An upper bound on the Chebotarev invariant of a finite group

Abstract

A subset {g1,.., gd} of a finite group G invariably generates {g1x1,..,gdxd} generates G for every choice of xi ∈ 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)≤β|G| 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 cε such that C(G)≤(1+∈)|G|+c

Untitled

“…As we said in the introduction, in [23] we improved Theorem 1, proving that for each > 0 there exists a constant c such that C(G) ≤ (1 + ) |G| + c . As we noticed in that paper (see in particular [23,Proposition 8]), the proof of Lemma 14 implies that the difference α U = C(G) − C(G/U ) is "small" if U is nonabelian and can be bounded in terms of δ, q, n, m, and |H| when U is abelian. This gives some hints on the structure of the finite groups G with C(G) ∼ |G|.…”

“…In this paper we don't give any kind of estimation for the constant β appearing in the statement of Theorem 1. More recently, in a joint paper with Gareth Tracey [23], we proved that the methods and results introduced in this paper can be employed to show that for each > 0 there exists a constant c such that C(G) ≤ (1+ ) |G|+c .…”

On the probability of generating invariably a finite simple group

“…Confirming a conjecture of Kowalski and Zywina [KZ12], Lucchini [Luc18] proved that, for every finite group G, picking O(|G| 1/2 ) random elements is sufficient, on average, in order to generate G invariably. The implied constant was estimated by Lucchini-Tracey [LT17].…”

Finite groups with large Chebotarev invariant

“…It was also conjectured in [12] that these are the "worst" cases: that is, that C(G) = O( |G|) as |G| → ∞. The conjecture was proved by the first author in [15], and was later improved in [17] where it is shown that for each ǫ > 0, there exists a constant c ǫ such that C(G) ≤ (1 + ǫ) |G| + c ǫ . Furthermore, one has C(G) ≤ 5 3 |G| when G is soluble.…”

“…[17, Proposition 8 and the Proof of Theorem 1] Let G be a finite group with trivial Frattini subgroup, and let U , V and R…”