Tag - Finite simple groups
Let G be a finite simple group and S a normal subset of G. If |G| is large enough in terms of |S|/|G|, can we deduce that every element of G can be expressed as x y-1 for x and y elements of S? Shalev, Tiep, and I have proven that this is true assuming G is an alternating group or a group of Lie type in bounded rank, but the question remains open for classical groups of high rank over small fields. I will say something about the methods of proof, which involve both character methods and geometric ideas and also say something about the more general question of covering G by ST where S and T are large normal subsets.
In this talk we will discuss the structure of maximal subgroups of finite simple groups, particularly groups of Lie type. We will discuss subgroups of exceptional groups of Lie type, and a version of Ennola duality that exists for groups of Lie type, which relates untwisted and twisted groups of Lie type.
Zoltan Halasi: Babai’s conjecture for classical groups with generating sets containing transvections
A well-known conjecture of Babai states that if G is any finite simple group and X is a generating set of G, then the diameter of the Cayley graph Cay(G, X) is bounded by a polylogarithmic function of |G|. The goal of the talk is to sketch a proof of such a bound in the case that X contains a transvection.
Given the current knowledge of complex representations of finite simple groups, obtaining good upper bounds for their characters values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We first discuss some such applications. Then we will report on recent results that produce such character bounds.
Babai's conjecture asserts that the diameter of the Cayley graph of any finite simple group G is bounded by (log |G|)O(1). This conjecture has been resolved for groups of bounded rank, but for groups of unbounded rank such as SLn(2) it is wide open. Even for random generators, only the case of alternating groups is resolved. In this talk we sketch the proof of Babai's conjecture for SLn(p), p = O(1), with at least three random generators. The proof extends to other classical groups over 𝔽q if we have at least q100 random generators. The heart of the proof consists of showing that the Schreier graph of SLn(q) acting on 𝔽qn with respect to q100 random generators is an expander graph.
A subset X of a group G invariably generates G if we are free to replace each element of X by an arbitrary conjugate, and we must always obtain a generating set of G. This concept was introduced by Dixon in the early nineties with motivations from computational Galois theory. We will review these motivations and their intimate connections with permutation groups. We will then present some new results concerning the probability of generating invariably a finite simple group. For instance, we will see that two random elements of a finite simple group of Lie type of bounded rank invariably generate with probability bounded away from zero.
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