An upper bound on the Chebotarev invariant of a finite group

A subset {g1,.., gd} of a finite group G invariably generates {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 first author recently showed that C(G)≤β|G| for some absolute constant β. In this paper we show that, when G is soluble, then β is at most 5/3. We also show that this is best possible. Furthermore, we show that, in general, for each ε > 0 there exists a constant cε such that C(G)≤(1+∈)|G|+c∈