AN UPPER BOUND ON THE CHEBOTAREV INVARIANT OF A FINITE GROUP

作者：Lucchini Andrea^*; Tracey Gareth

来源：Israel Journal of Mathematics, 2017, 219(1): 449-467.

DOI：10.1007/s11856-017-1507-x

A subset {g1,..., gd} of a finite group G invariably generates G if the set {g(1)(x1),..., g(d)(xd)} generates G for every choice of x(i) epsilon 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) <= beta root broken vertical bar G broken vertical bar for some absolute constant beta. In this paper we show that, when G is soluble, then beta is at most 5/3. We also show that this is best possible. Furthermore, we show that, in general, for each epsilon > 0 there exists a constant c(epsilon) such that C(G) <= (1 + epsilon) root broken vertical bar G broken vertical bar + c(epsilon).

• 出版日期2017-4