & 5 & 4 & 5 & 23 & 26 & 3 & 2 & 2 & 1 \\ 27 & 6 & 3 & 3 & 20 & 28 & 4 & 4 & 4 & 19 \\ 29 & 2 & 2 & 5 & 1 & 30 & 1 & 2 & 4 & ≥ 1.10^{35} \\ 31 & 6 & 5 & 5 & 22 & 32 & 3 & 5 & 4 & 20 \\ 33 & 2 & 2 & 4 & 1 & 34 & 4 & 4 & 4 & 17 \\ 35 & 3 & 4 & 2 & 1 & 36 & 3 & 4 & 1 & ≥ 3.10^{20} \\ 37 & 7 & 7 & 5 & 27 & 38 & 4 & 4 & 5 & 20 \\ 39 & 3 & 3 & 1 & ≥ 1.10^{23} & 40 & 6 & 5 & 5 & 9 \\ 41 & 2 & 3 & 3 & 1 & 42 & 3 & 4 & 6 & 22 \\ 43 & 6 & 5 & 2 & 22 & 44 & 1 & 2 & 2 & ≥ 5.10^{51} \\ 45 & 5 & 5 & 5 & 24 & 46 & 3 & 5 & 4 & 18 \\ 47 & 5 & 5 & 4 & 24 & 48 & 2 & 5 & 4 & 1 \\ 49 & 8 & 6 & 6 & 28 & 50 & 2 & 3 & 3 & 1 \\ 51 & 2 & 3 & 3 & 1 & 52 & 10 & 7 & 7 & 54 \\ 53 & 1 & 3 & 3 & ≥ 4.10^{62} & 54 & 2 & 3 & 3 & 1 \\ 55 & 6 & 6 & 4 & 25 & 56 & 5 & 4 & 1 & ≥ 2.10^{32} \\ 57 & 6 & 6 & 6 & 30 & 58 & 9 & 9 & 7 & 36 \\ 59 & 6 & 5 & 7 & 30 & 60 & 4 & 3 & 3 & 8 \end{matrix}

Table 3.12: The number of prime divisors of q − 1, α_{q} and β_{q}, and the number of non-isomorphic geometries of rank 2

.

We also studied the thin geometries of Sz(8) more deeply in [29]. This work is included in the thesis as Appendix B.

3.7 Observations on the results

While computing the exact number of geometries up to isomorphism for every Sz(2^{2e+1}) with e ≤ 60, we observed the following facts.

• If Sz(q) does not have proper subgroups isomorphic to a Suzuki group, then table 3.12 shows that its number of geometries up to isomorphism is either 1 or a number that grows very fast like 2411 for e = 3, more than 6 millions for e = 6, more than 10 billions for e = 9, etc.

• If Sz(q) has proper subgroups isomorphic to some Suzuki simple group, then its number of geometries up to isomorphism is more than reasonable.

This seems a little bit intriguing. We could have thought that when a Suzuki ovoid has subovoids, its collineation group should have more geometries. This is not the case.

Maybe we should add some condition in order to restrict the number of geometries that appear. If we impose (2T)_{1}, we see that when e > 2, there is only one geometry that survives this test (the case e = 1 gives 3 such geometries). Thus the huge number of geometries we obtained in these cases disappear and the number of geometries obtained is quite small.

Anyway, one which seems to play a central role is the geometry given by the Rwpri configuration (Sz(q), D_{2(q−1)}, q : (q − 1), q − 1). It is the geometry that appears in all cases. It satisfies (2T)_{1}.

The rank 3 geometries of Sz(q)

This chapter is taken from [27].

4.1 Introduction

In the present chapter, we determine all Rwpri geometries of rank three, on which a Suzuki simple group Sz(q), with q an odd power of 2, acts residually weakly primitively. Here we plan to determine the geometries that satisfy the Rwpri condition, be them flag-transitive or not. The main reason of this choice is that Rwpri is a property much easier to test than flag-transitivity. Also, it seems much more restrictive, so it is natural to test it first. Flag-transitivity will not be studied in this thesis.

Our main results are theorems 4.4.1, 4.4.2 and 4.4.3 which give all Rwpri geometries of rank 3 for any Suzuki simple group Sz(q), with q an odd power of two. As in the rank 2 classification (see chapter 3), we subdivide the results into three cases, namely the case q − 1 prime (and hence 2e + 1 prime), the case q − 1 not prime and 2e + 1 prime, and the case 2e + 1 not prime (and hence q − 1 not prime).

50 CHAPTER 4. THE RANK 3 GEOMETRIES OF SZ(Q)

As a corollary of the classification theorems for the rank 3 Rwpri geometries of the Suzuki groups, we show in theorem 4.6.1 that there is no pre-geometry (and hence no geometry) of rank ≥ 4, satisfying both Rwpri and (2T)_{1}. Moreover, we show that the only rank three geometries satisfying that property are thin, and that they arise only for Suzuki groups Sz(q) with q − 1 a prime number. Thus, as in the rank 2 case, the (2T)_{1} property seems to be a good candidate to enter our set of axioms. Anyway, we show in the next chapter that it is possible to classify all Rwpri pre-geometries of rank ≥ 4 for any Suzuki group Sz(q).

The chapter is organised as follows. In section 2, we prove some preliminary lemmas used in the next sections for the classification process. In section 3, we prove some lemmas that tend to restrict the possible maximal parabolic subgroups of the geometries we want to classify. In section 4, we classify all rank 3, Rwpri geometries of a Suzuki group Sz(q). In section 5, we show that for most of the geometries obtained, we can say whether the group from which they arise acts flag-transitively or not. Finally, in section 6, we prove that there is no Rwpri and (2T)_{1} pre-geometry of rank strictly greater than 3 for any Suzuki simple group Sz(q) and that rank 3 geometries appear only when q − 1 is prime.

4.2 Preliminary lemmas

Lemma 4.2.1 The group Sz(q) has a unique class of maximal subgroups isomorphic to a Suzuki group if and only if q = 2^{2e+1}, 2e + 1 = p^{n} with p a prime number, and n > 1 an integer.

Proof: It suffices to apply the classification of maximal subgroups of Sz(q) as given in [32]. □

Lemma 4.2.2 Let Sz(s) be a maximal subgroup of Sz(q). Then s − 1 |_{M} q − 1 implies s = 2^{p^{n}} and q = s^{p}, with p a prime number and n a positive integer.

Proof: Suppose that there is no prime number p and positive integer n such that q = 2^{p^{n}}. Then, by lemma 4.2.1, there are at least two different classes of Suzuki groups in Sz(q), say Sz(s) and Sz(t). Thus q = s^{p} = t^{p′} with p ≠ p′ two prime numbers. Also, we know that s − 1 and t − 1 divide q − 1. And (s − 1) · (t − 1) < (q − 1). This implies that \frac{q−1}{s−1} cannot be prime because q − 1 = (s − 1) · \frac{t−1}{(s−1,t−1)} · n with n > 1 an integer. □

Lemma 4.2.3 If Sz(q) has at least two different classes of maximal subgroups of Suzuki type, then s − 1 is not a prime divisor of q − 1.

Proof: Obvious thanks to lemmas 4.2.1 and 4.2.2. □

Lemma 4.2.4 Let Γ(G; G_{0}, G_{1}, G_{2}) be a rank 3 geometry. If the diagram of Γ is linear, then Γ is a flag-transitive geometry.

Proof: To check that Γ is a flag-transitive geometry, we have to see if G_{0}(G_{1} ∩ G_{2}) = G_{0}G_{1}∩G_{0}G_{2}. It is obvious that G_{0}(G_{1}∩G_{2}) ⊆ G_{0}G_{1}∩G_{0}G_{2}. We know that | G_{0}(G_{1}∩G_{2}) |= \frac{|G_{0}|·|G_{1}∩G_{2}|}{|G_{0}∩G_{1}∩G_{2}|}. We also know that | G_{0}G_{1} |= \frac{|G_{0}|·|G_{1}|}{|G_{0}∩G_{1}|}. If the diagram of Γ is linear, it means that one of the rank 2 residues is a generalized digon. We may suppose that this residue is the one corresponding to G_{1}. This implies that \frac{|G_{1}|}{|G_{0}∩G_{1}|} = \frac{|G_{1}∩G_{2}|}{|G_{0}∩G_{1}∩G_{2}|}. Thus | G_{0}(G_{1} ∩ G_{2}) |= \frac{|G_{0}|·|G_{1}∩G_{2}|}{|G_{0}∩G_{1}∩G_{2}|} = \frac{|G_{0}|·|G_{1}|}{|G_{0}∩G_{1}|} =| G_{0}G_{1} |. □

Remark that this lemma is a corollary of the main theorem proved in [1].

Lemma 4.2.5 Let Sz(s) < Sz(q). Then either α_{s} | α_{q} and β_{s} | β_{q} or α_{s} | β_{q} and β_{s} | α_{q}.

Proof: This is an immediate consequence of lemmas 3.2.4 and 3.2.5. □

It is possible to determine whether α_{s} divides α_{q} or not, as shown in the next lemma.

Lemma 4.2.6 5 divides q^{2} + 1 for all q = 2^{2e+1} with e a positive integer.

Proof: q^{2} + 1 = 2^{4e+2} + 1 = (2^{2} + 1) · ∑_{i=0}^{2e}(2^{4e−2i})(−1)^{i} □

Lemma 4.2.7 If q = 2^{2e+1}, then 5 | α_{q} (resp. β_{q}) if and only if 2e + 1 = 0 or 3 (resp. 1 or 2) modulo 4.

Proof: It is easy to see that 2^{n} = 2 (resp. 4, 3, 1) modulo 5 if n = 1 (resp. 2, 3,

0) modulo 4. Then the following table completes the proof. □

e modulo 4 2e + 1 modulo 4 e + 1 modulo 4 α_{q} modulo 5 β_{q} modulo 5
0  1  1  0  1
1  3  2  3  0
2  1  3  1  0
3  3  0  0  3

Lemma 4.2.8 Let m and n be odd positive integers. Then (2^{2m} + 1, 2^{2n} + 1) = 2^{2(m,n)} + 1.

Proof: The following argument is due to Wieb Bosma. It is well known that (s^{a} − 1, s^{b} − 1) = s^{(a,b)} − 1 for all a and b positive integers. Take s = 2. Applying the latter result with a = 4m and b = 4n, we get

(2^{4m} − 1, 2^{4n} − 1) = 2^{4(m,n)} − 1

and with a = 2m and b = 4n, we get

(2^{2m} − 1, 2^{4n} − 1) = 2^{2(m,n)} − 1

Since 2^{2m} − 1 and 2^{2m} + 1 are coprime, we have

2^{4(m,n)} − 1 = (2^{4m} − 1, 2^{4n} − 1) = (2^{2m} − 1, 2^{4n} − 1) · (2^{2m} + 1, 2^{4n} − 1)
= (2^{2(m,n)} − 1) · (2^{2m} + 1, 2^{2n} + 1) · (2^{2m} + 1, 2^{2n} − 1)

the latter because 2^{2n} − 1 and 2^{2n} + 1 are also coprime. Thus we still have to prove that (2^{2m} + 1, 2^{2n} − 1) = 1 to finish the proof. This must hold because n is odd. Indeed, suppose a prime number p divides 2^{2m} + 1 and 2^{2n} − 1. Then the order k of 2 mod p divides 4m but not 2m since 2^{2m} = −1 mod p. Thus the number of factors 2 in k is 2. On the other hand, 2^{2n} = 1 mod p yields that k divides 2n and the number of factors 2 in k cannot exceed 1 because n is odd. Thus we have a contradiction. □

4.3 Restricting the maximal parabolic subgroups

In this section, we prove lemmas that restrict the possible maximal parabolic subgroups to construct Rwpri pre-geometries of rank ≥ 3. Thanks to lemma 1.1.1, we know that every rank 3 pre-geometry is indeed a geometry, so we can use these lemmas for the classification of the rank 3 Rwpri geometries of Sz(q). We prefer to talk about pre-geometries here since the results obtained in this section are used also in chapter 5 to construct all Rwpri pre-geometries of rank ≥ 4.

Since we want to construct Rwpri pre-geometries, we know that at least one of the maximal parabolic subgroups of the pre-geometry must be a maximal subgroup of Sz(q). Not every maximal subgroup of Sz(q) can give rise to Rwpri pre-geometries of rank ≥ 3. We give here lemmas that discard some of them.

We already saw in chapter 3 that there is no Rwpri pre-geometry of rank ≥ 2 with some G_{i} isomorphic to (qˆ.q):(q-1) (see lemma 3.5.1).

Lemma 4.3.1 There is no Rwpri pre-geometry of rank ≥ 3 with some G_{i} isomorphic to α_{q}:4 or β_{q}:4.

Proof: We may assume without loss of generality that G_{0} = Z_{n} : 4 where n = α_{q} or β_{q}. Knowing the structure of these groups, we can say that G_{01} must be either n : 2 or n_{p} : 4 where p is a prime divisor of n. The case G_{01} = n : 2 implies that G_{1} must be a subgroup of n : 4. This possibility is rejected by the fact that n : 2 is a normal subgroup of n : 4. The case G_{01} = n_{p} : 4 is a little bit longer. Because n_{p} : 4 is self-normalized in Sz(q), G_{1} cannot be n : 4. Thus G_{1} must be a Sz(s) maximal in Sz(q). And because we have supposed G_{01} maximal in G_{0}, it must also be maximal in G_{1}. Thus n_{p} = α_{s} or β_{s}. Because the case G_{01} = n : 2 leads to a dead end, G_{02} = n_{r} : 4 with r, a prime divisor of n and r ≠ p. And this implies G_{2} = Sz(t). This already means that n cannot be a prime number and that q − 1 must have at least two prime divisors. We also know that G_{012} = n_{pr} : 4 because it must be maximal in G_{01} and it must also be a subgroup of G_{02}. Suppose q = 2^{2e+1}, s = 2^{2f+1} and t = 2^{2g+1}. Because Sz(s) and Sz(t) are maximal subgroups of Sz(q), we have q = s^{p_{s}} = t^{p_{t}} with p_{s} and p_{t} two distinct prime numbers. Thus, 2e + 1 = (2f + 1) · p_{s} = (2g + 1) · p_{t} and (p_{s}, p_{t}) = 1. This allows us to write 2e + 1 = (2h + 1) · p_{s} · p_{t}. We know that n is either α_{q} or β_{q} and that n_{p} is either α_{s} or β_{s}. Then, p = \frac{n}{n_{p}} > 2^{(2h+1)(p_{s}−1)p_{t}−1} > α_{t}> β_{t}. But p must divide α_{t} or β_{t}. So we have a contradiction. □

We show that if G_{0} is a D_{2(q−1)}, then the pre-geometry has rank at most 3.

Lemma 4.3.2 Every Rwpri pre-geometry of rank n > 3 with G_{0} = D_{2(q−1)} implies that at least two G_{0i} are dihedral groups.

Proof: We rely on the classification of the Rwpri geometries for the dihedral groups given in theorem 2.5.1, since all Rwpri pre-geometries are flag-transitive Rwpri geometries for these groups. The proof becomes obvious thanks to that classification. □

Lemma 4.3.3 Suppose q − 1 is not prime. In a Rwpri pre-geometry of rank n > 2 with G_{0} = D_{2(q−1)}, there can be at most one G_{0i} isomorphic to a dihedral group.

Proof: Assume, by way of contradiction and without loss of generality, that G_{01} and G_{02} are dihedrical groups and that G_{01} is maximal in G_{0}. Then G_{01} = D_{2x} with x |_{M} q−1. This implies G_{02} = D_{2y} with y |_{M} q−1. Then G_{1} (resp. G_{2}) must be a Sz(s) (resp. Sz(t)). These two subgroups have to be maximal subgroups of Sz(q) and G_{01} (resp. G_{02}) has to be maximal in Sz(s) (resp. Sz(t)). We have q = 2^{(2e+1)p_{s}p_{t}}, s = 2^{(2e+1)p_{t}} and t = 2^{(2e+1)p_{s}}. We also have \frac{q−1}{s−1} = p a prime number, which must be a divisor of t − 1. But

p = \frac{2^{(2e+1)p_{s}p_{t}} − 1}{2^{(2e+1)p_{t}} − 1} = ∑_{i=1}^{p_{t}} 2^{(2e+1)p_{s}(p_{t}−1)} > 2^{(2e+1)p_{s}} − 1

So p cannot divide t − 1, a contradiction. □

Lemma 4.3.4 The maximal rank of a Rwpri pre-geometry with G_{0} = D_{2(q−1)} is three.

Proof: Trivial by lemmas 4.3.2 and 4.3.3. □

This is a fairly strong result. It permits us to concentrate only on the rank 3 when we construct Rwpri pre-geometries with G_{0} = D_{2(q−1)}.

Lemma 4.3.5 If Γ is a Rwpri pre-geometry of rank ≥ 4 of a Suzuki group Sz(q), then all the maximal parabolic subgroups that are maximal in Sz(q) must be isomorphic to a Suzuki group Sz(s).

Proof: Apply lemmas 3.5.1, 4.3.1 and 4.3.4. □

4.4 The rank 3 geometries

Thanks to lemma 1.1.1, we know that any rank 3 pre-geometry constructed in this section is indeed a geometry. We determine now all the rank 3 Rwpri geometries of a Suzuki group Sz(q) with q = 2^{2e+1} and e a positive integer. As in the rank 2 classification given in theorem 3.6.1, we subdivide our discussion in three cases:

• q − 1 is prime.

• q − 1 is not prime and 2e + 1 is prime.

• 2e + 1 is not prime.

4.4.1 First case: q − 1 is a prime number

If q − 1 is a prime number, Sz(q) cannot have subgroups isomorphic to Sz(s). Remark that lemma 4.3.4 tells us then that Sz(q) does not have Rwpri pregeometries of rank ≥ 4 in that case. The only G_{0} candidate is D_{2(q−1)} by lemmas 3.5.1 and 4.3.1. Thanks to theorem 2.4.1, we know that there are two possible residues for G_{0}. The first one is with G_{01} = q − 1, G_{02} = 2 and G_{012} = 1. The only subgroups of Sz(q) containing G_{01} are then D_{2(q−1)}, (qˆ.q) : (q − 1) and q : (q − 1). Because N_{Sz(q)}(q − 1) = D_{2(q−1)}, and D_{2(q−1)} has only one subgroup isomorphic to q − 1, the subgroup G_{1} cannot be isomorphic to D_{2(q−1)}. Lemma 3.5.1 implies that G_{1} cannot be (qˆ.q) : (q − 1). So we may assume G_{1} = q : (q − 1). Because B = G_{012} = 1, we know that G_{12} must be a cyclic simple group. Here there are two possibilities. They are q − 1 and 2. Suppose first that G_{12} = q − 1. Then G_{2} must be either q : (q − 1) or D_{2(q−1)}. In both cases, G_{01} and G_{12} have two fixed points and it implies that G_{02} has a fixed point which is in the orbit of two points of G_{0}. This is a contradiction with the fact that all the Z_{2}’s in a D_{2(q−1)} have a fixed point which lies in the orbit of q − 1 points of this subgroup. Suppose that G_{12} = 2. It implies that G_{2} contains two cyclic groups of order 2. Thus G_{2} = D_{2p} with p | α_{q},β_{q} or q − 1. The case p = 2 is not possible because G_{02} and G_{12} do not have the same fixed point. The second possible residue is with G_{01} and G_{02} isomorphic to cyclic groups of order 2. There, either G_{1} is one of p : 2 (with p | α_{q}, β_{q}, 2 or q − 1) and G_{2} is one of p′ : 2 (with p′ | α_{q}, β_{q}or q − 1), or G_{1} = q : (q − 1) and G_{2} = q : (q − 1). Remark that there are no two maximal parabolic subgroups isomorphic to 2^{2}. Indeed, this should mean that all the G_{ij}’s have a common fixed point. Thus the group generated by G_{0}, G_{1} and G_{2} cannot be Sz(q) because, otherwise, it would be a subgroup of qˆ.q : (q − 1).

We resume the preceding discussion in a first theorem.

Theorem 4.4.1 Let G = Sz(q) with q = 2^{2e+1} and suppose 2e + 1 and q − 1 are prime numbers. Then all rank 3 Rwpri geometries of Sz(q) have a sublattice isomorphic to one of the following.

• \begin{aligned}G_{0} & G_{1} & G_{2} \\ D_{2(q−1)} & q : (q − 1) & q : (q − 1) \\ G_{01} & G_{02} & G_{12} \\ 2 & 2 & q − 1 \\ & G_{012} \\ & 1 \end{aligned}

• \begin{aligned}G_{0} & G_{1} & G_{2} \\ D_{2(q−1)} & q : (q − 1) & D_{2p} \\ G_{01} & G_{02} & G_{12} \\ q − 1 & 2 & 2 \\ & G_{012} \\ & 1 \end{aligned} with p | α_{q}, β_{q}or q − 1, and p is a prime

number.

• \begin{aligned}G_{0} & G_{1} & G_{2} \\ D_{2(q−1)} & D_{2p} & D_{2p′} \\ G_{01} & G_{02} & G_{12} \\ 2 & 2 & 2 \\ & G_{012} \\ & 1 \end{aligned} with p | α_{q}, β_{q}, 2 or q − 1, and p′ | α_{q}, β_{q}, or q − 1,

and p, p′ two prime numbers.

4.4.2 Second case: q − 1 is not prime and 2e + 1 is prime

In this case again, we have G_{0} = D_{2(q−1)}. Every possible rank 2 residue for G_{0} has a dihedral subgroup as one of its maximal parabolic subgroups. Thus we may assume that G_{01} = D_{2n} where n |_{M} q − 1. As D_{2n} is self-normalized in Sz(q), there is only one subgroup isomorphic to D_{2(q−1)} that contains G_{01}. Thus we cannot find a subgroup G_{1} to complete the geometry. Remark that in that case as in the preceding one, lemma 4.3.4 tells us that Sz(q) does not have a Rwpri pre-geometry of rank ≥ 4. Thus we can now state the following theorem.

Theorem 4.4.2 Let G = Sz(q) with q = 2^{2e+1} and suppose 2e + 1 is a prime number but q − 1 is not. Then G does not possess any Rwpri pre-geometry of rank ≥ 3.

4.4.3 Third case: 2e + 1 is not prime

This case is the most tricky one because the G_{0} candidates are D_{2(q−1)} and Sz(s) where s = 2^{2f+1} and \frac{2e+1}{2f+1} is a prime number.

The case G_{0} = D_{2(q−1)}

In this case, lemma 4.3.4 tells us that the maximal rank is 3. Also, lemma 4.3.3 and theorem 2.4.1 imply that one of G_{01} or G_{02} must be a cyclic group. We can choose G_{01} = Z_{q−1} without restricting generality. The sublattice is then partially filled as below, with n |_{M} q − 1.

G_{0}  G_{1} G_{2}
D_{2(q−1)}
G_{01} G_{02} G_{12}
q − 1 D_{2n}
G_{012}
n

The only subgroups that can contain G_{02} are then G_{0} and subgroups isomorphic to a Suzuki group. This implies G_{2} is a Suzuki group and its residue is well known thanks to the classification theorem given in theorem 3.6.1. We know then that G_{2} = Sz(s) with q = s^{p} and p a prime number. Also, we must have n = s − 1. This means \frac{q−1}{s−1} is a prime number. Lemma 4.2.2 yields then 2e + 1 = p^{a}. Now, looking at the rank 2 geometries of Sz(s) as given in theorem 3.6.1, we can complete our geometry. Indeed, G_{12} must be s : (s − 1) and thus G_{1} =< G_{01}, G_{12} >= q : (q − 1).

Lemma 4.4.1 Let G = Sz(q) with q = 2^{2e+1} and suppose 2e + 1 is not a prime number. Then G has a unique Rwpri geometry of rank 3 with G_{0} = D_{2(q−1)} if and only if 2e + 1 = p^{a} and p is a prime number. Its sublattice is as follows.

No. G_{0}  G_{1}  G_{01} (2T)_{1}
1  D_{2(s−1)} D_{2(s−1)}  2  NO
2  D_{2n} where n is a prime divisor of α_{s} or β_{s} 2  NO
3  2^{2}  2  NO
4  4  2  NO
5  s : (s − 1)  s − 1 YES

\underline{α_{s}prime}

6  α_{s}:4  α_{s}:4  4  NO
7  n:4 where n is a prime divisor of β_{s}  4  NO
8  4 × 2  4  NO

\underline{β_{s}prime}

9  β_{s}:4  β_{s}:4  4  YES iff s=8
10  n:4 where n is a prime divisor of α_{s}  4  NO
11  4 × 2  4  YES iff s=8

Table 4.1: The case s − 1 prime

G_{0}  G_{1}  G_{2}
D_{2(q−1)} q : (q − 1) Sz(s)
G_{01} G_{02} G_{12}
q − 1  D_{2(s−1)} s : (s − 1)
G_{012}
s − 1

The case G_{0} = Sz(s)

In order to have Sz(s) maximal in Sz(q), we must impose that q = s^{p} with p a prime number. The rank 2 residue of G_{0} is given by theorem 3.6.1. We subdivided the classification of the rank 2 geometries of the Suzuki groups into three cases. Here, we use this subdivision again.

Suppose first that s − 1 is a prime number. This means the residue of G_{0} is one of those appearing in table 4.1. We take each of these 11 possible residues and try to extend them to construct Rwpri rank 3 geometries.

Residue 1:

The sublattice looks as follows.

G_{0}  G_{1}  G_{2}
Sz(s)
G_{01}  G_{02} G_{12}
D_{2(s−1)} D_{2(s−1)}

G_{012} 2

Thus G_{1} must contain a D_{2(s−1)}. This yields G_{1} = Sz(t) or G_{1} ≤ D_{2(q−1)}. The first case implies t ≠ s because D_{2(s−1)} is self-normalized in Sz(q). Again, the residue is then known and we must have D_{2(s−1)} maximal in Sz(t). This means s − 1 = t − 1 and thus s = t, a contradiction. The second case yields G_{1} = D_{2(s−1)p} where p | \frac{q−1}{s−1} is a prime number. The same argument shows that G_{2} = D_{2(s−1)p′} where p′ | \frac{q−1}{s−1} is also a prime number. Thanks to theorem 2.4.1, we know that G_{12} = D_{2p} and p = p′. Thus G_{1} = G_{2}, a contradiction.

Residue 2:

The sublattice looks as follows (with n |_{P} α_{s} or β_{s}).

G_{0}  G_{1} G_{2}
Sz(s)
G_{01} G_{02} G_{12}
D_{2(s−1)} D_{2n}

G_{012} 2

As in the preceding case, we know that G_{1} < D_{2(q−1)}. This means G_{1} = D_{2(s−1)p} with p | q − 1 a prime number and G_{12} = D_{2p} thanks theorem 2.4.1. Thus G_{2} =< G_{02}, G_{12} > is a Suzuki simple group Sz(t) with t − 1 a prime number. Because of G_{02}, we must have t ≠ s. This means q = s^{a} = t^{b} and because s − 1 (resp. t − 1) is a prime number, we know that s = 2^{x} (resp. t = 2^{y}) with x (resp.

y) a prime number. Thus q = 2^{xy}, with x, y two different odd prime numbers. Thanks to lemma 4.2.8, we then get n = 5.

Residue 3:

The sublattice looks as follows.

G_{0}  G_{1} G_{2}
Sz(s)
G_{01} G_{02} G_{12}
D_{2(s−1)} 2^{2}

G_{012} 2

Again G_{1} must be a subgroup of D_{2(q−1)}. Thus G_{12} = D_{2p} with p | q − 1 a prime number as in the preceding case, and G_{2} =< 2^{2}, D_{2p} > must be a subgroup of Suzuki type. So G_{2} = Sz(t) with t ≠ s and p = t − 1. Also, q = 2^{xy}, s = 2^{x} and t = 2^{y} and s − 1 and t − 1 must be prime numbers. Then the geometry is the following one.

G_{0}  G_{1}  G_{2}
Sz(s) D_{2(s−1)(t−1)} Sz(t)
G_{01}  G_{02}  G_{12}
D_{2(s−1)}  2^{2}  D_{2(t−1)}

G_{012} 2

Residue 4:

Almost the same discussion as in the preceding case leads to the following geometry.

G_{0}  G_{1}  G_{2}
Sz(s) D_{2(s−1)(t−1)} Sz(t)
G_{01}  G_{02}  G_{12}
D_{2(s−1)}  4  D_{2(t−1)}

G_{012} Residue 5:

The sublattice looks as follows.

G_{0}  G_{1}  G_{2}
Sz(s)
G_{01} G_{02} G_{12}
D_{2(s−1)} s : (s − 1)
G_{012}
s − 1

Again, G_{1} must be a subgroup of D_{2(q−1)}. Thus G_{1} is a subgroup D_{2(s−1)p} where p | q − 1 is a prime number thanks to theorem 2.4.1. This implies that G_{12} may be isomorphic to either Z_{(s−1)p} or D_{2(s−1)}. Suppose G_{12} = D_{2(s−1)}. This yields G_{2} = Sz(s). Then, G_{02} = s : (s − 1) must be contained in two distinct Sz(s), a contradiction. So G_{12} = Z_{(s−1)p}. Then G_{2} =< s : (s − 1), Z_{(s−1)p} > is a subgroup of q : (q − 1). In order to have a Rwpri geometry, we must impose G_{2} = q : ((s − 1)p). A priori, the number p can be any prime divisor of \frac{q−1}{s−1}. So we get more than one geometry most of the time. Remark that, when p = \frac{q−1}{s−1}, we get the geometry given in lemma 4.4.1.

Residue 6:

The sublattice looks as follows.

G_{0}  G_{1} G_{2}
Sz(s)
G_{01} G_{02} G_{12}
α_{s}: 4 α_{s}: 4

G_{012} 4

Because α_{s} is self-normalized in Sz(q), the subgroup G_{1} must be a subgroup of α_{q}: 4 or β_{q}: 4 depending whether α_{s} divides α_{q} or β_{q} . This means G_{1} = (α_{s}p) : 4 and G_{12} = p : 4 with p a prime number such that pα_{s} divides either α_{q} or β_{q} as α_{s} does, and p ≠ α_{s}. Thus G_{2} =<α_{s}: 4, p : 4 > is isomorphic to G_{1}. This implies a contradiction because G_{12} is self-normalized in Sz(q) and thus cannot be contained in two different (α_{s}p) : 4.

Residue 7:

The sublattice looks as follows (with n |_{P} β_{s}).

G_{0}  G_{1} G_{2}
Sz(s)
G_{01} G_{02} G_{12}
α_{s}: 4 n : 4

G_{012} 4

The same argument as in the preceding case leads to G_{1} =α_{s}p : 4 and G_{12} = p : 4, with p a prime number such that pα_{s} divides either α_{q} or β_{q} as α_{s} does and p ≠ α_{s}. Then G_{2} =< n : 4, p : 4 >= Sz(q) because of lemma 4.2.5, which is not permitted.

Residue 8:

The sublattice looks as follows.

G_{0}  G_{1} G_{2}
Sz(s)
G_{01} G_{02} G_{12}
α_{s}: 4 4 × 2

G_{012} 4

The same argument as in the preceding case leads to G_{1} =α_{s}p : 4 and G_{12} = p : 4, with p a prime number such that pα_{s} divides α_{q} or β_{q} and p ≠ α_{s}. Then G_{2} =< 4 × 2, p : 4 >= Sz(t) with t ≠ s and we must have p =α_{t} or β_{t}. Such a geometry exists, for example, with s = 2^{3}, t = 2^{5} and q = 2^{15}.

Residues 9 to 11:

The discussion for residue 9 (resp. 10, 11) is almost the same as for residue 6 (resp. 7, 8). We just have to read β_{s} instead of α_{s} in these cases.

A resume of the preceding discussion is given at the end of this section in theorem 4.4.3.

No. G_{0}  G_{1}  G_{01}  (2T)_{1}
12 D_{2(s−1)} s : (s − 1)  s-1  YES
13  Sz(t) if t-1 is a maximal divisor of s-1  D_{2(t−1)}  NO
14 α_{s}:4  Sz(t) if α_{t} or β_{t} is a maximal divisor of α_{s}  α_{t}:4 or β_{t}:4 NO
15 β_{s}:4  Sz(t) if α_{t} or β_{t} is a maximal divisor of β_{s}  α_{t}:4 or β_{t}:4 NO
16 Sz(t) (sˆ.t):(t-1)  (tˆ.t):(t-1) NO
17  Sz(t’) when it is a maximal subgroup and t’ ≠ t Sz(u)  NO
18  D_{2n} where t-1 |_{M} n |_{P} s-1  D_{2(t−1)}  NO
19  n:4 where α_{t} |_{M} n |_{P} α_{s} or β_{s}  α_{t}:4  NO
20  n:4 where β_{t} |_{M} n |_{P} α_{s} or β_{s}  β_{t}:4  NO

Table 4.2: The case 2e + 1 not prime (and thus s − 1 not prime)

We now consider the case s − 1 not prime and s = 2^{2e+1} with 2e + 1 a prime number. This means that the residue of G_{0} is one of numbers 5, 6, 7, 8, 9, 10 or 11 appearing in table 4.1. The discussions of the preceding cases corresponding to those residues is independent from the fact that s − 1 is prime or not. So we can rely on the results obtained in the preceding discussion.

Finally, we have to consider the case where s = 2^{2e+1} and 2e + 1 is not a prime number. In this case, the possible residues for G_{0} are given in table 4.2.

Residue 12:

The same discussion as in the case of residue 5 leads to the following geometry.

G_{0}  G_{1}  G_{2}
Sz(s) D_{2(s−1)p} q : ((s − 1)p)
G_{01} G_{02} G_{12}
D_{2(s−1)} s : (s − 1)  (s − 1)p
G_{012}
s − 1

Remark that, when p = \frac{q−1}{s−1}, we get the geometry given in lemma 4.4.1.

Residue 13:

The sublattice looks as follows.

G_{0}  G_{1}  G_{2}
Sz(s)
G_{01}  G_{02} G_{12}
D_{2(s−1)} Sz(t)
G_{012}
D_{2(t−1)}

Since t − 1 |_{M} s − 1 for this residue, we have t = 2^{p^{a}} and s = t^{p}. Again, G_{1} is a subgroup of D_{2(q−1)}, thus G_{1} = D_{2(s−1)n} and G_{12} = D_{2n} with n a prime number that divides \frac{q−1}{s−1}. Then G_{2} =< Sz(t), D_{2n} > is a subgroup of Suzuki type. Let G_{2} = Sz(s′). We must have s′ ≠ s, thus q = 2^{(p^{a}·p′)} and s′ = 2^{(p^{a−1}p′)}. This implies that the residue of G_{2} is of type 18, and n | s′ − 1. Here n can take more than one value. So there might be more than one geometry there.

Residue 14:

The sublattice looks as follows (where n is either α_{t} or β_{t}).

G_{0}  G_{1} G_{2}
Sz(s)
G_{01} G_{02} G_{12}
α_{s}: 4 Sz(t)
G_{012}
n : 4

Because of lemma 4.2.5, we know that Sz(t) is a maximal subgroup of Sz(s). This implies that G_{2} = Sz(s′) must also contain Sz(t) as a maximal subgroup. Thus its residue is such that G_{12} = n′ : 4 where n′ divides s^{′2} + 1 and n |_{M} n′. Also, G_{1} = \frac{α_{s}n′}{n} : 4.

Residue 15:

Almost the same discussion as the preceding one leads to the following geometry (n = α_{t} or β_{t}, n |_{M} β_{s}, n′ | s^{′2} + 1 and n |_{M} n′).

G_{0}  G_{1}  G_{2}
Sz(s) (\frac{α_{s}n′}{n}) : 4 Sz(s′)
G_{01} G_{02} G_{12}
β_{s}: 4 Sz(t) n′ : 4
G_{012}
n : 4

Residue 16:

The sublattice looks as follows.

G_{0}  G_{1}  G_{2}
Sz(s)
G_{01} G_{02} G_{12}
Sz(t) sˆ.t : (t − 1)
G_{012}
tˆ.t : (t − 1)

Because of G_{01}, the subgroup G_{1} must be a subgroup of Suzuki type, say Sz(s′) with s′ ≠ s. Thus its residue is known. It implies that G_{12} = s′ˆ.t : (t − 1). Then G_{2} =< sˆ.t : (t−1), s′ˆ.t : (t−1) > is a subgroup of qˆ.q : (q −1). In order to have a Rwpri geometry, we must be sure that either G_{02} or G_{12} is a maximal subgroup of G_{2}. This implies G_{2} = qˆ.t : (t − 1).

Residue 17:

The sublattice looks as follows.

G_{0}  G_{1} G_{2}
Sz(s)
G_{01} G_{02} G_{12}
Sz(t) Sz(t′)
G_{012}
Sz(u)

It is easy to see that the only geometry we can obtain here is with all subgroups of the sublattice isomorphic to subgroups of Suzuki type.

Residue 18:

The sublattice looks as follows (with t − 1 |_{M} n |_{P} s − 1).

G_{0}  G_{1}  G_{2}
Sz(s)
G_{01}  G_{02} G_{12}
Sz(t) D_{2n}
G_{012}
D_{2(t−1)}

Because G_{01} is of Suzuki type, we have G_{1} = Sz(s′) with s′ ≠ s. Thus the residue of G_{1} is well known. Then G_{12} = D_{2m} where t − 1 |_{M} m |_{P} s′ − 1. And G_{2} =< D_{2n}, D_{2m} >= D_{2 \frac{nm}{t−1}} . Here n ≠ m thanks to theorem 2.4.1.

Residue 19:

The sublattice looks as follows (with α_{t} |_{M} n |_{P} α_{s} or β_{s}).

G_{0}  G_{1} G_{2}
Sz(s)
G_{01} G_{02} G_{12}
Sz(t) n : 4
G_{012}
α_{t} : 4

Again, G_{1} = Sz(s′) with s′ ≠ s. Thus G_{12} = m : 4 with α_{t} |_{M} m |_{P} α_{s′} or β_{s′} and G_{2} = \frac{nm}{α_{t}} : 4.

Residue 20:

Almost the same discussion as in the preceding case leads to the following geometry.

G_{0}  G_{1}  G_{2}
Sz(s) Sz(s′) \frac{nm}{β_{t}} : 4
G_{01} G_{02} G_{12}
Sz(t) n : 4 m : 4
G_{012}
β_{t} : 4

In that case, β_{t} |_{M} n |_{P} α_{s} or β_{s} and β_{t} |_{M} m |_{P} α_{s′} or β_{s′}

This ends the classification of all Rwpri geometries of rank 3 for the Suzuki groups.

We now state a theorem that summarizes the discussion of this section.

Theorem 4.4.3 Let G = Sz(q) with q = 2^{2e+1} and suppose 2e + 1 = p_{1}^{e_{1}} … p_{n}^{e^{n}} with p_{i} ≠ p_{j}, ∀i ≠ j, and ∑_{i=1}^{n} e_{i} ≥ 2. Suppose q = s^{p_{i}} for some i ∈ {1 … n}. If s − 1 is a prime number (which implies that ∑_{i=1}^{n} e_{i} = 2) then all rank 3 Rwpri geometries of G have a sublattice isomorphic to one of the following.

1. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & D_{2(s−1)(t−1)} & Sz(t) \\ G_{01} & G_{02} & G_{12} \\ D_{2(s−1)} & D_{10} & D_{2(t−1)} \\ & G_{012} \\ & 2 \end{aligned} with s = 2^{p_{1}}, t = 2^{p_{2}}, 2e + 1 = p_{1} · p_{2}, and

p_{1} ≠ p_{2} two prime numbers.

2. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & D_{2(s−1)(t−1)} & Sz(s′) \\ G_{01} & G_{02} & G_{12} \\ D_{2(s−1)} & 2^{2} & D_{2(t−1)} \\ & G_{012} \\ & 2 \end{aligned} with s = 2^{p_{1}}, s′ = 2^{p_{2}}, 2e + 1 = p_{1} · p_{2},

and p_{1} ≠ p_{2} two prime numbers.

3. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & D_{2(s−1)(t−1)} & Sz(s′) \\ G_{01} & G_{02} & G_{12} \\ D_{2(s−1)} & 4 & D_{2(t−1)} \\ & G_{012} \\ & 2 \end{aligned} with s = 2^{p_{1}}, s′ = 2^{p_{2}}, 2e + 1 = p_{1} · p_{2}, are

p_{1} ≠ p_{2} two prime numbers.

4. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & D_{2(s−1)p} & q : ((s − 1)p) \\ G_{01} & G_{02} & G_{12} \\ D_{2(s−1)} & s : (s − 1) & (s − 1)p \\ & G_{012} \\ & s − 1 \end{aligned} with p | \frac{q−1}{s−1} a prime number.

5. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & (ap) : 4 & Sz(t) \\ G_{01} & G_{02} & G_{12} \\ a : 4 & 4 × 2 & p : 4 \\ & G_{012} \\ & 4 \end{aligned} with a = α_{s} or β_{s} (resp. p = α_{t} or β_{t}) prime

numbers, n = 2, q = s^{p_{1}} = t^{p_{2}} and p_{1} ≠ p_{2}.

If s − 1 is not a prime number but ∑_{i=1}^{n} e_{i} = 2 then all rank 3 Rwpri geometries of G have a sublattice isomorphic to sublattices number 4 or 5 of the case s − 1 prime. Finally, if ∑_{i=1}^{n} e_{i} > 2 then all rank 3 Rwpri geometries of G have a sublattice isomorphic to one of the following.

6. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & D_{2(s−1)p} & q : ((s − 1)p) \\ G_{01} & G_{02} & G_{12} \\ D_{2(s−1)} & s : (s − 1) & (s − 1)p \\ & G_{012} \\ & s − 1 \end{aligned} with p | \frac{q−1}{s−1} a prime number.

4.5. ABOUT THE FLAG-TRANSITIVITY 69

7. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & D_{2(t−1)pp′} & Sz(s′) \\ G_{01} & G_{02} & G_{12} \\ D_{2(t−1)p} & Sz(t) & D_{2(t−1)p′} \\ & G_{012} \\ & D_{2(t−1)} \end{aligned} with p | \frac{s−1}{t−1}, p′ | \frac{s′−1}{t−1} two distinct prime

numbers, s = t^{p_{i}} and s′ = t^{p_{j}}, with i ≠ j.

8. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & (mpp′) : 4 & Sz(s′) \\ G_{01} & G_{02} & G_{12} \\ mp : 4 & Sz(t) & mp′ : 4 \\ & G_{012} \\ & m : 4 \end{aligned} with m = α_{t} or β_{t}, mp | α_{s} or β_{s}, mp′ | α_{s′}

or β_{s′}, and p, p′ two prime numbers.

9. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & qˆ .t : (t − 1) & Sz(s′) \\ G_{01} & G_{02} & G_{12} \\ sˆ .t : (t − 1) & Sz(t) & s′ˆ .t : (t − 1) \\ & G_{012} \\ & tˆ .t : (t − 1) \end{aligned} with q = s^{p_{i}} = s^{′p_{j}} = t^{p_{i}p_{j}}, and

i ≠ j.

10. \begin{aligned}G_{0} & G_{1} & G_{2} \\ Sz(s) & Sz(s′) & Sz(s”) \\ G_{01} & G_{02} & G_{12} \\ Sz(t) & Sz(t′) & Sz(t”) \\ & G_{012} \\ & Sz(u) \end{aligned} with ∑_{i=1}^{n} e_{i} ≥ 4, n ≥ 3, q = s^{p_{i}} = s^{′p_{j}} = s”^{p_{k}} =

t^{p_{i}p_{j}} = t^{′p_{i}p_{k}} = t”^{p_{j}p_{k}} = u^{p_{i}p_{j}p_{k}}, and i, j, k three pairwise distinct numbers.

4.5 About the flag-transitivity

Most of the time, we are interested in knowing whether a Rwpri (pre-)geometry is a flag-transitive geometry or not. We do not intend to solve that problem in the present thesis. Anyway, lemma 4.2.4 permits us to give the answer for a lot of geometries obtained in the previous section.

For the case q − 1 prime, looking at theorem 4.4.1, we see that geometries corresponding to the second type, and geometries corresponding to the third type with p = 2 are indeed flag-transitive.

For the case 2e+1 not prime, looking at theorem 4.4.3, we see that all geometries, except number 9 and 10, have a linear diagram and are thus surely flag-transitive.

We intend to study the flag-transitivity of those geometries that do not have a linear diagram in future work.

4.6 About the (2T)_{1} property

If we look at the rank 3 Rwpri geometries that satisfy the (2T)_{1} property, we see quite easily that in theorem 4.4.1, the only geometries that survive this test are the thin geometries, and in theorem 4.4.3, no geometry satisfies this test. This means that, if one wants to construct a rank 4 Rwpri pre-geometry satisfying (2T)_{1} for a Suzuki group Sz(q), one has to choose q = 2^{pp′} with p, p′ two prime numbers. Then, G_{0} being a subgroup of Suzuki type because of lemma 4.3.5, it means that G_{0} = Sz(s) must have a rank 3 Rwpri geometry satisfying (2T)_{1}. This implies that s − 1 is a prime number and we may assume that G_{01} = D_{2(s−1)}. Thus G_{1} is either a group of Suzuki type or a dihedral group. It cannot be a dihedral group D_{2n} with n an odd number, because no such dihedral group has Rwpri pre-geometries of rank 3 or higher satisfying (2T)_{1} thanks to the classification of the Rwpri geometries for these groups given in theorem 2.5.1 (we remind again here that all Rwpri pre-geometries of a dihedral group are flag-transitive geometries). So it must be a Sz(t) with t ≠ s. But then s − 1 = t − 1 and s ≠ t, a contradiction. This permits to state the following theorem.

Theorem 4.6.1 Let G = Sz(q) with q = 2^{2e+1}. If q − 1 is not prime, Sz(q) has only one Rwpri and (2T)_{1} pre-geometry up to isomorphism. It is a rank 2 F, RC, FT geometry with G_{0} = D_{2(q−1)}, G_{1} = q : (q − 1) and G_{01} = q − 1. If q − 1 ≠ 7 is prime, the Rwpri and (2T)_{1} pre-geometries of G are of rank ≤ 3, one of them only being of rank 2, the others being thin rank 3 geometries. Finally, if q = 8, the Rwpri and (2T)_{1} pre-geometries of G are of rank ≤ 3, three of them only being of rank 2, the others being thin rank 3 geometries.

The pre-geometries of rank ≥ 4 of Sz(q)

This chapter is taken from [28].

5.1 Introduction

In this chapter, we determine all Rwpri pre-geometries of rank ≥ 4 for any Suzuki simple group Sz(q). To do that, we start from the classification theorems stated in chapter 4. We get, as final result, eleven constructions leading to all Rwpri pre-geometries of rank ≥ 4 for any Sz(q) (see theorem 5.3.1). We plan to determine in future work which of these pre-geometries are indeed flagtransitive geometries. The results obtained in this chapter strenghten our belief that the Rwpri condition is an efficient condition to impose on the subgroups forming the geometries.

72 CHAPTER 5. THE PRE-GEOMETRIES OF RANK ≥ 4 OF SZ(Q)

The chapter is organised as follows. In section 2, we determine the pre-geometries of rank 4. This section and the next one are there to show that even without imposing the (2T)_{1} property (as suggested in chapter 4), it is possible to determine theoretically all the Rwpri pre-geometries of any rank. Also, nice families of pre-geometries arise, all with the same kind of diagram. In section 3, we resume the long discussion made in section 2 by giving eleven constructions that lead to all Rwpri pre-geometries of rank ≥ 4. Finally, in section 4, we give some concluding remarks about the results obtained in this chapter and mention some future work we plan to do in this area.

5.2 The rank ≥ 4 pre-geometries of Sz(q)

We remind the reader that lemma 1.1.1 permits us to rely on the rank 3 classification given in chapter 4 when we determine all rank ≥ 4 Rwpri pre-geometries since in the rank three case, every pre-geometry is a geometry. Thanks to lemma 4.3.5, we may assume G_{0} = Sz(s) with q = s^{p} and p a prime number. Throughout this section we assume s = 2^{2f+1} and q = s^{p} = 2^{2e+1}. We now concentrate on the rank 4. The higher ranks will appear clearly as generalizations of the extension process we use to construct the rank 4 pre-geometries from the rank 3 residues. These generalizations are summarized in the next section in eleven constructions of Rwpri pre-geometries.

5.2.1 s − 1 is prime

In this case, the rank 3 residues for G_{0} are those given in theorem 4.4.1. We take each of the 3 different types of residue one by one and try to extend it to a rank 4 Rwpri pre-geometry.

Residue 1:

The sublattice is partly filled as in table 5.1.

Because G_{01} = D_{2(s−1)}, the subgroup G_{1} is either a subgroup of D_{2(q−1)} or a group of Suzuki type. Suppose first that G_{1} < D_{2(q−1)}. Then G_{1} is a dihedral group. Thanks to theorem 2.5.1, we know how the G_{1}-residue looks like, and we also know that G_{1} = D_{2(s−1)n} with n | \frac{q−1}{s−1} a prime number. Table 5.2 gives a more complete sublattice. Now, G_{2} =< s : (s − 1), D_{2n}, G_{23} > implies that G_{2} is a subgroup of Suzuki type, let’s say G_{2} = Sz(s′), with q = s′^{p′} and s′ − 1, p′ two prime numbers. Thus the rank 3 residue of G_{3} must be one of those appearing in theorem 4.4.1. Because all pre-geometries given in this theorem have at least two subgroups isomorphic to Z_{2} in their sublattice, we see that the pre-geometry we have here is not Rwpri.

5.2. THE RANK ≥ 4 PRE-GEOMETRIES OF SZ(Q) 73

G_{0}  G_{1}  G_{2}  G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
D_{2(s−1)} s : (s − 1) s : (s − 1)
G_{012}  G_{013}  G_{023}  G_{123}
2  2  s − 1

G_{0123} 1

Table 5.1: The sublattice partly filled by residue 1

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s) D_{2(s−1)n}
G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
D_{2(s−1)} s : (s − 1) s : (s − 1) D_{2n} D_{2n}
G_{012}  G_{013}  G_{023}  G_{123}
2  2  s − 1  n

G_{0123} 1

Table 5.2: The residue 1 and G_{1} a dihedral group

G_{0}  G_{1}  G_{2} G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
D_{2(s−1)} s : (s − 1) D_{2n}
G_{012}  G_{013}  G_{023} G_{123}
s − 1  2  2

G_{0123} 1

Table 5.3: The sublattice partly filled by residue 2

We may thus assume G_{1} = Sz(s′) with q = s′^{p′} and s′ − 1 and p′ two prime numbers. This yields s′ − 1 = s − 1. Thus we have a contradiction because D_{2(s−1)} is self-normalized in Sz(q) and hence cannot be contained in two subgroups isomorphic to Sz(s).

Residue 2:

The sublattice is partly filled as in table 5.3. Again, the subgroup G_{1} is either a dihedral group or a Suzuki simple group. It cannot be a Suzuki group for the same reason as in the previous case. So we may assume G_{1} = D_{2(s−1)n′} with n′ | \frac{q−1}{s−1} a prime number. Thanks to theorem 2.5.1, we know that there are two possible residues for G_{1}.

The first one is with G_{12} = D_{2(s−1)}. It implies that G_{2} =< s : (s−1), D_{2(s−1)}, G_{23} > must be a subgroup of Suzuki type. Thus G_{2} = Sz(s′) with q = s′^{p′}, and s′−1 and p′ two prime numbers. Then s − 1 = s′ − 1. Because s : (s − 1) is self-normalized in Sz(q), we have G_{0} = G_{2}, a contradiction.

The second residue for G_{1} completes a little more our sublattice (see table 5.4). Suppose n | q − 1. Then n = s − 1 and G_{3} is a dihedral group. Thanks to theorem 2.5.1, we have G_{23} = D_{2n′} and hence G_{2} =< s : (s − 1), (s − 1)n′, D_{2n′} > must be a subgroup of Suzuki type. Then, G_{2} = Sz(s) and because s : (s − 1) is self-normalized in Sz(q), we get G_{0} = G_{1}, a contradiction. Suppose then that n | s^{2} + 1. Then G_{3} =< D_{2n}, D_{2n′}, G_{23} > must be a subgroup of Suzuki type, say Sz(s′) with q = s′^{p′} and s′ − 1 and p′ two prime numbers. Again, p′ ≠ p otherwise, s = s′ and then G_{0} = G_{2} as in the preceding case.

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s) D_{2(s−1)n′}
G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
D_{2(s−1)} s : (s − 1) D_{2n} (s − 1)n′ D_{2n′}
G_{012}  G_{013}  G_{023} G_{123}
s − 1  2  2  n’

G_{0123} 1

Table 5.4: The residue 2 and G_{1} a dihedral group.

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s) D_{2(s−1)(s′−1)} q : ((s − 1)(s′ − 1))  Sz(s′)
G_{01}  G_{02}  G_{03}  G_{12}  G_{13}  G_{23}
D_{2(s−1)} s : (s − 1)  D_{10}  (s − 1)(s′ − 1) D_{2(s′−1)} s′ : (s′ − 1)
G_{012}  G_{013}  G_{023}  G_{123}
s − 1  2  2  s’-1

G_{0123} 1

Table 5.5: The only sublattice for residue 2.

This yields G_{23} = s′ : (s′ − 1) and G_{2} = q : ((s − 1)(s′ − 1)). Finally, thanks to lemma 4.2.8, we know that n = 5. The sublattice is then fully determined (see table 5.5). We don’t know whether the group Sz(q) acts flag-transitively on this pre-geometry, but if it does, the diagram of this pre-geometry looks as in figure 5.1.

The generalization to higher ranks is obvious. It is described in construction 5.3.1 of the next section.

Residue 3:

The sublattice is partly filled as in table 5.6. Again, we know that G_{1} must be a dihedral group, and because G_{012} ≅ G_{013} ≅ Z_{2}, we have G_{123} ≅ Z_{n”} and G_{12} ≅ G_{13} ≅ D_{2n”} thanks to theorem 2.5.1.

[figure: ⊃; ⊃; 5; s − 2; Sz(s); s′ − 2; Sz(s′); 1; q : ((s − 1)(s′ − 1)); 1; D_{2(s−1)(s′−1)}]

Figure 5.1: The diagram obtained for residue 2.

G_{0}  G_{1} G_{2} G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
D_{2(s−1)} D_{2n} D_{2n′}
G_{012} G_{013} G_{023} G_{123}
2 2 2

G_{0123} 1

Table 5.6: The sublattice partly filled by residue 3 Suppose n = s − 1. Then G_{2} ≅ G_{1} and because N_{Sz(q)}(D_{2n”}) = D_{2n”}, we get G_{1} = G_{2}, a contradiction. Suppose then that n ≠ s − 1 ≠ n′. Then G_{3} ≅ Sz(s′) (resp. G_{4} ≅ Sz(s”)), with q = s′^{p′} (resp. q = s”^{p”}) and s′ − 1 and p′ (resp. s” − 1 and p”) two prime numbers. This yields s′ = s” and thus G_{2} = G_{3}, a contradiction.

5.2.2 s − 1 not prime and 2f + 1 prime

In this case, lemma 4.3.5 and theorem 4.4.2 show that there is no Rwpri pregeometry of rank greater than 3.

5.2.3 2f + 1 not prime

This is the most complicated case to analyse. The rank 3 residues of G_{0} are those given in theorem 4.4.3. They are subdivided into three cases depending of the primality of t − 1 and 2g + 1, where s = t^{p′} with p′ a prime number and thus 2f + 1 = (2g + 1)p′.

The case t − 1 is prime

In this case, the G_{0}-residues are numbers 1 to 5 of theorem 4.4.3. We analyse them one by one.

Residue 1:

The sublattice is partly filled as in table 5.7. In that table, 2f + 1 = p_{1} · p_{2} with p_{1} ≠ p_{2}, s = t^{p_{1}} = t^{′p_{2}}, and t − 1, t′ − 1 two prime numbers. Because Sz(s) is a maximal subgroup of Sz(q), we have 2e + 1 = p_{1}p_{2}p_{3} with p_{i} prime for i = 1, 2, and 3. Also, G_{1} (resp. G_{3}) contains a subgroup of Suzuki type. Thus G_{1} (resp. G_{3}) is of Suzuki type. This yields G_{1} ≅ Sz(s′) (resp. G_{3} ≅ Sz(s”)) with s ≠ s′ (resp. s ≠ s”) and hence p_{3} ≠ p_{1} (resp. p_{3} ≠ p_{2}). They both have a residue of type 1 as G_{0}. Now, G_{2} must be a dihedral group and thanks to theorem 2.5.1, we can determine the full sublattice for this pre-geometry (see table 5.8). The generalization to higher ranks is obvious. It is described in construction 5.3.2 of the next section.

G_{0}  G_{1}  G_{2}  G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) D_{2(t−1)(t′−1)} Sz(t′)
G_{012}  G_{013}  G_{023} G_{123}
D_{2(t−1)}  D_{10}  D_{2(t′−1)}

G_{0123} 2

Table 5.7: The sublattice partly filled by residue 1

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s)  Sz(s′)  D_{2(t−1)(t′−1)(t”−1)}  Sz(s”)
G_{01}  G_{02}  G_{03}  G_{12}  G_{13}  G_{23}
Sz(t) D_{2(t−1)(t′−1)}  Sz(t′)  D_{2(t−1)(t”−1)} Sz(t”) D_{2(t′−1)(t”−1)}
G_{012}  G_{013}  G_{023}  G_{123}
D_{2(t−1)}  D_{10}  D_{2(t′−1)}  D_{2(t”−1)}

G_{0123} 2

Table 5.8: The only full sublattice for residue 1

G_{0}  G_{1}  G_{2}  G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) D_{2(t−1)n} s : ((t − 1)n)
G_{012} G_{013} G_{023} G_{123}
D_{2(t−1)} t : (t − 1)  (t − 1)n

G_{0123} t − 1

Table 5.9: The sublattice partly filled by residue 4

Residue 2:

We can make the same analysis as in the previous case, just by changing D_{10} by 2^{2}.

The generalization to higher ranks is obvious. It is described in construction 5.3.3 of the next section.

Residue 3:

We can make the same analysis as in the previous case, just by changing 2^{2} by Z_{4}.

The generalization to higher ranks is obvious. It is described in construction 5.3.4 of the next section.

Residue 4:

The sublattice is partly filled as in table 5.9. In that table, n | \frac{s−1}{t−1} is a prime number, and s = 2^{p_{1}p_{2}} with p_{1} and p_{2} two prime numbers (not necessarily distinct). Because G_{1} contains a subgroup of Suzuki type, it must be a subgroup of Suzuki type. Thus G_{1} = Sz(s′) with s′ ≠ s and q = 2^{p_{1}p_{2}p_{3}} with p_{2} ≠ p_{3}, and s′ = 2^{p_{1}p_{3}}. The Borel subgroup of Γ_{1} being t − 1, the residue of G_{1} is as the residue of G_{0}. The sublattice is then a little more complete (see table 5.10). Again here, n′ | \frac{q−1}{s′−1} is a prime number. Now, G_{2} =< D_{2(t−1)n}, D_{2(t−1)n′} > must be a dihedral group. This yields G_{23} ≅ (t − 1)nn′ and G_{2} = D_{2(t−1)nn′}. Finally, G_{4} =< s : ((t − 1)n), s′ : ((t − 1) : n′), (t − 1)nn′ >= q : ((t − 1)nn′). Thus the sublattice is fully known (see table 5.11).

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s) Sz(s′)
G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) D_{2(t−1)n} s : ((t − 1)n) D_{2(t−1)n′} s′ : ((t − 1)n′) (t − 1)nn′
G_{012}  G_{013}  G_{023}  G_{123}
D_{2(t−1)} t : (t − 1)  (t − 1)n  (t − 1)n′
G_{0123}
t − 1

Table 5.10: The residue 4 and G_{1} known

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s) Sz(s′)  D_{2(t−1)nn′} q : ((t − 1)nn′)
G_{01}  G_{02}  G_{03}  G_{12}  G_{13}  G_{23}
Sz(t) D_{2(t−1)n} s : ((t − 1)n)  D_{2(t−1)n′}  s′ : ((t − 1)n′) (t − 1)nn′
G_{012}  G_{013}  G_{023}  G_{123}
D_{2(t−1)} t : (t − 1)  (t − 1)n  (t − 1)n′

G_{0123} t − 1

Table 5.11: The only full sublattice for residue 4

G_{0}  G_{1}  G_{2}  G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) (γ_{t}γ_{t′}) : 4 Sz(t′)
G_{012} G_{013} G_{023} G_{123}
γ_{t} : 4 4 × 2 γ_{t′} : 4

G_{0123} 4

Table 5.12: The sublattice partly filled by residue 5

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s) Sz(s′) γ_{t}γ_{t′}γ_{t”} : 4 Sz(s”)
G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) (γ_{t}γ_{t′}) : 4 Sz(t′) (γ_{t}γ_{t”}) : 4 Sz(t”) (γ_{t′}γ_{t”}) : 4
G_{012}  G_{013}  G_{023}  G_{123}
γ_{t} : 4 4 × 2  γ_{t′} : 4  γ_{t”} : 4

G_{0123} 4

Table 5.13: The completed sublattice for residue 5

The generalization to higher ranks is obvious. It is described in construction 5.3.5 of the next section.

Residue 5:

The sublattice is partly filled as in table 5.12. In that table, γ_{t} = α_{t} or β_{t} is prime and γ_{t′} = α_{t′} or β_{t′} is also prime. Because G_{01} and G_{03} are Suzuki groups, G_{1} and G_{3} must also be Suzuki groups. Thus q = 2^{p_{1}p_{2}p_{3}}, s = 2^{p_{1}p_{2}}, t = 2^{p_{1}}, t′ = 2^{p_{2}}, and G_{1} ≅ Sz(s′), G_{3} ≅ Sz(s”) with s′ = 2^{p_{1}p_{3}} and s” = 2^{p_{2}p_{3}}. The G_{1} and G_{3} residues are the same as the one of G_{0}. This yields G_{13} ≅ Sz(t”) with t” = 2^{p_{3}}, G_{123} ≅ γ_{t”} : 4 with γ_{t”} = α_{t”} or β_{t”} a prime number, and G_{23} ≅ γ_{t′}γ_{t”} : 4. Finally, G_{2} =< γ_{t}γ_{t′} : 4, γ_{t}γ_{t”} : 4 > yields G_{2} ≅ γ_{t}γ_{t′}γ_{t”} : 4. The sublattice is completely determined (see table 5.13).

G_{0}  G_{1}  G_{2}  G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) D_{2(t−1)n} s : ((t − 1)n)
G_{012} G_{013} G_{023} G_{123}
D_{2(t−1)} t : (t − 1)  (t − 1)n

G_{0123} t − 1

Table 5.14: The sublattice partly filled by residue 6

The generalization to higher ranks is obvious. It is described in construction 5.3.6 of the next section.

The case t − 1 not prime and 2g + 1 prime.

Theorem 4.4.3 says that residues 4 and 5 are the only possible G_{0}-residues. In the preceding discussion for residues 4 and 5, the primality of t − 1 is not important. So these two cases are exactly the same, whether t − 1 is prime or not.

The case 2g + 1 not prime.

In this case, the residues are numbers 6 to 10 of theorem 4.4.3. We analyse them one by one.

Residue 6:

The sublattice is partly filled as in table 5.14. In that table, n | \frac{s−1}{t−1} is prime. Because of G_{01}, the subgroup G_{1} must be a subgroup of Suzuki type. Its residue can only be of type 6. Also, G_{1} ≅ Sz(s′) with s′ ≠ s. The sublattice is then filled a little more (see table 5.15, where n′ | \frac{s′−1}{t−1} is prime). Now, because G_{2} =< G_{02}, G_{12}, G_{23} >, we know that G_{2} must be a dihedral group. Thanks to theorem 2.5.1, we then have G_{23} ≅ (t − 1)nn′. Thus G_{3} ≅ q : ((t − 1)nn′) and the sublattice is completely determined (see table 5.16).

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s) Sz(s′)
G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) D_{2(t−1)n} s : ((t − 1)n) D_{2(t−1)n′} s′ : ((t − 1)n′)
G_{012}  G_{013}  G_{023}  G_{123}
D_{2(t−1)} t : (t − 1)  (t − 1)n  (t − 1)n′

G_{0123} t − 1

Table 5.15: Residue 6 and G_{1} known

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s) Sz(s′)  D_{2(t−1)nn′} q : ((t − 1)nn′)
G_{01}  G_{02}  G_{03}  G_{12}  G_{13}  G_{23}
Sz(t) D_{2(t−1)n} s : ((t − 1)n)  D_{2(t−1)n′}  s′ : ((t − 1)n′) (t − 1)nn′
G_{012}  G_{013}  G_{023}  G_{123}
D_{2(t−1)} t : (t − 1)  (t − 1)n  (t − 1)n′

G_{0123} t − 1

Table 5.16: The only sublattice for residue 6

G_{0}  G_{1}  G_{2}  G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) D_{2(u−1)nn′} Sz(t′)
G_{012} G_{013} G_{023} G_{123}
D_{2(u−1)n} Sz(u) D_{2(u−1)n′}

G_{0123} D_{2(u−1)}

Table 5.17: The sublattice partly filled by residue 7

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s)  Sz(s′)  Sz(s”) D_{2(u−1)nn′n”}
G_{01}  G_{02}  G_{03}  G_{12}  G_{13}  G_{23}
Sz(t) D_{2(u−1)nn′} Sz(t′) D_{2(u−1)nn”} Sz(t”) D_{2(u−1)n′n”}
G_{012} G_{013} G_{023} G_{123}
D_{2(u−1)n} Sz(u) D_{2(u−1)n′} D_{2(u−1)n”}
G_{0123}
D_{2(u−1)}

Table 5.18: The only sublattice for residue 7

of the next section.

Residue 7:

The sublattice is partly filled as in table 5.17. In that table, n | \frac{t−1}{u−1} and n′ | \frac{t′−1}{u−1} are prime. Thanks to subgroups G_{01} and G_{03}, we know that subgroups G_{1} and G_{3} are isomorphic to Suzuki simple groups. Their residue has the same form as that of G_{0}. Thus the sublattice is almost completed. And we readily see that G_{2} must be a dihedral group. The sublattice is then completely determined (see table 5.18). There we have t = u^{p_{i}}, t′ = u^{p_{j}} and t” = u^{p_{k}} with i, j, k three pairwise distinct integers in \{1, …, m\}. Also, n” | \frac{t”−1}{u−1} is a prime number.

The generalization to higher ranks is obvious. It is described in construction 5.3.8 of the next section.

G_{0}  G_{1}  G_{2}  G_{3}

Sz(s)

G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) (γ_{u}nn′) : 4 Sz(t′)
G_{012} G_{013} G_{023} G_{123}
(γ_{u}n) : 4 Sz(u) (γ_{u}n′) : 4

G_{0123} γ_{u} : 4

Table 5.19: The sublattice partly filled by residue 8

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s)  Sz(s′) (γ_{u}nn′n”) : 4 Sz(s”)
G_{01}  G_{02}  G_{03}  G_{12}  G_{13}  G_{23}
Sz(t) (γ_{u}nn′) : 4  Sz(t′)  (γ_{u}nn”) : 4 Sz(t”) (γ_{u}n′n”) : 4
G_{012}  G_{013}  G_{023}  G_{123}
(γ_{u}n) : 4 Sz(u)  (γ_{u}n′) : 4  (γ_{u}n”) : 4
G_{0123}
γ_{u} : 4

Table 5.20: The only sublattice for residue 8

Residue 8:

The sublattice is partly filled as in table 5.19. In that table γ_{u} is either α_{u} or β_{u} and n (resp. n′) is a prime number dividing \frac{α_{t}}{γ_{u}} or \frac{β_{t}}{γ_{u}} (resp. \frac{α_{t′}}{γ_{u}} or \frac{β_{t′}}{γ_{u}}) depending whether γ_{u} divides α_{t} (resp. α_{t′}) or β_{t} (resp. β_{t′}). Because of G_{01} and G_{03}, we know that G_{1} and G_{3} must be Suzuki simple groups. So we may assume G_{1} ≅ Sz(s′) and G_{3} ≅ Sz(s”) with q = s^{p} = s^{′p′} = s”^{p”} and p, p′, p” three pairwise distinct prime numbers. These two subgroups have residues of the same kind as G_{0}. Thus G_{13} ≅ Sz(t”) with t” = q^{\frac{1}{p′p”}} , G_{123} ≅ (γ_{u}n”) : 4 with n” a prime number dividing \frac{α_{t}”}{γ_{u}} or \frac{β_{t}”}{γ_{u}}. Also, G_{12} ≅ (γ_{u}nn”) : 4 and G_{23} ≅ (γ_{u}n′n”) : 4. Finally, because G_{2} =< G_{02}, G_{12}, G_{23} > we have G_{2} ≅ (γ_{u}nn′n”) : 4. The sublattice is then fully known (see table 5.20).

G_{0}  G_{1}  G_{2}  G_{3}

Sz(s)

G_{01}  G_{02}  G_{03}  G_{12} G_{13} G_{23}
Sz(t)  sˆ.u : (u − 1)  Sz(t′)
G_{012}  G_{013}  G_{023}  G_{123}
tˆ.u : (u − 1)  Sz(u)  t′ˆ.u : (u − 1)

G_{0123}

uˆ.u : (u − 1)

Table 5.21: The sublattice partly filled by residue 9

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s)  Sz(s′)  qˆ.u : (u − 1)  Sz(s”)
G_{01}  G_{02}  G_{03}  G_{12}  G_{13}  G_{23}
Sz(t)  sˆ.u : (u − 1)  Sz(t′)  s′ˆ.u : (u − 1) Sz(t”) s”ˆ.u : (u − 1)
G_{012}  G_{013}  G_{023}  G_{123}
tˆ.u : (u − 1)  Sz(u)  t′ˆ.u : (u − 1) t”ˆ.u : (u − 1)

G_{0123} uˆ.u : (u − 1)

Table 5.22: The only sublattice for residue 9

of the next section.

Residue 9:

The sublattice is partly filled as in table 5.21. Because of G_{01} (resp. G_{03}), the subgroup G_{1} (resp. G_{3}) must be a Suzuki group, Sz(s′) (resp. Sz(s”)) with q = s^{′p′} = s”^{p”} and p′ ≠ p” two prime numbers. Also, thanks to G_{0123}, we know that their residues are as the one of G_{0}. This yields G_{12} ≅ s′ˆ.u : (u − 1), G_{13} ≅ Sz(t”) with q = t”^{p′p”}, G_{23} ≅ s”ˆ.u : (u − 1), G_{123} ≅ t”ˆ.u : (u − 1). We then know that G_{2} ≅ qˆ.u : (u − 1). Thus the sublattice is fully determined (see table 5.22).

The generalization to higher ranks is obvious. It is described in construction 5.3.10 of the next section.

G_{0}  G_{1}  G_{2}  G_{3}
Sz(s)
G_{01} G_{02} G_{03} G_{12} G_{13} G_{23}
Sz(t) Sz(t′) Sz(t”)
G_{012} G_{013}  G_{023} G_{123}
Sz(u) Sz(u′) Sz(u”)
G_{0123}
Sz(v)

Table 5.23: The sublattice partly filled by residue 10

Residue 10:

The sublattice is partly filled as in table 5.23. This is the easiest pre-geometry to complete. It is obvious that all subgroups appearing in the sublattice must be Suzuki groups.

The generalization to higher ranks is obvious. It is described in construction 5.3.11 of the next section.

This ends the determination of all the Rwpri pre-geometries for a Suzuki simple group Sz(q). We now resume the long discussion of this section in the next one.

5.3 Constructions and classification theorem

We give now the eleven constructions mentioned in the previous discussion. These are all generalizations of constructions of rank 4 pre-geometries described above. Of course, in order to use these constructions to build a Rwpri pre-geometry, the reader has to take care of the whole sublattice which is not completely detailed here, but which can be easily guessed by looking at the corresponding rank 4 pregeometry from which it arises. After these constructions, we state a classification theorem for the Rwpri pre-geometries of rank greater than 3 for a Suzuki simple group Sz(q).

Construction 5.3.1 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1} … p_{m} with p_{i} ≠ p_{j} for all i ≠ j. Suppose n_{i} = 2^{p_{i}} − 1 is prime for all i = 1, …, m.

Take G_{i−1} ≅ Sz(q^{\frac{1}{p_{i}}}) for all i = 1, …, m;
G_{m} ≅ D_{2n_{1}...n_{m}};
G_{m+1} ≅ q : (n_{1} … n_{m}),

with ∩_{j=0}^{m−1}G_{j} = D_{10}. Then Γ(G; G_{0}, …, G_{m+1}) is a rank m + 2 Rwpri pregeometry.

Construction 5.3.2 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1} … p_{m} with p_{i} ≠ p_{j} for all i ≠ j. Suppose n_{i} = 2^{p_{i}} − 1 is prime for all i = 1, …, m.

Take G_{i−1} ≅ Sz(q^{\frac{1}{p_{i}}}) for all i = 1, …, m;
G_{m} ≅ D_{2n_{1}...n_{m}},

with ∩_{j=0}^{m}G_{j} = D_{10}. Then Γ(G; G_{0}, …, G_{m}) is a rank m+1 Rwpri pre-geometry.

Remark that the pre-geometries obtained in construction 5.3.2 are truncations of those obtained in construction 5.3.1.

Construction 5.3.3 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1} … p_{m} with p_{i} ≠ p_{j} for all i ≠ j. Suppose n_{i} = 2^{p_{i}} − 1 is prime for all i = 1, …, m.

Take G_{i−1} ≅ Sz(q^{\frac{1}{p_{i}}}) for all i = 1, …, m;
G_{m} ≅ D_{2n_{1}...n_{m}},

with ∩_{j=0}^{m}G_{j} = D_{4}. Then Γ(G; G_{0}, …, G_{m}) is a rank m+1 Rwpri pre-geometry.

Construction 5.3.4 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1} … p_{m} with p_{i} ≠ p_{j} for all i ≠ j. Suppose n_{i} = 2^{p_{i}} − 1 is prime for all i = 1, …, m.

Take G_{i−1} ≅ Sz(q^{\frac{1}{p_{i}}}) for all i = 1, …, m;
G_{m} ≅ D_{2n_{1}...n_{m}},

with ∩_{j=0}^{m}G_{j} = Z_{4}. Then Γ(G; G_{0}, …, G_{m}) is a rank m + 1 Rwpri pre-geometry.

Remark that the only difference between constructions 5.3.2, 5.3.3 and 5.3.4 is the intersection of the Suzuki subgroups.

Construction 5.3.5 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1} … p_{m−1} · p_{m}^{ϵ} with p_{i} ≠ p_{j} for all i ≠ j and ϵ = 1 or 2. We distinguish the case ϵ = 1 and the case ϵ = 2.

Case 1: ϵ = 1

Suppose t − 1 = 2^{p_{i}} − 1 is prime for some i ∈ {1, …, m}. We construct a pre-geometry as follows.

Take G_{j−1} ≅ Sz(q^{\frac{1}{p_{j}}}) for all j = 1, …, m, j ≠ i;

G_{i} ≅ D_{2Π_{j∈\{1,...,m\}\\{i\}}n_{j}} where n_{j} | \frac{2^{p_{i}p_{j}} −1}{t−1} are pairwise distinct prime numbers;

G_{m} ≅ q : ((t − 1)Π_{j∈{1,...,m}\{i}}n_{j}).

Then Γ(G; G_{0}, …, G_{m}) is a rank m + 1 Rwpri pre-geometry. The choice of i might lead to m pairwise distinct pre-geometries.

Case 2: ϵ = 2

Suppose t − 1 = 2^{p_{m}} − 1 is prime. We construct a pre-geometry as follows.

Take G_{j−1} ≅ Sz(q^{\frac{1}{p_{j}}}) for all j = 1, …, m;

G_{m} ≅ D_{2n_{1}...n_{m}} where n_{j} | \frac{2^{p_{j} pm}−1}{t−1} are pairwise distinct prime numbers;

G_{m+1} ≅ q : ((t − 1)Π_{j∈{1,...,m}\{i}}n_{j}).

Then Γ(G; G_{0}, …, G_{m}) is a rank m + 2 Rwpri pre-geometry.

Construction 5.3.6 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1} … p_{m} with p_{i} ≠ p_{j} for all i ≠ j. Suppose that for all i = 1, …, m, at least one of α_{2^{p_{i}}} or β_{2^{p_{i}}} is prime and denote that number by γ_{i}.

Take G_{i−1} ≅ Sz(q^{\frac{1}{p_{i}}}) for all i = 1, …, m;
G_{m} ≅ (γ_{1} … γ_{m}) : 4.

Then Γ(G; G_{0}, …, G_{m}) is a rank m + 1 Rwpri pre-geometry.

Construction 5.3.7 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1}^{e_{1}} … p_{m}^{e_{m}} with p_{i} ≠ p_{j} for all i ≠ j. Let α ∈ {3, … min(m + 2, ∑_{i=1}^{m} e_{i})}.

Take α−2 pairwise non-isomorphic maximal subgroups G_{i} ≅ Sz(s_{i}) (i = 1, …, α− 2) and put a_{i} = log_{2}(\frac{q}{s_{i}}). Let t = q^{\frac{1}{a_{1}...a_{α−2}}} and take n_{i} | \frac{t^{a_{i}}−1}{t−1} pairwise distinct prime numbers (i = 1, …, α − 2). Take G_{α−2} ≅ D_{2(t−1)n_{1}...n_{α−2}} and G_{α−1} ≅ q : ((t − 1)n_{1} … n_{α−2}).

Then Γ(G; G_{0}, …, G_{α−1}) is a rank α Rwpri pre-geometry.

Construction 5.3.8 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1}^{e_{1}} … p_{m}^{e_{m}} with p_{i} ≠ p_{j} for all i ≠ j. Let α ∈ {3, … min(m + 1, ∑_{i=1}^{m} e_{i})}.

Take α − 1 pairwise non-isomorphic maximal subgroups G_{i} ≅ Sz(s_{i}) with i = 1, …, α − 1. Put a_{i} = log_{2}(\frac{q}{s_{i}}). Let t = q^{\frac{1}{a_{1}...a_{α−1}}} and take n_{i} | \frac{t^{a_{i}}−1}{t−1} pairwise distinct prime numbers (i = 1, …, α − 1). Take G_{α−1} ≅ D_{2(t−1)n_{1}...n_{α−1}}.

Then Γ(G; G_{0}, …, G_{α−1}) is a rank α Rwpri pre-geometry.

Construction 5.3.9 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1}^{e_{1}} … p_{m}^{e_{m}} with p_{i} ≠ p_{j} for all i ≠ j. Let α ∈ {3, … min(m + 1, ∑_{i=1}^{m} e_{i})}.

Take α − 1 pairwise non-isomorphic maximal subgroups G_{i} ≅ Sz(s_{i}) with i = 1, …, α − 1. Put a_{i} = log_{2}(\frac{q}{s_{i}}). Let t = q^{\frac{1}{a_{1}...a_{α−1}}} . Take γ = α_{t} or β_{t} and take n_{i} | \frac{α_{t^{a_{i}}}}{γ} or \frac{β_{t^{a_{i}}}}{γ} pairwise distinct prime numbers (i = 1, …, α − 1) such that γn_{1} … n_{α−1} divides either α_{q} of β_{q}. Take G_{α−1} ≅ (γn_{1} … n_{α−1}) : 4.

Then Γ(G; G_{0}, …, G_{α−1}) is a rank α Rwpri pre-geometry.

Construction 5.3.10 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1}^{e_{1}} … p_{m}^{e_{m}} with p_{i} ≠ p_{j} for all i ≠ j. Let α ∈ {3, … min(m + 1, ∑_{i=1}^{m} e_{i})}.

Take α − 1 pairwise non-isomorphic maximal subgroups G_{i} ≅ Sz(s_{i}) with i = 1, …, α − 1. Put a_{i} = log_{2}(\frac{q}{s_{i}}). Let t = q^{\frac{1}{a_{1}...a_{α−1}}} . Take G_{α−1} ≅ qˆ .t : (t − 1).

Then Γ(G; G_{0}, …, G_{α−1}) is a rank α Rwpri pre-geometry.

Construction 5.3.11 Let G ≅ Sz(q) with q = 2^{2e+1} and e > 1 a positive integer. Let 2e + 1 = p_{1}^{e_{1}} … p_{m}^{e_{m}} with p_{i} ≠ p_{j} for all i ≠ j. Let α ∈ {2, … m − 1}.

Take α pairwise non-isomorphic maximal subgroups G_{i} ≅ Sz(s_{i}) with i = 1, …, α.

Then Γ(G; G_{0}, …, G_{α−1}) is a rank α Rwpri pre-geometry.

Theorem 5.3.1 The Rwpri pre-geometries of rank ≥ 4 are those given by constructions 5.3.1 to 5.3.11.

Proof: This is an immediate consequence of the discussion given in section 5.2 and the eleven constructions described in this section. □

5.4 Observations and future work

A first observation is that for almost all Rwpri pre-geometries of rank greater than 3 of a Suzuki simple group Sz(q), if they are flag-transitive geometries, then, their diagram is a star-diagram. Indeed, only those obtained by construction 5.3.11 will not have a star-diagram.

The pre-geometries given by construction 5.3.1 look very attractive. Indeed, if they are flag-transitive geometries, their diagram is fully known. It is a star diagram with one rank 2 residue being a generalized 5-gon and all other nontrivial rank 2 residues being circles. It looks as follows.

[figure: ⋮; ⊃; ⊃; ⊃; 5]

As future work, we plan to study the flag-transitivity of these Rwpri pre-geometries, and also of all those given by theorem 5.3.1.

Appendix A

The RWPRI geometries of Sz(8)

This appendix is taken from [30].

A.1 Introduction

The present appendix gives a complete classification of the firm and residually connected geometries on which the group Sz(8) acts flag-transitively and residually weakly primitively.

This work continues a systematic investigation of groups that has started some years ago (see [7, 8, 9, 17, 18, 19, 20, 23]).

Here we study the smallest Suzuki simple group, namely Sz(8). One of the reasons for this choice is that the infinite class of Suzuki groups Sz(q), with q = 2^{2e+1} and e ≥ 1, looks particularly attractive for a general study and that a previous treatment of the smallest case may help in guessing the way to follow in general. Such a work is described in chapters 3, 4 and 5 of this thesis.

At the end of the appendix, we give the subgroup pattern of Sz(8). This pattern has been computed by Michel Dehon in August 1994, using the help of the computer algebra package Cayley [10, 11].

For an introduction on the Suzuki groups, we refer to [33]. Facts about Sz(8) can also be found in the Atlas of Finite Groups [14]. For a historical survey of the Suzuki groups, we refer to [6].

Electronic Availability: A Magma file containing the subgroups forming the 180 geometries obtained in this appendix, is available at the following address:

http://cso.ulb.ac.be/∼dleemans/abstracts/sz8.html

A.2 The geometries of the Suzuki group Sz(8)

In this section we mention all the F, RC, FT, RWPRI geometries that can be constructed from Sz(8).

Besides the numbering of each geometry, there is a number between parenthesis giving the number of non-conjugate geometries that are fused under the action of Aut(G). We mention when a geometry satisfies the conditions PRI, QPRI, RPRI, (2T)_{1}. We also give diagrams of the geometries that do not satisfy the intersection property (IP)_{2} as there are only 5 such geometries.

Up to isomorphism, there are 29 geometries of rank 2 and 151 of rank 3. There is no geometry of rank ≥ 4.

A.2.1 Rank 2 geometries

The rank 2 geometries have been determined using the classification theorem stated in chapter 3 (see theorem 3.6.1). Another way to obtain these results is the following. We look at the subgroup pattern of Sz(8). We choose a subgroup G_{0} maximal in G. Then we find all its maximal subgroups. Finally, all the subgroups containing a maximal subgroup of G_{0} as a maximal subgroup are taken as G_{1} subgroups. These couples of subgroups give us the rank 2 geometries of G. Then we still have to compute how many of them are pairwise non-isomorphic, which is a quite long (but not hard) task. And when we have all the geometries, we can compute their diagrams using the computer algebra package Magma. Table A.1 gives for each rank 2 geometry Γ, the rank 3 geometries containing it as a truncation, Aut(Γ, G) and Cor(Γ, G). The first entry is the number of the rank 2 geometry that is analysed. The second entry is a list of numbers corresponding to the rank 3 geometries that contain the corresponding rank 2 geometry as a truncation. When G_{0} is not isomorphic to G_{1}, it is not possible to find nontype preserving automorphisms. Thus in these cases, we write nothing in the Cor(Γ, G) column.

Theorem A.2.1 Up to isomorphism, there are 29 firm, residually connected geometries of rank 2 on which Sz(8) acts flag-transitively and residually weakly primitively. Their diagrams are given below.

Proof : This is a direct application of theorem 3.6.1. □

[figure: 1. (3); 7; 2; 7; 6; 6; 2080; 2080; D_{14}; D_{14}; B = 2; RPRI]

[figure: 2. (3); 3. (3); 4. (3); 5. (3); 6. (1); 7; 3; 7; 6; 6; 2080; 2080; D_{14}; D_{14}; B = 2; RPRI]

[figure: 7. (3); 7; 4; 7; 6; 6; 2080; 2080; D_{14}; D_{14}; B = 2; RPRI]

[figure: 8. (3); 9. (3); 10. (3); 6; 2; 6; 12; 6; 2080; 1120; D_{14}; D_{26}; B = 2; QPRI]

[figure: 11. (3); 12. (3); 13. (1); 6; 3; 6; 12; 6; 2080; 1120; D_{14}; D_{26}; B = 2; QPRI]

[figure: 14. (3); 8; 2; 8; 4; 6; 2080; 2912; D_{14}; D_{10}; B = 2; QPRI]

[figure: 15. (3); 8; 3; 8; 4; 6; 2080; 2912; D_{14}; D_{10}; B = 2; QPRI]

[figure: 16. (3); 17. (3); 18. (3); 8; 4; 8; 4; 6; 2080; 2912; D_{14}; D_{10}; B = 2; QPRI]

[figure: 19. (1); 9; 3; 10; 4; 6; 2080; 2912; D_{14}; D_{10}; B = 2; QPRI]

[figure: 20. (3); 13; 5; 14; 1; 6; 2080; 7280; D_{14}; 2^{2}; B = 2]

[figure: 21. (3); 13; 5; 14; 1; 6; 2080; 7280; D_{14}; 4; B = 2]

[figure: 22. (1); 12; 6; 12; 1; 6; 2080; 7280; D_{14}; 4; B = 2]

[figure: 23. (1); 10; 5; 10; 7; 1; 2080; 520; D_{14}; 2^{3}:7; B = 7; (2T)_{1}; QPRI]

[figure: 24. (3); 5; 3; 5; 12; 12; 560; 560; 13:4; 13:4; B = 4; RPRI]

[figure: 25. (1); 5; 3; 6; 12; 4; 560; 1456; 13:4; 5:4; B = 4; RPRI]

[figure: 26. (3); 6; 4; 6; 12; 4; 560; 1456; 13:4; 5:4; B = 4; RPRI]

[figure: 27. (3); 8; 5; 8; 1; 12; 560; 3640; 13:4; 4 × 2; B = 4; QPRI]

[figure: 28. (3); 8; 5; 8; 4; 4; 1456; 1456; 5:4; 5:4; B = 4; (2T)_{1}; RPRI]

[figure: 29. (3); 14; 7; 14; 1; 4; 1456; 3640; 5:4; 4 × 2; B = 4; (2T)_{1}; QPRI]

A.2.2 Rank 3 geometries

For the rank 3 case, we use the fact that all the rank 2 geometries of the subgroups of Sz(8) can easily be constructed using Magma.

Theorem A.2.2 Up to isomorphism, there are 151 firm, residually connected geometries of rank 3 on which Sz(8) acts flag-transitively and residually weakly primitively. Their diagrams are given below.

Sketch of the proof: We do not give a complete proof here since it is quite lenghty. The idea consists in looking at the rank 2 geometries of the maximal subgroups of Sz(8) which are respectively 2^{3+3} : 7, 13 : 4, 5 : 4, D_{14} ≅ 7 : 2 (see picture of the subgroup pattern at the end of the appendix). It is easy to see that if we take 2^{3+3} : 7 as one of the subgroups, say G_{0}, forming the geometry, we are not able to construct a geometry satisfying the Rwpri condition. If we take 13 : 4 (resp. 5 : 4) as G_{0}, then there must be one of the G_{0i} isomorphic to 13 : 2 ≅ D_{26} (resp. 5 : 2 ≅ D_{10}). By looking at the subgroup pattern we readily see that there is no possibility to find a subgroup G_{i} that does not contradict the Rwpri condition. Thus the only maximal subgroup that can be taken as G_{0} is 7 : 2 ≅ D_{14}. And the geometries of rank two of this group are known (see [9]). We then start from these geometries to construct configurations satisfying the Rwpri condition. And then, by using Magma, we test the firmness, the residual connectedness and the flag-transitivity on our configurations. The diagrams are computed with Magma while they could be computed by hand. □

[figure: 1. to; 5. (3); 7; 5; 5; 1; 1; 1; 2912; 2912; 2080; D_{10}; D_{10}; D_{14}; B = 1; (2T)_{1}; QPRI; 1-4: Aut(Γ, G) = Cor(Γ, G) = Sz(8); 5: Aut(Γ, G) = Sz(8) and Cor(Γ, G) = Sz(8) × 2]

[figure: 6. to; 19. (3); 7; 5; 7; 1; 1; 1; 2080; 2912; 2080; D_{14}; D_{10}; D_{14}; B = 1; (2T)_{1}; QPRI; 6-18: Aut(Γ, G) = Cor(Γ, G) = Sz(8); 19: Aut(Γ, G) = Sz(8) and Cor(Γ, G) = Sz(8) × 2]

[figure: 20. to; 45. (3); 7; 5; 13; 1; 1; 1; 1120; 2912; 2080; D_{26}; D_{10}; D_{14}; B = 1; (2T)_{1}; QPRI; 20-45: Aut(Γ, G) = Sz(8)]

[figure: 46. to; 58. (3); 59. (1); 7; 7; 7; 1; 1; 1; 2080; 2080; 2080; D_{14}; D_{14}; D_{14}; B = 1; (2T)_{1}; RPRI; 46-56: Aut(Γ, G) = Cor(Γ, G) = Sz(8); 57-58: Aut(Γ, G) = Sz(8) and Cor(Γ, G) = Sz(8) × 2; 59: Aut(Γ, G) = Sz(8) and Cor(Γ, G) = Sz(8) : 3]

[figure: 60. to; 101. (3); 7; 7; 13; 1; 1; 1; 1120; 2080; 2080; D_{26}; D_{14}; D_{14}; B = 1; (2T)_{1}; QPRI; 60-98: Aut(Γ, G) = Cor(Γ, G) = Sz(8); 99-101: Aut(Γ, G) = Sz(8) and Cor(Γ, G) = Sz(8) × 2]

[figure: 102. to; 142. (3); 7; 13; 13; 1; 1; 1; 1120; 1120; 2080; D_{26}; D_{26}; D_{14}; B = 1; (2T)_{1}; QPRI; 102-139: Aut(Γ, G) = Cor(Γ, G) = Sz(8); 140-142: Aut(Γ, G) = Sz(8) and Cor(Γ, G) = Sz(8) × 2]

[figure: 143. (3); 7; ⊂; 1; 1; 6; 520; 2080; 2080; 2^{3} : 7; D_{14}; D_{14}; B = 1; QPRI; Aut(Γ, G) = Sz(8)]

[figure: 144. (1); 5; ⊂; 1; 1; 6; 520; 2080; 2912; 2^{3} : 7; D_{14}; D_{10}; B = 1; QPRI; Aut(Γ, G) = Sz(8) : 3]

[figure: 145. (3); 13; ⊂; 1; 1; 6; 520; 2080; 1120; 2^{3} : 7; D_{14}; D_{26}; B = 1; QPRI; Aut(Γ, G) = Sz(8)]

[figure: 146. (3); 7; 7; 1; 1; 1; 2080; 7280; 2080; D_{14}; 2^{2}; D_{14}; B = 1; (2T)_{1}; Aut(Γ, G) = Cor(Γ, G) = Sz(8)]

[figure: 147. (3); 5; 7; 1; 1; 1; 2080; 7280; 2912; D_{14}; 2^{2}; D_{10}; B = 1; (2T)_{1}; Aut(Γ, G) = Sz(8)]

[figure: 148. (3); 149. (3); 150. (3); 13; 7; 1; 1; 1; 2080; 7280; 1120; D_{14}; 2^{2}; D_{26}; B = 1; (2T)_{1}; 148-150: Aut(Γ, G) = Sz(8)]

[figure: 151. (3); 7; ⊂; ⊃; 1; 1; 6; 520; 520; 2080; 2^{3}:7; 2^{3}:7; D_{14}; B = 1; QPRI; Aut(Γ, G) = Sz(8) and Cor(Γ, G) = Sz(8) × 2]

A.2.3 Rank ≥ 4 geometries

Theorem A.2.3 There is no geometry of rank ≥ 4 on which Sz(8) acts residually weakly primitively.

Proof: The subgroup G_{0} forming a geometry must be maximal in Sz(8). Because 13:4, 5:4 and D_{14} do not have geometries of rank > 2, they cannot be taken as G_{0}. So the only candidate as G_{0} is 2^{3+3}:7. It has two maximal subgroups which are 2^{3+3} and 2^{3}:7. But the only subgroup of Sz(8) containing either one of these 102 APPENDIX A. THE RWPRI GEOMETRIES OF SZ(8) two subgroups is G_{0}. So it is not possible to construct a geometry of rank > 1 with this subgroup. □

A.2. THE GEOMETRIES OF THE SUZUKI GROUP SZ(8) 103 104 APPENDIX A. THE RWPRI GEOMETRIES OF SZ(8)

No. Rank 3 geometries containing it as a truncation  Aut(Γ, G) Cor(Γ, G)
1  10,11,18,48,52,54,55,57(2),58(2),61,63,68,69,74,81,84,86,146 Sz(8)  Sz(8)
2  13-15,54,55,56(2),85,94  Sz(8)  Sz(8) × 2
3  6,46-50,59(3),60,62,64,72,73,75,80,82,83,87,89  Sz(8)  Sz(8)
4  7,17,46,47,50(2),51,52(2),53,65,66,71,76,78,79,91,98,143  Sz(8)  Sz(8)
5  8,12,47,48,49(2),51,53,55,56,77,88,90,92,93,95,97  Sz(8)  Sz(8)
6  46,51,67  Sz(8) : 3 Sz(8) : 3 × 2
7  9,16,19,53,54,57,58,70,96,99,100,101  Sz(8)  Sz(8) × 2
8  20-27,60,62,63(2),64-71,99(2),102,104-106,  Sz(8)

107(2),108-113,140(2),150

9 35-38,68,72,73(2),74(2),75-79,81,100(2),102,106,108,114, Sz(8)

116(2),117-126,141(2),148

10 39-41,61,67,75,77-79,80(2),81,84,85,90,95,97,98,101(2),110, Sz(8)

113,115,121,122,126,127,130,133,135,136,138,139,149

11 28-34,65,69,71,72,76,83,87,92,94,95,96(2),97,98,103,105,  Sz(8)

109,111,119,123,128,131,134-136,137(2),138,139,142(2),145

12 42-44,60,64,66,70,82(2),83-85,86(2),87,88,89(2),90-93,  Sz(8)

104,112,114,118,124,125,127-131,132(2),133,134

13 45,61,62,88,91,93,94,103,115,117,120,129  Sz(8) : 3
14 4(2),5(2),11,17,18(2),19(2),22,25,31,147  Sz(8)
15 6(2),7,8,21,24,33,35,38,39,42,43  Sz(8)
16 1,2(2),3,14(2),15,16,20,26,32,36  Sz(8)
17 7,8,10,12,15,16,28,29,40,41,45  Sz(8)
18 3,9(2),10-13,23,27,30,37,44  Sz(8)
19 1,13,17,34,144  Sz(8) : 3
20 4,146(2),147-150  Sz(8)
21  Sz(8)
22  Sz(8) : 3
23 143-145,151(2)  Sz(8) : 3
24 Sz(8) Sz(8) × 2
25  Sz(8) : 3
26  Sz(8)
27  Sz(8)
28 Sz(8) Sz(8) × 2
29  Sz(8)

Table A.1: The rank 2 geometries seen as truncations of some rank 3 geometries.

Appendix B

Thin geometries for Sz(8)

This appendix is taken from [29].

B.1 Introduction

An observation that arises while classifying all firm, residually connected geometries on which a group G acts flag-transitively and fulfils some primitivity condition (as for example Pri, Rpri or Rwpri) is that when we get thin geometries, we generally get lots of them. The classification of all the firm, residually connected geometries on which Sz(8) acts flag-transitively and residually weakly primitively (Rwpri) (see Appendix A) gave us a lot of thin geometries (147 out of the 151 rank 3 ones). The classification of all the thin geometries of Sz(8) might help us to guess what is happening in the general case.

This work has first been done ”by hand”, that is by looking at the dihedral subgroups of Sz(8) and seeing which of these subgroups could give rise to thin geometries, using Magma [12]. During a stay at the University of Sydney, we made a series of Magma programs that classify all thin residually connected geometries on which a group G acts flag-transitively. This permitted us to crosscheck the results. And this is also the reason why we prefer to mention our results as a fact instead of a theorem, because no proof is given for the classification itself.

The appendix is organized as follows. In section 2, we list all the thin residually connected geometries on which Sz(8) acts flag-transitively. We also give their correlation group and their rank 2 truncations. Some of these geometries can be derived from others by using a construction described in [31]. We mention when such relations occur. In section 3, we give some observations we made while looking at the results. We define a new diagram with more structure and we introduce a new property that seems interesting to impose. In section 4, we apply the Neumaier construction to obtain other geometries from those given in section 3. We mention whether they are chiral or regular. The geometries mentioned in this section can be seen as abstract polytopes which are a particular case of thin geometries. Much work has already been accomplished on abstract regular polytopes. The spherical case has been well detailed by Coxeter [15]. The toroidal case has been studied by McMullen and Shulte in a series of papers (see [35] for references on the subject). Chiral polytopes have been studied recently by Shulte and Weiss [38, 39] and Nostrand and Schulte [35].

This appendix is also a prelude to a project quite similar to [9], [7], that is, to build a new Atlas of thin geometries [24].

Electronic Availability: A Magma file, that contains the maximal parabolic subgroups of the 183 thin geometries given in this paper, is available at the following address.

http://cso.ulb.ac.be/∼dleemans/abstracts/sz8thin.html

B.2 The thin geometries of Sz(8)

Fact B.2.1 Up to isomorphism, there are 183 thin residually connected geometries on which the group Sz(8) acts flag-transitively.

Table B.1 lists, up to isomorphism, the thin geometries on which Sz(8) acts flagtransitively. The columns a, b, and c correspond to those a, b and c appearing in the diagram given in figure B.1. The column ”#CC” gives the number of conjugacy classes of geometries. These conjugacy classes are fused under Aut(G). The correlation groups are given in the column Cor(Γ, G). We do not mention the automorphism groups because Aut(Γ, G) = G for every thin geometry Γ. We mention also when a geometry satisfies the Rwpri condition. If this condition is satisfied, the corresponding geometry was already mentioned in Appendix A.

B.2. THE THIN GEOMETRIES OF SZ(8) 107

Nr.  a b c #CC Cor(Γ, G) Rwpri
1  2 5 7  3  Yes
2  2 5 13 3  No
3  2 7 7  3  Sz(8)  Yes
4-6  2 7 13 3  Yes
7  2 13 13 3  Sz(8)  No
8  5 5 5  1  Sz(8):3  No
9-12 5 5 7  3  Sz(8)  Yes
13  5 5 7  3  Sz(8)×2  Yes
14-17 5 5 13 3  Sz(8)  No
18  5 5 13 3  Sz(8)×2  No
19-31 5 7 7  3  Sz(8)  Yes
32  5 7 7  3  Sz(8)×2  Yes
33-58 5 7 13 3  Yes
59-71 5 13 13 3  Sz(8)  No
72  5 13 13 3  Sz(8)×2  No
73-83 7 7 7  3  Sz(8)  Yes
84-85 7 7 7  3  Sz(8)×2  Yes
86  7 7 7  1  Sz(8):3  Yes
87-125 7 7 13 3  Sz(8)  Yes
126-128 7 7 13 3  Sz(8)×2  Yes
129-166 7 13 13 3  Sz(8)  Yes
167-169 7 13 13 3  Sz(8)×2  Yes
170-179 13 13 13 3  Sz(8)  No
180-181 13 13 13 3  Sz(8)×2  No
182-183 13 13 13 1  Sz(8):3  No

Table B.1: The thin geometries of Sz(8)

[figure: c; a; b; 1; 1; 1; \frac{14560}{b}; \frac{14560}{a}; \frac{14560}{c}; D_{2b}; D_{2a}; D_{2c}; B = 1]

Figure B.1: The diagrams of the rank 3 thin geometries

Γ  Γ’
1  13 (replacing 2 by 5) and 32 (replacing 2 by 7)
2  18 (replacing 2 by 5) and 72 (replacing 2 by 13)
3  84 and 85 (replacing 2 by one of the two 7)
4-6 126-128 (replacing 2 by 7) and 167-169 (replacing 2 by 13)
7  180 and 181 (replacing 2 by one of the two 13)

Table B.2: The derived geometries

In [31], a concept of derived geometry is defined. It would be too long to recall that definition here. Roughly speaking, the construction is applyable to geometries having points and pairs of points as some of their varieties. If these geometries satisfy some additional conditions, we can replace the pairs of points by a copy of the points to obtain a new geometry. This process goes in the opposite way of the Neumaier construction (see section B.4) which is a linearization process. Table B.2 gives for each geometry having one of a, b or c = 2, the number of the derived geometry in the sense of [31].

If we decide to group geometries with their derived geometries and just count the ”primitive” ones, then we have 176 geometries instead of 183.

Table B.3 lists the rank 2 geometries appearing as truncations of the thin geometries. The first column gives a number to each of them. The next two columns give

Nr. G_{0} G_{1} d_{01} g_{01} d_{10}
1  D_{4} D_{10} 16 7 15
2  D_{4} D_{14} 14 5 13
3  D_{4} D_{26} 10 5 10
4-5 D_{10} D_{10} 9 4 10
6  10 5 10
7 D_{10} D_{14} 8 2 8
8  8 3 8
9-11  8 4 8
12  10 3 9
13 D_{10} D_{26} 6 3 6
14  8 2 8
15  8 3 8
16  8 4 8
17 D_{14} D_{14} 7 2 7
18-22  7 3 7
23  7 4 7
24-26 D_{14} D_{26} 6 2 6
27-29  6 3 6
30-31 D_{26} D_{26} 6 2 6
32  6 3 6

Table B.3: Rank 2 geometries of Sz(8) appearing as truncations of the thin geometries

the structure of the maximal parabolic subgroups. And the last three columns give the diameters and the gonality of these geometries.

Table B.4 gives, for each of the 183 thin geometries, its rank 2 truncations. The first column contains the number of the geometry that is analysed. The next three columns give the three truncations Γ^{ij} = Γ(G; G_{i}, G_{j}). Sometimes, two truncations that are non-isomorphic have the same parameters. For these, when they appear as truncations in table B.4, we precise their number. For example, geometry 4 of table B.4 has a truncation Γ^{12} isomorphic to geometry number 25 of table B.3. A ∗ at the end of the information given in one of these three columns means that the corresponding truncation is isomorphic to the dual of the geometry given in table B.3. For example, geometry 3 of table B.4 has a truncation Γ^{12} which is isomorphic to the dual of geometry 17 appearing in table B.3.

B.3 Some observations

We give in this section some observations arising from the tables given in the preceding section.

First, let us remark that we get only 15 distinct diagrams for 183 non-isomorphic geometries. It might be a good idea to add some information on our diagrams. We now decide to add three parameters on each edge of the diagram, that are the three parameters of the rank 2 truncation obtained by the two vertices of an edge.

Definition B.3.1 The extended diagram of a firm, residually connected, flagtransitive geometry is a diagram (as defined in section 2), whose structure is completed by adding between every pair of vertices three parameters that are the diameters and the gonality of the corresponding rank 2 truncation.

For example, geometry number 9 has an extended diagram as follows.

[figure: 7; 5; 5; 9; 4; 10; 9; 3; 10; 8; 4; 8; 1; 1; 1; 2912; 2912; 2080; D_{10}; D_{10}; D_{14}; B = 1]

Looking at the truncations of table B.4, we obtain 75 distinct extended diagrams.

It is of course not sufficient to look at the truncations in order to say whether two geometries are isomorphic or not. If two geometries have distinct rank 2 B.3. SOME OBSERVATIONS 111

\begin{matrix}Nr. & Γ^{01} & Γ^{02} & Γ^{12} \\ 1 & 16 − 7 − 15 & 14 − 5 − 13 & 8 − 2 − 8 \\ 2 & 16 − 7 − 15 & 10 − 5 − 10 & 8 − 2 − 8 \\ 3 & 14 − 5 − 13 & 14 − 5 − 13 & 7 − 2 − 7∗ \\ 4 & 14 − 5 − 13 & 10 − 5 − 10 & 6 − 2 − 6^{(25)} \\ 5 & 14 − 5 − 13 & 10 − 5 − 10 & 6 − 2 − 6^{(26)} \\ 6 & 14 − 5 − 13 & 10 − 5 − 10 & 6 − 2 − 6^{(24)} \\ 7 & 10 − 5 − 10 & 10 − 5 − 10 & 6 − 2 − 6^{(31)} \\ 8 & 9 − 4 − 10^{(4)}∗ & 9 − 4 − 10^{(4)} & 9 − 4 − 10^{(4)}∗ \\ 9 & 9 − 4 − 10^{(4)}∗ & 8 − 4 − 8^{(9)} & 10 − 3 − 9 \\ 10 & 9 − 4 − 10^{(4)}∗ & 8 − 4 − 8^{(9)} & 8 − 4 − 8^{(9)} \\ 11 & 9 − 4 − 10^{(4)}∗ & 8 − 4 − 8^{(9)} & 8 − 4 − 8^{(11)} \\ 12 & 9 − 4 − 10^{(4)}∗ & 8 − 4 − 8^{(10)} & 8 − 2 − 8 \\ 13 & 10 − 5 − 10 & 8 − 2 − 8 & 8 − 2 − 8 \\ 14 & 9 − 4 − 10^{(4)} & 6 − 3 − 6 & 8 − 2 − 8 \\ 15 & 10 − 5 − 10 & 6 − 3 − 6 & 6 − 3 − 6 \\ 16 & 9 − 4 − 10^{(4)}∗ & 6 − 3 − 6 & 8 − 2 − 8 \\ 17 & 9 − 4 − 10^{(4)} & 8 − 2 − 8 & 8 − 2 − 8 \\ 18 & 10 − 5 − 10 & 8 − 2 − 8 & 8 − 2 − 8 \\ 19 & 8 − 3 − 8 & 8 − 3 − 8 & 7 − 3 − 7^{(19)} \\ 20 & 8 − 4 − 8^{(10)} & 8 − 3 − 8 & 7 − 3 − 7^{(20)}∗ \\ 21 & 8 − 4 − 8^{(10)} & 8 − 3 − 8 & 7 − 3 − 7^{(21)} \\ 22 & 8 − 4 − 8^{(11)} & 8 − 4 − 8^{(11)} & 7 − 4 − 7 \\ 23 & 8 − 4 − 8^{(10)} & 8 − 4 − 8^{(11)} & 7 − 2 − 7∗ \\ 24 & 8 − 2 − 8 & 8 − 4 − 8^{(11)} & 7 − 2 − 7 \\ 25 & 8 − 4 − 8^{(10)} & 8 − 4 − 8^{(11)} & 7 − 3 − 7^{(21)} \\ 26 & 10 − 3 − 9 & 8 − 4 − 8^{(11)} & 7 − 3 − 7^{(18)} \\ 27 & 8 − 4 − 8^{(9)} & 8 − 4 − 8^{(9)} & 7 − 3 − 7^{(18)} \\ 28 & 8 − 4 − 8^{(10)} & 8 − 4 − 8^{(9)} & 7 − 3 − 7^{(18)} \\ 29 & 8 − 4 − 8^{(10)} & 8 − 4 − 8^{(9)} & 7 − 4 − 7 \\ 30 & 10 − 3 − 9 & 8 − 2 − 8 & 7 − 3 − 7^{(20)} \\ 31 & 8 − 2 − 8 & 8 − 2 − 8 & 7 − 2 − 7 \\ 32 & 8 − 2 − 8 & 8 − 2 − 8 & 7 − 4 − 7 \end{matrix}

Table B.4: The truncations of the thin geometries

\begin{matrix}Nr. & Γ^{01} & Γ^{02} & Γ^{12} \\ 33 & 8 − 4 − 8^{(9)} & 8 − 2 − 8 & 6 − 2 − 6^{(24)} \\ 34 & 8 − 3 − 8 & 6 − 3 − 6 & 6 − 2 − 6^{(24)} \\ 35 & 8 − 2 − 8 & 6 − 3 − 6 & 6 − 2 − 6^{(24)} \\ 36 & 8 − 4 − 8^{(11)} & 8 − 2 − 8 & 6 − 2 − 6^{(24)} \\ 37 & 8 − 3 − 8 & 8 − 2 − 8 & 6 − 2 − 6^{(24)} \\ 38 & 8 − 2 − 8 & 8 − 2 − 8 & 6 − 2 − 6^{(24)} \\ 39 & 8 − 4 − 8^{(9)} & 8 − 2 − 8 & 6 − 2 − 6^{(24)} \\ 40 & 8 − 4 − 8^{(11)} & 8 − 3 − 8 & 6 − 2 − 6^{(24)} \\ 41 & 8 − 4 − 8^{(10)} & 8 − 2 − 8 & 6 − 3 − 6^{(27)} \\ 42 & 8 − 4 − 8^{(10)} & 8 − 3 − 8 & 6 − 3 − 6^{(27)} \\ 43 & 8 − 4 − 8^{(11)} & 6 − 3 − 6 & 6 − 3 − 6^{(27)} \\ 44 & 8 − 2 − 8 & 6 − 3 − 6 & 6 − 3 − 6^{(27)} \\ 45 & 8 − 4 − 8^{(9)} & 6 − 3 − 6 & 6 − 3 − 6^{(27)} \\ 46 & 8 − 3 − 8 & 6 − 3 − 6 & 6 − 3 − 6^{(27)} \\ 47 & 10 − 3 − 9 & 8 − 2 − 8 & 6 − 3 − 6^{(27)} \\ 48 & 8 − 3 − 8 & 6 − 3 − 6 & 6 − 2 − 6^{(25)} \\ 49 & 8 − 4 − 8^{(9)} & 8 − 2 − 8 & 6 − 2 − 6^{(25)} \\ 50 & 8 − 4 − 8^{(11)} & 6 − 3 − 6 & 6 − 2 − 6^{(25)} \\ 51 & 8 − 3 − 8 & 8 − 4 − 8 & 6 − 2 − 6^{(25)} \\ 52 & 8 − 3 − 8 & 8 − 3 − 8 & 6 − 2 − 6^{(26)} \\ 53 & 8 − 4 − 8^{(10)} & 6 − 3 − 6 & 6 − 2 − 6^{(26)} \\ 54 & 8 − 4 − 8^{(10)} & 6 − 3 − 6 & 6 − 2 − 6^{(26)} \\ 55 & 8 − 3 − 8 & 8 − 3 − 8 & 6 − 3 − 6^{(28)} \\ 56 & 8 − 3 − 8 & 8 − 3 − 8 & 6 − 3 − 6^{(28)} \\ 57 & 8 − 4 − 8^{(11)} & 8 − 3 − 8 & 6 − 3 − 6^{(28)} \\ 58 & 8 − 4 − 8^{(10)} & 8 − 3 − 8 & 6 − 3 − 6^{(29)} \\ 59 & 6 − 3 − 6 & 8 − 3 − 8 & 6 − 2 − 6^{(31)} \\ 60 & 8 − 2 − 8 & 6 − 3 − 6 & 6 − 2 − 6^{(31)} \\ 61 & 8 − 2 − 8 & 8 − 2 − 8 & 6 − 2 − 6^{(31)} \\ 62 & 6 − 3 − 6 & 8 − 2 − 8 & 6 − 2 − 6^{(31)} \\ 63 & 8 − 3 − 8 & 8 − 2 − 8 & 6 − 2 − 6^{(31)} \\ 64 & 6 − 3 − 6 & 8 − 2 − 8 & 6 − 2 − 6^{(31)} \end{matrix}

\begin{matrix}Nr. & Γ^{01} & Γ^{02} & Γ^{12} \\ 65 & 8 − 2 − 8 & 8 − 4 − 8 & 6 − 2 − 6^{(31)} \\ 66 & 6 − 3 − 6 & 8 − 2 − 8 & 6 − 2 − 6^{(31)} \\ 67 & 6 − 3 − 6 & 6 − 3 − 6 & 6 − 2 − 6^{(31)} \\ 68 & 8 − 4 − 8 & 8 − 3 − 8 & 6 − 2 − 6^{(31)} \\ 69 & 6 − 3 − 6 & 8 − 3 − 8 & 6 − 2 − 6^{(31)} \\ 70 & 8 − 4 − 8 & 6 − 3 − 6 & 6 − 3 − 6 \\ 71 & 6 − 3 − 6 & 8 − 3 − 8 & 6 − 3 − 6 \\ 72 & 8 − 2 − 8 & 8 − 2 − 8 & 6 − 3 − 6 \\ 73 & 7 − 3 − 7^{(22)} & 7 − 3 − 7^{(19)}∗ & 7 − 3 − 7^{(20)} \\ 74 & 7 − 3 − 7^{(20)} & 7 − 3 − 7^{(19)}∗ & 7 − 3 − 7^{(21)} \\ 75 & 7 − 3 − 7^{(21)} & 7 − 3 − 7^{(19)}∗ & 7 − 2 − 7∗ \\ 76 & 7 − 3 − 7^{(21)}∗ & 7 − 3 − 7^{(19)}∗ & 7 − 3 − 7^{(21)} \\ 77 & 7 − 3 − 7^{(20)}∗ & 7 − 3 − 7^{(19)}∗ & 7 − 3 − 7^{(20)}∗ \\ 78 & 7 − 3 − 7^{(22)} & 7 − 3 − 7^{(20)}∗ & 7 − 3 − 7^{(21)} \\ 79 & 7 − 3 − 7^{(20)}∗ & 7 − 2 − 7∗ & 7 − 3 − 7^{(20)} \\ 80 & 7 − 3 − 7^{(21)}∗ & 7 − 4 − 7 & 7 − 3 − 7^{(20)} \\ 81 & 7 − 2 − 7∗ & 7 − 4 − 7 & 7 − 3 − 7^{(18)} \\ 82 & 7 − 3 − 7^{(21)} & 7 − 3 − 7^{(18)} & 7 − 2 − 7 \\ 83 & 7 − 3 − 7^{(21)}∗ & 7 − 3 − 7^{(18)} & 7 − 3 − 7^{(18)} \\ 84 & 7 − 2 − 7 & 7 − 4 − 7 & 7 − 2 − 7∗ \\ 85 & 7 − 2 − 7∗ & 7 − 4 − 7 & 7 − 2 − 7 \\ 86 & 7 − 3 − 7^{(19)} & 7 − 3 − 7^{(19)}∗ & 7 − 3 − 7^{(19)} \\ 87 & 7 − 3 − 7^{(19)}∗ & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(24)} \\ 88 & 7 − 2 − 7∗ & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(29)} \\ 89 & 7 − 3 − 7^{(19)} & 6 − 3 − 6^{(29)} & 6 − 2 − 6^{(24)} \\ 90 & 7 − 3 − 7^{(21)}∗ & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(24)} \\ 91 & 7 − 3 − 7^{(19)} & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(24)} \\ 92 & 7 − 3 − 7^{(20)} & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(24)} \\ 93 & 7 − 3 − 7^{(20)}∗ & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(24)} \\ 94 & 7 − 3 − 7^{(22)} & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(24)} \\ 95 & 7 − 2 − 7∗ & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(24)} \\ 96 & 7 − 2 − 7∗ & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(24)} \end{matrix}

\begin{matrix}Nr. & Γ^{01} & Γ^{02} & Γ^{12} \\ 97 & 7 − 4 − 7 & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(24)} \\ 98 & 7 − 3 − 7^{(20)}∗ & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(24)} \\ 99 & 7 − 3 − 7^{(19)}∗ & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(25)} \\ 100 & 7 − 3 − 7^{(19)}∗ & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(25)} \\ 101 & 7 − 2 − 7∗ & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(25)} \\ 102 & 7 − 3 − 7^{(19)}∗ & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(25)} \\ 103 & 7 − 3 − 7^{(20)} & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(25)} \\ 104 & 7 − 3 − 7^{(21)} & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(25)} \\ 105 & 7 − 3 − 7^{(20)}∗ & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(25)} \\ 106 & 7 − 3 − 7^{(20)} & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(25)} \\ 107 & 7 − 3 − 7^{(19)} & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(26)} \\ 108 & 7 − 2 − 7∗ & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(25)} \\ 109 & 7 − 3 − 7^{(19)}∗ & 6 − 3 − 6^{(28)} & 6 − 3 − 6^{(28)} \\ 110 & 7 − 3 − 7^{(19)} & 6 − 3 − 6^{(27)} & 6 − 3 − 6^{(28)} \\ 111 & 7 − 2 − 7∗ & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(28)} \\ 112 & 7 − 3 − 7^{(18)} & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(28)} \\ 113 & 7 − 2 − 7 & 6 − 3 − 6^{(28)} & 6 − 3 − 6^{(28)} \\ 114 & 7 − 3 − 7^{(19)}∗ & 6 − 3 − 6^{(27)} & 6 − 3 − 6^{(28)} \\ 115 & 7 − 3 − 7^{(21)} & 6 − 3 − 6^{(29)} & 6 − 3 − 6^{(28)} \\ 116 & 7 − 3 − 7^{(19)} & 6 − 3 − 6^{(28)} & 6 − 3 − 6^{(28)} \\ 117 & 7 − 3 − 7^{(21)} & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(28)} \\ 118 & 7 − 3 − 7^{(20)}∗ & 6 − 3 − 6^{(29)} & 6 − 3 − 6^{(28)} \\ 119 & 7 − 3 − 7^{(21)} & 6 − 3 − 6^{(27)} & 6 − 3 − 6^{(28)} \\ 120 & 7 − 3 − 7^{(21)} & 6 − 3 − 6^{(29)} & 6 − 3 − 6^{(28)} \\ 121 & 7 − 3 − 7^{(18)} & 6 − 3 − 6^{(29)} & 6 − 3 − 6^{(27)} \\ 122 & 7 − 3 − 7^{(21)} & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(27)} \\ 123 & 7 − 4 − 7 & 6 − 3 − 6^{(27)} & 6 − 3 − 6^{(27)} \\ 124 & 7 − 3 − 7^{(21)}∗ & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(27)} \\ 125 & 7 − 3 − 7^{(20)}∗ & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(27)} \\ 126 & 7 − 4 − 7 & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(24)} \\ 127 & 7 − 4 − 7 & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(25)} \\ 128 & 7 − 4 − 7 & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(26)} \end{matrix}

\begin{matrix}Nr. & Γ^{01} & Γ^{02} & Γ^{12} \\ 129 & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(31)} \\ 130 & 6 − 3 − 6^{(29)} & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(30)} \\ 131 & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(30)} \\ 132 & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(31)} \\ 133 & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(30)} \\ 134 & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(30)} \\ 135 & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(31)} \\ 136 & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(31)} \\ 137 & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(31)} \\ 138 & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(24)} & 6 − 3 − 6 \\ 139 & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(31)} \\ 140 & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(31)} \\ 141 & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(30)} \\ 142 & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(29)} & 6 − 2 − 6^{(31)} \\ 143 & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(31)} \\ 144 & 6 − 3 − 6^{(29)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(30)} \\ 145 & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(25)} & 6 − 3 − 6 \\ 146 & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(30)} \\ 147 & 6 − 3 − 6^{(29)} & 6 − 2 − 6^{(25)} & 6 − 3 − 6 \\ 148 & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(30)} \\ 149 & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(31)} \\ 150 & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(25)} & 6 − 3 − 6 \\ 151 & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(30)} \\ 152 & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(30)} \\ 153 & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(30)} \\ 154 & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(31)} \\ 155 & 6 − 3 − 6^{(27)} & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(31)} \\ 156 & 6 − 3 − 6^{(29)} & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(31)} \\ 157 & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(28)} & 6 − 3 − 6 \\ 158 & 6 − 3 − 6^{(27)} & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(31)} \\ 159 & 6 − 3 − 6^{(28)} & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(31)} \\ 160 & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(28)} & 6 − 2 − 6^{(30)} \end{matrix}

\begin{matrix}Nr. & Γ^{01} & Γ^{02} & Γ^{12} \\ 161 & 6 − 3 − 6^{(27)} & 6 − 3 − 6^{(28)} & 6 − 3 − 6 \\ 162 & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(31)} \\ 163 & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(31)} \\ 164 & 6 − 3 − 6^{(27)} & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(30)} \\ 165 & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(31)} \\ 166 & 6 − 2 − 6^{(26)} & 6 − 3 − 6^{(27)} & 6 − 2 − 6^{(30)} \\ 167 & 6 − 2 − 6^{(24)} & 6 − 2 − 6^{(24)} & 6 − 3 − 6 \\ 168 & 6 − 2 − 6^{(25)} & 6 − 2 − 6^{(25)} & 6 − 3 − 6 \\ 169 & 6 − 2 − 6^{(26)} & 6 − 2 − 6^{(26)} & 6 − 3 − 6 \\ 170 & 6 − 2 − 6^{(30)} & 6 − 2 − 6^{(31)} & 6 − 3 − 6 \\ 171 & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(30)} \\ 172 & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} \\ 173 & 6 − 3 − 6 & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(30)} \\ 174 & 6 − 3 − 6 & 6 − 2 − 6^{(31)} & 6 − 3 − 6 \\ 175 & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} & 6 − 3 − 6 \\ 176 & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} \\ 177 & 6 − 2 − 6^{(30)} & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} \\ 178 & 6 − 2 − 6^{(30)} & 6 − 2 − 6^{(31)} & 6 − 3 − 6 \\ 179 & 6 − 3 − 6 & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} \\ 180 & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} & 6 − 3 − 6 \\ 181 & 6 − 3 − 6 & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(31)} \\ 182 & 6 − 2 − 6^{(30)} & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(30)} \\ 183 & 6 − 2 − 6^{(30)} & 6 − 2 − 6^{(31)} & 6 − 2 − 6^{(30)} \end{matrix}

truncations, they are non-isomorphic. But the converse is not necessarily true. There are several examples appearing here. The geometries having three rank 2 truncations in common are (33,39), (53,54), (55,56), (59,69), (62,64,66), (84,85), (105,106), (109,116), (115,120), (129,135), (132,136), (137,140), (141,151,152), (148,153), (155,158), (162,163,165).

If we observe table B.4, we see that a lot of truncations do not satisfy the (IP)_{2} condition. A new condition to impose on our geometries is the following.

Condition B.3.1 Γ satisfies the strong intersection property of rank 2 (SIP)_{2} if Γ is (IP)_{2} and for every edge of the diagram of Γ, the corresponding truncation satisfies the (IP)_{2} condition.

Remark that when a geometry has a rank 2 residue which is a generalized digon, the rank 2 truncation obtained by the two vertices that are not joined by an edge does not satisfy (IP)_{2} or is a generalized digon. This is why we restrict ourselves to truncations corresponding to edges of the diagram. Out of the 183 thin geometries of Sz(8), only 50 are (SIP)_{2}. This new condition seems attractive because the intersection property (IP) implies it for every geometry of finite rank.

Theorem B.3.1 Let Γ be a geometry of rank n < ∞. If Γ satisfies (IP), then Γ satisfies (SIP)_{2}.

Proof: Let I be the set of types of Γ. Suppose Γ is not (SIP)_{2}. To show that Γ cannot be (IP), we suppose it is (IP) and we derive a contradiction. Because Γ is not (SIP)_{2}, there exist two types 0 and 1 in I such that the residue of a flag of type I \ {0, 1} has gonality at least 3 and the truncation Γ^{{0,1}} has a circuit of length 4. Hence there are two 1-elements, say e and e′, such that | σ_{0}(e) ∩ σ_{0}(e′) |≥ 2. Because Γ satisfies (IP), it must have a flag F such that e ∗ F ∗ e′ and σ_{0}(F) = σ_{0}(e) ∩ σ_{0}(e′). Thus σ_{0}(F) has at least two 0-elements. It implies that F is a flag of type t(F) ⊆ I \ {0, 1}. If t(F) = I \ {0, 1}, then the geometry Γ_{F} is not (IP)_{2} and thus Γ cannot be (IP), a contradiction. Otherwise, the geometry Γ_{F}, whose rank is smaller than the rank of Γ, satisfies (IP) and does not satisfy (SIP)_{2}. We may then suppose Γ = Γ_{F} and start the discussion again until we get the contradiction. □

In a first version of this paper, we stated theorem B.3.1 for flag-transitive geometries. Antonio Pasini showed us that it was not necessary to impose the flag-transitivity in the hypotheses of this theorem.

Because buildings satisfy (IP), it might be interesting to impose this new property. Also, theorem B.3.1 shows that (SIP)_{2} is a necessary condition for (IP). This is very helpful because (IP) is not easy to test with a computer whereas (SIP)_{2} is. For example, the second rank 4 geometry mentionned in [23] for the Janko group J_{1} cannot satisfy (IP) because one of its rank 2 truncations, namely the one with P SL(2, 11) and S_{3} × D_{10} is not (IP)_{2} and hence this geometry is not (SIP)_{2}.

B.4 Applying the Neumaier construction

This construction is detailed in [34]. It would be too long to recall it here. We just give a rough idea of how it works. We start from a diagram as the one given below.

[figure: b; a; a; 1; 1; 1; D_{2a}; D_{2a}; D_{2b}; B = 1]

Applying the Neumaier construction to a geometry corresponding to the diagram given above, we obtain another geometry whose diagram is given below, for the group Sz(8)×2. This group acts chirally or regularly on the geometry, depending on the correlation group of the starting geometry. Roughly speaking, if the starting geometry has a duality involving the two D_{2a}, then Sz(8)×2 acts regularly on the new geometry. Otherwise, it acts chirally. Remark that the starting diagram was a triangle whereas the diagram of the geometry obtained by the Neumaier construction is linear. This construction may be seen as a linearisation process of the diagram of a geometry.

Nr.  a b c a’ b’ Chiral or Regular
3  2 7 7 4 7  Chiral
7  2 13 13 4 13  Chiral
8  5 5 5 3 5  Chiral
9-12  5 5 7 5 14  Chiral
13  5 5 7 5 14  Regular
14-17 5 5 13 5 26  Chiral
18 5 5 13 5 26 Regular
19-31 5 7 7 7 10  Chiral
32 5 7 7 7 10 Regular
59-71 5 13 13 10 13  Chiral
72 5 13 13 10 13 Regular
73-83 7 7 7 7 14  Chiral
84-85 7 7 7 7 14  Regular
86 7 7 7 3 7 Chiral
87-125 7 7 13 7 26  Chiral
126-128 7 7 13 7 26  Regular
129-166 7 13 13 13 14  Chiral
167-169 7 13 13 13 14  Regular
170-179 13 13 13 13 26  Chiral
180-181 13 13 13 13 26  Regular
182-183 13 13 13 3 13  Chiral

Table B.5: The geometries obtained by the Neumaier construction

[figure: a = a’; 2b = b’; 1; 1; 1; \frac{29120}{2b}; 14560; \frac{29120}{a}; D_{4b}; 2^{2}; D_{2a}; B = 1]

In table B.5 we give for each thin geometry obtained in section B.2 the geometry obtained by applying this construction (when the construction is applyable) and we mention whether the group Sz(8)×2 acts chirally or regularly.

120 APPENDIX B. THIN GEOMETRIES FOR SZ(8) Remark that, according to Francis Buekenhout, Jacques Tits arrived at the same construction independently.

Bibliography

[1] M. Aschbacher. Flag structures on Tits geometries. Geom. Dedicata, 14:21– 31, 1983.

[2] W. Bosma, J. Cannon, and C. Playoust. The Magma Algebra System I: the user language. J. Symbolic Comput., (3/4):235–265, 1997.

[3] F. Buekenhout. Diagrams for geometries and groups. J. Combin. Theory Ser. A, 27:121–151, 1979.

[4] F. Buekenhout. (g, d, d*)-gons. In Johnson N.L., Kallaher M.J., and Long C.T., editors, Finite Geometries, pages 93–102, Marcel Dekker, New York, 1983.

[5] F. Buekenhout, editor. Handbook of Incidence Geometry. Elsevier, Amsterdam, 1995.

[6] F. Buekenhout. About the history of the Suzuki Groups. Preprint, 1996.

[7] F. Buekenhout, P. Cara, and M. Dehon. Geometries of small almost simple groups based on maximal subgroups. Bull. Belg. Math. Soc. - Simon Stevin Suppl., 1998.

[8] F. Buekenhout, M. Dehon, and D. Leemans. All geometries of the Mathieu group M_{11} based on maximal subgroups. Experiment. Math., 5:101–110, 1996.

[9] F. Buekenhout, M. Dehon, and D. Leemans. An Atlas of residually weakly primitive geometries for small groups. Mém. Acad. Royale Belg., Classe des Sciences, 1996. To appear.

[10] J. Cannon. Cayley: A language for group theory. Department of Pure Mathematics, University of Sydney, December 1982.

[11] J. Cannon and W. Bosma. Cayley: quick reference guide. Department of Pure Mathematics, University of Sydney, 1991.

[12] J. Cannon and W. Bosma. Handbook of Magma functions. Department of Pure Mathematics, University of Sydney, November 1994.

[13] J. Cannon and C. Playoust. An introduction to Magma. Department of Pure Mathematics, University of Sydney, 1993.

[14] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson. Atlas of Finite Groups. Oxford U.P., 1985.

[15] H.S.M. Coxeter. Regular Polytopes. 3rd. Ed. Dover, New York, 1973.

[16] H.S.M. Coxeter and W.O.J. Moser. Generators and relations for discrete groups. Springer-Verlag, second edition, 1965.

[17] M. Dehon. Classifying geometries with Cayley. J. Symbolic Comput., 17:259– 276, 1994.

[18] M. Dehon and X. Miller. The residually weakly primitive and (IP)_{2} geometries of M_{11}. In preparation.

[19] M. Dehon and X. Miller. The residually weakly primitive and (IP)_{2} geometries of U(4,2). In preparation.

[20] H. Gottschalk. A classification of geometries associated with PSL(3,4). Diplomarbeit, Giessen, 1995.

[21] H. Gottschalk. A connection between six distinguished diagrams. Atti Sem. Mat. Fis. Univ. Modena, 1996.

[22] H. Gottschalk. Rank three geometries associated with PSL(3,4). Bull. Belg. Math. Soc. Simon Stevin, 3:147–160, 1996.

[23] H. Gottschalk and D. Leemans. The residually weakly primitive geometries of the Janko group J_{1}. In A. Pasini et al., editor, Groups and Geometries, pages 65–79. Birkhäuser, 1998.

[24] D. Leemans. An atlas of regular thin geometries for small groups. Preprint, [25] D. Leemans. The residually weakly primitive geometries of the dihedral

groups. Preprint, 1997.

[26] D. Leemans. The rank 2 geometries of the simple Suzuki groups Sz(q). Beiträge Algebra Geom., 39(1):97–120, 1998.

[27] D. Leemans. The rank 3 geometries of the simple Suzuki groups Sz(q). Preprint, 1998.

[28] D. Leemans. The residually weakly primitive pre-geometries of the simple Suzuki groups Sz(q). Preprint, 1998.

[29] D. Leemans. Thin geometries for the Suzuki simple group Sz(8). In F. De Clerck et al., editor, Proc. Third Int. Conf. on Finite Geometries and Combinatorics, volume 5, pages 373–387. Bull. Belg. Math. Soc. - Simon Stevin, 1998.

[30] D. Leemans. The residually weakly primitive geometries of the Suzuki simple group Sz(8). In Campbell C.M., Robertson E.F, Ruskuc N., and Smith G.C., editors, Proceedings of Groups St Andrews 1997 in Bath. CUP, To appear.

[31] C. Lefèvre-Percsy, N. Percsy, and D. Leemans. New geometries for finite groups and polytopes. Preprint, 1998.

[32] H. Lüneburg. Die Suzukigruppen und ihre Geometrien. Lecture Notes in Math., Vol.10, Springer, Berlin, 1965.

[33] H. Lüneburg. Translation Planes. Springer-Verlag, New York, 1980.

[34] A. Neumaier. Rectagraphs, diagrams, and Suzuki’s sporadic simple group. In E. Mendelsohn, editor, Algebraic and Geometric Combinatorics, volume 15 of Ann. Discrete Math., pages 305–318, 1982.

[35] B. Nostrand and E. Schulte. Chiral polytopes from hyperbolic honeycombs. Discrete Comput. Geom., 13:17–39, 1995.

[36] K. Rosenbaum. Die Untergruppen der Diedergruppe. Math. Semesterber., 42:133–137, 1995.

[37] P. Samuel. Théorie algébrique des nombres. Hermann, Paris, second edition, [38] E. Schulte and A.I.Weiss. Chiral polytopes. In Applied Geometry and Dis-

crete Mathematics. The Victor Klee Festschrift. Editors: P. Gritzmann and B. Sturmfels, DIMACS Series, AMS-ACM, 4:493–516, 1991.

[39] E. Schulte and A.I.Weiss. Chirality and projective linear groups. Discrete Math., 131:221–261, 1994.

[40] M. Suzuki. On a class of doubly transitive groups. Ann. of Math., 75:105– 145, 1962.

[41] J. Tits. Géométries polyédriques et groupes simples. Atti 2a Riunione Groupem. Math. Express. Lat. Firenze, pages 66–88, 1962.

[42] J. Tits. Ovoïdes et groupes de Suzuki. Arch. Math., 13:187–198, 1962.