Finite groups with large Chebotarev invariant

A subset {g1, …, gd} of a finite group G is said to invariably generate G if the set {g1x1,…,gdxd} generates G for every choice of xi ∈ 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 cϵ such that C(G)≤(1+ϵ)|G|+cϵ. 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)>α|G|, then G has a section X/Y such that |X/Y|≥δα|G|, and X/ Y≅ Fq⋊ H for some prime power q, with H≤Fq×.