The Chebotarev invariant of a finite group: A conjecture of Kowalski and Zywina
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Abstract:
A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. 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 $C(G) \leq \beta \sqrt {|G|}$ for all finite groups $G.$References
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Additional Information
- Andrea Lucchini
- Affiliation: Dipartimento di Matematica, University of Padova, Via Trieste 63, 35121 Padova, Italy
- MR Author ID: 233594
- Email: lucchini@math.unipd.it
- Received by editor(s): February 19, 2016
- Received by editor(s) in revised form: November 7, 2016, and May 7, 2017
- Published electronically: August 10, 2018
- Additional Notes: The author was partially supported by Università di Padova (Progetto di Ricerca di Ateneo: “Invariable generation of groups”).
- Communicated by: Pham Huu Tiep
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4549-4562
- MSC (2010): Primary 20P05
- DOI: https://doi.org/10.1090/proc/13805
- MathSciNet review: 3856127