Martin W. Liebeck

A classification of the maximal subgroups of the finite alternating and symmetric groups

Following the classification of finite simple groups, one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups-that is, of groups X such that X0 u X< Aut X, for some finite non-abelian simple group X0. The problem has been solved for most sporadic groups and groups of Lie type of low rank (see [ 19, Sects. 3, 43 for discussion and references). This paper is a contribution to the case where X0 is an alternating group A, (so that for n # 6, X is A, or S,). The maximal subgroups of A, and S, are known for several classes of degrees n:

The probability of generating a finite simple group

We show that two random elements of a finite simple groupG generateG with probability → 1 as |G| → ∞. This settles a conjecture of Dixon.

Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras

This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.

Simple groups, permutation groups, and probability

In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in other cases the use of probability, or counting, is not entirely anticipated by the nature of the problem (see [LiSh2], [GSSh]). In this paper we study a variety of problems in finite simple groups and finite permutation groups using a unified method, which often involves probabilistic arguments. We obtain new bounds on the minimal degrees of primitive actions of classical groups, and prove the Cameron-Kantor conjecture that almost simple primitive groups have a base of bounded size, apart from various subset or subspace actions of alternating and classical groups. We use the minimal degree result to derive applications in two areas: the first is a substantial step towards the Guralnick-Thompson genus conjecture, that for a given genus g, only finitely many non-alternating simple groups can appear as a composition factor of a group of genus g (see below for definitions); and the second concerns random generation of classical groups. Our proofs are largely based on a technical result concerning the size of the intersection of a maximal subgroup of a classical group with a conjugacy class of elements of prime order. We now proceed to describe our results in detail.

Classical groups, probabilistic methods, and the

We study the probability that randomly chosen elements of prescribed type in a finite simple classical group G generate G; in particular, we prove a conjecture of Kantor and Lubotzky in this area. The probabilistic approach is then used to determine the finite simple classical quotients of the modular group PSL2(Z), up to finitely many exceptions.

(2,3)

Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph r(G, S) associated with G and S, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant c such that every element of G is a product of c involutions (and we generalize this to elements of arbitrary order). We also show that for any word w = w(xl,..., xd), there is a constant c = c(w) such that for any simple group G on which w does not vanish, every element of G is a product of c values of w. From this we deduce that every verbal subgroup of a semisimple profinite group is closed. Other applications concern covering numbers, expanders, and random walks on finite simple groups.

-generation problem

The maximal closed subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields are determined, subject to certain restrictions on the characteristic. This result is used to prove a reduction theorem for maximal subgroups of finite exceptional groups of Lie type: such a subgroup is either of known type, or it is almost simple. Finally, the “generic” almost simple maximal subgroups are determined for large characteristic.

Diameters of finite simple groups: sharp bounds and applications

For a finite group

Maximal subgroups of exceptional groups of Lie type, finite and algebraic

H

Character degrees and random walks in finite groups of Lie type

, let