Aner Shalev

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.

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

Lie methods in the theory of pro-p groups, A. Shalev on the classification of prop-p groups and finite p-groups, C.R. Leedham-Green, S. McKay prop-p trees and applications, L. Ribes, P. Zalesskii just infinite branch groups, R. I. Grigorchuk on just infinite abstract and profinite groups, J.S. Wilson the Nottingham group, R. Camina on groups satisfying the Golod-Shafarevich condition, E. Zelmanov sub-group growth in prop-p groups, A. Mann zeta functions of groups, M. du Sautoy, D. Segal where the wild things are - ramification groups and the Nottingham group, M. du Sautoy, I. Fesenko p-adic Galois representations and prop-p Galois groups, N. Boston the cohomology of p-adic analytic groups, P. Symonds, T. Weigel. Appendix: further problems.

Diameters of finite simple groups: sharp bounds and applications

For a finite group

New horizons in pro-p groups

H

Character degrees and random walks in finite groups of Lie type

, let

COMMUTATOR MAPS, MEASURE PRESERVATION, AND T-SYSTEMS

Irr(H)

HAUSDORFF DIMENSION, PRO-P GROUPS, AND KAC-MOODY ALGEBRAS

denote the set of irreducible characters of