On the probability of generating invariably a finite simple group

Abstract

Let G be a finite simple group. In this paper we consider the existence of small subsets A of G with the property that, if y∈G is chosen uniformly at random, then with high probability y invariably generates G together with some element of A. We prove various results in this direction, both positive and negative. As a corollary, we prove that two randomly chosen elements of a finite simple group of Lie type of bounded rank invariably generate with probability bounded away from zero. Our method is based on the positive solution of the Boston–Shalev conjecture by Fulman and Guralnick, as well as on certain connections between the properties of invariable generation of a group of Lie type and the structure of its Weyl group.