The Chebotarev invariant of a finite group: a conjecture of Kowalski and Zywina

TL;DR

This paper proves that the Chebotarev invariant of any finite group is bounded above by a constant times the square root of its size, confirming a conjecture by Kowalski and Zywina.

Contribution

It establishes a universal bound on the Chebotarev invariant for all finite groups, confirming a conjecture and advancing understanding of group generation properties.

Findings

C(G) is bounded by a constant times √|G| for all finite groups

Confirmed a conjecture of Kowalski and Zywina

Provides a universal bound applicable to all finite groups

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 $G$. Confirming a conjecture of Kowalski and Zywina, we prove that there exists an absolute constant $\beta$ such that $C(G) \leq \beta\sqrt{|G|}$ for all finite groups $G.$