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Seminario: “Finite groups with large Chebotarev invariant”

Venerdì 8 Marzo 2019, ore 11:30 - Aula 1AD100 - Gareth Tarcey

ARGOMENTI: Seminars

Venerdì 8 Marzo 2019 alle ore 11:30 in Aula 1AD100, Gareth Tarcey (University of Bath) terrà un seminario dal titolo “Finite groups with large Chebotarev invariant”.

Abstract
A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ is said to invariably generate $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ 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 $G$. We recently showed that for each $\epsilon>0$, there exists a constant $c_{\epsilon}$ such that $C(G)\le (1+\epsilon)\sqrt{|G|}+c_{\epsilon}$. This bound is asymptotically best possible. We can prove a partial converse: namely, for each $\alpha>0$ there exists an absolute constant $\delta_{\alpha}$ such that if $G$ is a finite group and $C(G)>\alpha\sqrt{|G|}$, then $G$ has a section $X/Y$ such that $|X/Y|\geq \delta_{\alpha}\sqrt{|G|}$, and $X/Y\cong \mathbb{F}_q\rtimes H$ for some prime power $q$, with $H\le \mathbb{F}_q^{\times}$.