The Chebotarev invariant for direct products of nonabelian finite simple groups

TL;DR

This paper studies the Chebotarev invariant for direct products of nonabelian finite simple groups, showing it is bounded or grows logarithmically with the number of factors, and provides bounds on the expected number of generators.

Contribution

It establishes bounds on the Chebotarev invariant for products of nonabelian finite simple groups, extending understanding of their generation properties.

Findings

C(G) is bounded for nonabelian finite simple groups.

C(G) grows logarithmically with the number of simple factors.

Sharp bounds are provided for the expected number of generators.

Abstract

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. In this paper, we show that if $G$ is a nonabelian finite simple group, then $C(G)$ is absolutely bounded. More generally, we show that if $G$ is a direct product of $k$ nonabelian finite simple groups, then $C(G)=\log{k}/\log{\alpha(G)}+O(1)$, where $\alpha$ is an invariant completely determined by the proportion of derangements of the primitive permutation actions of the factors in $G$. It follows from the proof of the Boston-Shalev conjecture that $C(G)=O(\log{k})$. We also derive sharp bounds on the expected number of generators for $G$.