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

Lucchini, Andrea

doi:10.1090/proc/13805

The Chebotarev invariant of a finite group: A conjecture of Kowalski and Zywina
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Proc. Amer. Math. Soc. 146 (2018), 4549-4562 Request permission

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 $G.$

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