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The Chebotarev invariant of a finite group: A conjecture of Kowalski and Zywina


Author: Andrea Lucchini
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 20P05
DOI: https://doi.org/10.1090/proc/13805
Published electronically: August 10, 2018
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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 {\vert G\vert}$ for all finite groups $ G.$


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Additional Information

Andrea Lucchini
Affiliation: Dipartimento di Matematica, University of Padova, Via Trieste 63, 35121 Padova, Italy
Email: lucchini@math.unipd.it

DOI: https://doi.org/10.1090/proc/13805
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
Article copyright: © Copyright 2018 American Mathematical Society

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