On the probability of generating invariably a finite simple group

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