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Adolfo Ballester-Bolinches, Ramón Esteban-Romero, Paz Jiménez-Seral, Hangyang Meng.
The number of maximal subgroups and probabilistic generation of finite groups.
Int. J. Group Theory, 9(1):31–42, 2020.

doi:10.22108/ijgt.2019.114469.1521

Abstract

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, Ann. Math.‎, 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.

2020 Mathematics Subject Classification: 20P05

Keywords: finite group; maximal subgroup; probabilistic generation; primitive group