This paper is a survey of some results by the author in the study of the subgroup structure of the finite simple groups of Lie type. Throughout the paper G is a simple algebraic group; if G is denned over a finite field Fq then σ is some Steinberg endomorphism of G. We shall omit index G in notations like NG(X), CG(X).

Subgroups of simple groups of exceptional type.

The first result of the paper is a reduction theorem for the maximal subgroups of finite exceptional groups similar to the well-known result of M. Aschbacher for finite classical groups. In the case of classical groups Theorem 1 doesn't give any new information, but it may be useful in the study of simple groups of exceptional type. Another and more explicit version of a reduction theorem was obtained recently by M. W. Liebeck and G. M. Seitz.

Theorem 1Let G be defined over the finite field Fq, Gσ ≅ G(Fq) and. Let G0 ≤ G1 ≤ Aut G0and let M be a subgroup in G1. Then one of the following statements is valid:

1. (a) for some proper connected nontrivial σ-invariant subgroup H ≤ G;

2. (b) M is an almost simple group, i.e. S ≤ M ≤ Aut S for some simple group S;

3. (c) M ≤ NG(J) for some Jordan subgroup J in G;

4. (d) G is of type E8, charFq = p > 5 and M ≤ NGl(X), where X ≅ Alt5 × Alt6.

Introduction

McLaughlin's sporadic simple group McL was originally constructed (see) as a permutation group on 275 letters. It is a simple group of order 898128000 = 27.36.53.7.11. It is now known to be the pointwise stabilizer of a 2-dimensional sublattice in the Leech lattice. Its maximal subgroups were found by Finkelstein (see). The modular character tables for the relevant primes p = 2, 7 and 11 were found by Thackray. The 5-modular character tables were found by Hiss, Lux and Parker, up to a few ambiguities (see). These ambiguities together with others in the values of the 5- modular characters 560 and 3038 of the automorphism group of McL, denoted by McL.2, were resolved by Suleiman (see). The main purpose of this paper is to complete the 3-modular character table of McL and to find the 3-modular character table of McL.2.

The 3-modular character table of McL

In this section we are going to complete what has been done by R. Parker on the 3-modular characters of McL. To do so we have to work out again most of the 3-modular characters using the techniques of the ‘Meat-Axe’ which is the main tool in our work. We then use the method of ‘condensation’ (see) to complete the 3-modular character table of McL.

The central characters modulo 3 give the block distribution of the ordinary irreducible characters. There are three blocks of defect zero. These blocks are B1 = {5103}, B2 = {8019a} and B3 = {8019b}. Hence, 5103, 8019a and 8019b are three 3-modular irreducible characters in McL.

Introduction.

The first theorem given here asserts that a geometric hyperplane H of a near hexagon, which intersects each quad at a star must be a generalized hexagon. The second theorem tells us that if a finite near hexagon with parameters possesses such a geometric hyperplane, then that near hexagon Γ must be the dual of a rank 3 polar space Δ. Moreover, there is a bijection H ↔ quads of Γ, which induces an embedding of the hexagon H into the polar space A which is an epimorphism on points. Conceivably, there is a possibility that generalized hexagons might be represented as geometric hyperplanes of some of the “other” dual polar spaces, such as Ω(n, ℝ) (with signature (n – 3,3)), Sp(6, κ), Ω−(8, κ), U(6, κ) or U(7, κ). But the final theorem shows that if Γ is finite, such possibilities cannot happen; that in fact Γ (and Δ) are type Ω(7, q) (or Sp(6, q) if q is even) and H is the hexagon of type G2(q) associated with the standard embedding of G2(q) (either as the stabilizer of an appropriate hyperplane in the 8-dimensional spin module for Ω(7, q) or as the stabilizer of a trilinear form in its natural 7-dimensional module – or the factor of this 7-space module by a 1-dimensional radical when q is even).

The author thanks Professor J. Tan for a valuable discussion, Queen Mary College, U. of London, and the Mathematisches Institute, Albert-Ludwigs Universität Freiburg for their kind hospitality during the writing of this work, and the Alexander von Humboldt Stiftung whose support made the research possible.

INTRODUCTION

A conjugacy class D of 3-transpositions in the group G is a class of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. Such groups were introduced and studied by Bernd Fischer who classified all finite 3-transposition groups with no nontrivial normal, solvable subgroups. His work was of great importance in the classification of finite simple groups.

The basic example of a class of 3-transpositions is the class of transpositions in any symmetric group. This was the only class which Fischer originally considered, but Roger Carter pointed out that examples could be found in several of the classical groups as well. The transvections of symplectic groups over GF(2) form a class of 3-transpositions, so additionally any subgroup of the symplectic group generated by a class of transvections is also a 3-transposition group. The symmetric groups arise in this way as do the orthogonal groups over GF(2). Symplectic transvections over GF(2) are special cases of unitary transvections over GF(4), and this unitary class is still a class of 3-transpositions. The final classical examples are given by the reflection classes of orthogonal groups over GF(3).

Introduction

In, the “bimonster” (the wreathed square of the Fischer-Griess monster group) was studied in terms of its representation as a quotient of a certain infinite Coxeter group. Here we shall use the representation of this Coxeter group as a hyperbolic reflection group to investigate both the bimonster and its subgroup 3Fi24.

Throughout the paper, we shall use the notation of for group structures. In section 1, we give a simple axiomatic definition of a group G, and deduce that G is generated by 16 involutions that satisfy the Coxeter relations of Figure 1. This allows us to represent them in Section 2 by reflections in certain vectors of a hyperbolic space (that is, a space with a Lorentzian metric).

This notation makes it easy to perform calculations with these elements. In Section 2, we shall find some relations that must hold in G, but are not consequences of the Coxeter relations, and will use these to establish many identities in G, which we express in terms of alias groups.

Our section 4 contains a short proof of the 26 node theorem of.

The remainder of the paper is devoted to the subgroup Y552 of G, which we shall show has the structure 3Fi24. By way of introduction, Section 5 is used to show that the smaller group Y551 has structure, by completely enumerating its root vectors. In Section 6, we describe the root vectors for 3Fi24, and compute the corresponding alias groups.

A certain element ωis defined in Section 7, and shown to generate a normal subgroup of order 3 in Y552.

Abstract

Introduction

Recently the author published a paper which showed how progress had been made towards proving that Y555 (which we redefine below) is a presentation for the wreath square of the Fischer-Griess Monster (which we call the Bimonster) and outlined a possible method of completing the proof. Since then the proof has indeed been completed, but by a different method: results announced by A. Ivanov at the 1990 Durham Conference, proved by showing the simple connectedness of a certain simplicial complex, meant that a slight strengthening of the results of was sufficient to complete the proof. This was achieved during the conference, and it therefore seems appropriate to publish it here in the conference proceedings.

We also take the opportunity to present proofs of two other results needed for which no full published version currently exists.

Summary of

We start by recalling some of the notation, terminology and (without proof) results of. Note that the numbering of the theorems has been changed. References contain many other useful results about subgroups of Y555.

We recall that a Coxeter group is generated by involutions corresponding to the nodes of a (Coxeter) diagram. The product of two generators has order 2 or 3 according as the corresponding nodes are unjoined or joined by a single unlabelled edge. (Other product orders are possible and correspond to other types of join.)

INTRODUCTION

Let G be a simple, connected algebraic group over an algebraically closed field F of characteristic p ≥ 0. Let P = LQ be a parabolic subgroup of G, where Q is the unipotent radical of P and L is a Levi subgroup of P. Here L acts on Q via conjugation. This induces an L-action on consecutive subquotients of the lower central series of Q. Provided with a suitable F-vector space structure these quotients can be regarded as L-modules. They are called internal Chevalley modules for L.

There exists a unique parabolic subgroup P− of G such that P∪P− = L. We refer to P− as the opposite of P and Q− = Ru(P−) is called the opposite unipotent radical of P. The internal Chevalley modules that occur in Q− are dual to the ones in Q.

In this note we describe some results from regarding the structure of orbits for the action of L on these modules and give some information on the associated stabilizers for arbitrary characteristic.

(1.1) A motivation for this is a result of R. Richardson asserting that L has only finitely many orbits on each of its internal Chevalley modules. We show that there is a close connection between the L-orbits on Q−/(Q−)′ and (P, P)-cosets of G. For details and further information we refer to.

We say that p is a ‘very bad’ prime for G, if p occurs as a structure constant in Chevalley's commutator relations for G. In this situation there are degeneracies in these relations affecting the structure of orbits. We assume throughout these notes that p is not very bad for G.