THE CHEBOTAREV INVARIANT OF A FINITE GROUP: A CONJECTURE OF KOWALSKI AND ZYWINA

A subset {g(1), . . . ,g(d)} of a finite group G invariably generates G if {g(1)(1)(x), . . .,g(d)(xd)} generates G for every choice of x(i )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. Confirming a conjecture of Kowalski and Zywina, we prove that there exists an absolute constant beta such that C(G) <= beta root vertical bar G vertical bar for all finite groups G.