Invariable generation of finite classical groups

TL;DR

This paper investigates the probability of random elements invariably generating finite classical groups, showing a positive lower bound for four elements and a zero limit for three as parameters grow, extending known results from symmetric groups.

Contribution

It extends the concept of invariable generation probabilities from symmetric groups to finite classical groups, establishing bounds and asymptotic behaviors for different numbers of elements.

Findings

Probability that four random elements invariably generate is bounded away from zero for large q.

Probability that three random elements invariably generate tends to zero as q and r grow.

Most elements in G_r(q) are separable, aiding the analysis.

Abstract

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements invariably generate $S_n$ is bounded away from zero by an absolute constant for all $n$. Subsequently, Eberhard, Ford and Green have shown that the probability that three randomly selected elements invariably generate $S_n$ tends to zero as $n \rightarrow \infty$. In this paper, we prove an analogous result for the finite classical groups. More precisely, let $G_r(q)$ be a finite classical group of rank $r$ over $\mathbb{F}_q$. We show that for $q$ large enough, the probability that four randomly selected elements invariably generate $G_r(q)$ is bounded away from zero by an absolute constant for all $r$, and for three elements the probability tends to zero as $q \rightarrow \infty$ and $r \rightarrow \infty$. We use the fact that most elements in $G_r(q)$ are separable and the well-known correspondence between classes of maximal tori containing separable elements in classical groups and conjugacy classes in their Weyl groups.