In this paper we obtain a classification of those subgroups of the finite general linear group GLd (q) with orders divisible by a primitive prime divisor of qe − 1 for some e>12d. In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross‐characteristic representations of some of the finite classical groups. 1991 Mathematics Subject Classification: primary 20G40; secondary 20C20, 20C33, 20C34, 20E99.

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 x 1 , x 2 are nontrivial elements of G, then there exists y ∈ C such that G = x 1 , y = x 2 , 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 x 1 , x 2 , x 3 are nontrivial elements of G, then there exists y ∈ C such that G = x 1 , y = x 2 , y = x 3 , y . We also prove analogous but weaker results for almost simple groups.

FOR HELMUT WIELANDT ON HIS 90TH BIRTHDAYFor each finite simple group G there is a conjugacy class C such that each G nontrivial element of G generates G together with any of more than 1r10 of the members of C . Precise asymptotic results are obtained for the probability implicit

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We prove several results on products of conjugacy classes in finite simple groups. The first result is that for any finite non-abelian simple groups, there exists a triple of conjugate elements with product 1 which generate the group. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).

Abstract. We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third).