## Abstract

A subset {*g*_{1}, …, *g*_{d}} of a finite group *G* is said to invariably generate *G* if the set \({\rm{\{ }}g_1^{{x_1}}{\rm{,}} \ldots {\rm{,}}g_d^{{x_d}}{\rm{\}}}\) 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 authors recently showed that for each ϵ > 0, there exists a constant *c*_{ϵ} such that \(C(G) \le (1 + \epsilon)\sqrt {{\rm{|}}G{\rm{|}}} + {c_{\epsilon}}\). This bound is asymptotically best possible. In this paper we prove a partial converse: namely, for each α > 0 there exists an absolute constant δ_{α} such that if *G* is a finite group and \(C(G) > \alpha \sqrt {{\rm{|}}G{\rm{|}}}\), then *G* has a section *X*/*Y* such that \({\rm{|}}X/Y{\rm{|}} \ge {\delta _\alpha}\sqrt {{\rm{|}}G{\rm{|}}} \), and \(X/Y \cong {\mathbb{F}_q}\rtimes H\) for some prime power *q*, with \(H \le \mathbb{F}_q^\times \).

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Research partially supported by MIUR-Italy via PRIN Group theory and applications and by the EPSRC standard grant EP/P0231QX/1.

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Lucchini, A., Tracey, G. Finite groups with large Chebotarev invariant.
*Isr. J. Math.* **235**, 169–182 (2020). https://doi.org/10.1007/s11856-019-1953-8

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DOI: https://doi.org/10.1007/s11856-019-1953-8