THE CHEBOTAREV INVARIANT FOR DIRECT PRODUCTS OF NONABELIAN FINITE SIMPLE GROUPS

A subset {g1, ... , gd} of a finite group G invariably generates G if {gx1 1 ,... , gxd d } generates G for every choice of xi is an element of 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. In this paper, we show that if G is a nonabelian finite simple group, then C(G) is absolutely bounded. More generally, we show that if G is a direct product of k nonabelian finite simple groups, then C(G) = log k/ log alpha(G) + O (1), where alpha is an invariant completely determined by the proportion of derangements of the primitive permutation actions of the factors in G. It follows from the proof of the Boston-Shalev conjecture that C(G) = O(log k). We also derive sharp bounds on the expected number of generators for G.