The number of maximal subgroups and probabilistic generation of‎ ‎finite groups

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{۱۸۳} (۲۰۱۱) ۷۶۹--۸۱۴) 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.

Keywords:

‎Finite group‎ , ‎maximal subgroup‎ , ‎probabilistic generation‎ , ‎primitive group