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

A subset $\{g_1,\dots ,g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1},\dots ,g_d^{x_d}\}$ 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 $C(G)\le \beta \sqrt{\mathrm{\mid }G\mathrm{\mid }}$ for all finite groups $G\mathrm{.}$