The number of maximal subgroups and probabilistic generation of‎ ‎finite groups

Document Type : Ischia Group Theory 2018


1 Departament de Matematiques‎, ‎Universitat de Valencia‎, ‎Spain

2 Departamento de Matematicas‎, ‎Universidad de Zaragoza‎, ‎Pedro Cerbuna‎, Spain


In this survey we present some significant bounds for the‎ ‎number of maximal subgroups of a given index of a finite group‎. ‎As a‎ ‎consequence‎, ‎new bounds for the number of random‎ ‎generators needed to generate a finite $d$-generated group with high‎ ‎probability which are significantly tighter than the ones obtained in‎ ‎the paper of Jaikin-Zapirain and Pyber (Random generation of finite‎ ‎and profinite groups and group enumeration‎, ‎\emph{Ann.\ Math.}‎, ‎\textbf{183} (2011) 769--814) are obtained‎. ‎The results of‎ ‎Jaikin-Zapirain and Pyber‎, ‎as well as other results of Lubotzky‎, ‎Detomi‎, ‎and Lucchini‎, ‎appear as particular cases of our theorems‎.


Main Subjects

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Volume 9, Issue 1 - Serial Number 1
Proceedings of the Ischia Group Theory 2018
March 2020
Pages 31-42
  • Receive Date: 10 December 2018
  • Revise Date: 19 January 2019
  • Accept Date: 24 January 2019
  • Published Online: 01 March 2020