On the probability of generating invariably a finite simple group

TL;DR

This paper investigates the probability that small subsets of finite simple groups can, with high likelihood, invariable generate the entire group when combined with a randomly chosen element, revealing new probabilistic generation properties.

Contribution

It establishes new bounds and conditions for invariable generation of finite simple groups, leveraging recent solutions to the Boston--Shalev conjecture and Weyl group structures.

Findings

Two random elements of a finite simple Lie type group of bounded rank invariable generate with positive probability.

Existence of small subsets that, with high probability, invariable generate the group when combined with a random element.

Connections between invariable generation properties and Weyl group structures.

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 $y \in G$ 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.