Finite groups with large Chebotarev invariant

Abstract

A subset {g₁, …, g_d} of a finite group G is said to invariably generate G if the set ${\rm{\{ }}g_1^{{x_1}}{\rm{,}} \ldots {\rm{,}}g_d^{{x_d}}{\rm{\}}}$ generates G for every choice of x_i ∈ 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. The authors recently showed that for each ϵ > 0, there exists a constant c_ϵ such that $C(G) \le (1 + \epsilon)\sqrt {{\rm{|}}G{\rm{|}}} + {c_{\epsilon}}$. This bound is asymptotically best possible. In this paper we prove a partial converse: namely, for each α > 0 there exists an absolute constant δ_α such that if G is a finite group and $C(G) > \alpha \sqrt {{\rm{|}}G{\rm{|}}}$, then G has a section X/Y such that ${\rm{|}}X/Y{\rm{|}} \ge {\delta _\alpha}\sqrt {{\rm{|}}G{\rm{|}}}$, and $X/Y \cong {\mathbb{F}_q}\rtimes H$ for some prime power q, with $H \le \mathbb{F}_q^\times$.