Adolfo Ballester Bolinches - Departament de Matematiques‎, ‎Universitat de Valencia‎, ‎Spain
Ramón Esteban-Romero - Departament de Matematiques‎, ‎Universitat de Valencia‎, ‎Spain
Paz Jiménez-Seral - Departamento de Matematicas‎, ‎Universidad de Zaragoza‎, ‎Pedro Cerbuna‎, Spain
Hangyang Meng - Departament de Matematiques‎, ‎Universitat de Valencia‎, ‎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{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.

‎Finite group‎, ‎maximal subgroup‎, ‎probabilistic generation‎, ‎primitive group