An upper bound on the Chebotarev invariant of a finite group

A subset {g ₁,..., g _d } of a finite group G invariably generates (left{ {g_1^{{x_1}}, ldots ,g_d^{{x_d}}} right}) 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 first author recently showed that (Cleft( G right) leqslant beta sqrt {left| G right|} ) 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 c _ε such that (Cleft( G right) leqslant left( {1 + in } right)sqrt {left| G right|} + {c_ in }).