Israel Journal of Mathematics

, Volume 219, Issue 1, pp 449–467 | Cite as

An upper bound on the Chebotarev invariant of a finite group

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Abstract

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 \(C\left( G \right) \leqslant \beta \sqrt {\left| G \right|} \) 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 \(C\left( G \right) \leqslant \left( {1 + \in } \right)\sqrt {\left| G \right|} + {c_ \in }\).

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Università degli Studi di PadovaDipartimento di MatematicaPadovaItaly
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

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