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2018 The expected number of elements to generate a finite group with $d$-generated Sylow subgroups
Andrea Lucchini, Mariapia Moscatiello
Rocky Mountain J. Math. 48(6): 1963-1982 (2018). DOI: 10.1216/RMJ-2018-48-6-1963

Abstract

Given a finite group $G,$ let $e(G)$ be the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa $, where $\kappa $ is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, $\kappa $ equals 2.752394. If $G$ is a permutation group of degree $n,$ then either $G={Sym} (3)$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor +\kappa ^*$ with $\kappa ^* \sim 1.606695.$ These results improve weaker bounds recently obtained by Lucchini.

Citation

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Andrea Lucchini. Mariapia Moscatiello. "The expected number of elements to generate a finite group with $d$-generated Sylow subgroups." Rocky Mountain J. Math. 48 (6) 1963 - 1982, 2018. https://doi.org/10.1216/RMJ-2018-48-6-1963

Information

Published: 2018
First available in Project Euclid: 24 November 2018

zbMATH: 06987235
MathSciNet: MR3879312
Digital Object Identifier: 10.1216/RMJ-2018-48-6-1963

Subjects:
Primary: 20P05

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 6 • 2018
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