On the probability of generating invariably a finite simple group

Abstract

Let G be a finite simple group. We look for small subsets A of G with the property that, if y ∈ G is chosen uniformly at random, then with good 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 of one of these results, 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.

Normal $2$-coverings of the finite simple groups and their generalizations

“…The invariably generation of groups has tight connection with groups admitting Beauville structures and with the spread of a group. It is an interesting invariant associated to a group that has recently received some attention [42,43,44] and its investigation requires some deep results, like the Fulman-Guralnick solution [35,36,37,38] of the Boston-Shalev [7] conjecture on derangements in finite simple groups.…”