In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1 ̸ = x ∈ G, the probability is greater than 1/10 that G =〈x,y 〉 for a random y ∈ C. Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound 1/10 can be replaced by 13/42; and, excluding an explicitly listed set of simple groups, the bound 2/3 holds. We use these results to show that any nonabelian finite simple group G has a conjugacy class C such that, if x1, x2 are nontrivial elements of G, then there exists y ∈ C such that G =〈x1,y〉=〈x2,y〉. Similarly, aside from one infinite family and a small, explicit finite set of simple groups, G has a conjugacy class C such that, if x1, x2, x3 are nontrivial elements of G, then there exists y ∈ C such that G =〈x1,y〉= 〈x2,y〉=〈x3,y〉. We also prove analogous but weaker results for almost simple groups.

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,

Abstract. We present some highlights of the 110-year project to classify the finite simple groups.

The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups. 1.

Given a set r of permutations of an n-set, let G be the group of permutations generated by r. If p is any prime, it is known that a Sylow p-subgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow p-subgroups of G, all elements conjugating one to the other can be found (as a coset of the normalizer of one of the Sylow p-subgroups). Analogous results are obtained in the case of Hall subgroups of solvable groups.

Abstract. We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X(q) ofLie type in the natural characteristic. Our approach is to show that for sufficiently large q (usually q>9 suffices), X(q) is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.

Abstract. We investigate the proportion of fixed point free permutations (derangements) in finite transitive permutation groups. This article is the first in a series where we prove a conjecture of Shalev that the proportion of such elements is bounded away from zero for a simple finite group. In fact, there are much stronger results. This article focuses on finite Chevalley groups of bounded rank. We also discuss derangements in algebraic groups and in more general primitive groups. These results have applications in questions about probabilistic generation of finite simple groups and maps between varieties over finite fields. 1.

Let G be a finite exceptional group of Lie type acting transitively on a set Ω. For x ∈ G, the fixed point ratio of x is the proportion of elements of Ω which are fixed by x. We obtain new bounds for such fixed point ratios. When a pointstabilizer is parabolic we use character theory; and in other cases, we use results on an analogous problem for algebraic groups in Lawther, Liebeck & Seitz, 2002. These give dimension bounds on fixed point spaces of elements of exceptional algebraic groups, which we apply by passing to finite groups via a Frobenius morphism. Introduction. If G is a finite group acting transitively on a set Ω, and x ∈ G, we define the fixed point ratio of x to be the proportion of points fixed by x; that is, denoting this quantity by fpr(x, Ω),

Abstract. A complete classification is given for finite vertex-primitive and vertex-biprimitive s-transitive graphs for s ≥ 4. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group M, and an infinite family of graphs of valency 5 admitting projective symplectic groups PSp(4,p)withp prime and p ≡±1 (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved. 1.

Introduction According to a long-standing conjecture in model theory, simple groups of finite Morley rank should be algebraic. The present paper is part of a series aimed ultimately at proving the following: Conjecture 1 (Even Type Conjecture) Let G be a simple group of finite Morley rank of even type, with no infinite definable simple section of degenerate type. Then G is algebraic. An infinite simple group G of finite Morley rank is said to be of even type if its Sylow 2subgroups are of bounded exponent. It is of degenerate type if its Sylow 2-subgroups are finite. If the main conjecture is correct, then there should be no groups of degenerate type. So the flavor of the Even Type Conjecture is that the classification in the even type case reduces to an extended Feit-Thompson Theorem. Those who are skeptical about the main conjecture would expect degenerate type groups to exis