University of Isfahan International Journal of Group Theory 2251-7650 9 1 2020 03 01 The number of maximal subgroups and probabilistic generation of‎ ‎finite groups 31 42 23260 10.22108/ijgt.2019.114469.1521 EN 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 Journal Article 2018 12 10 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. https://ijgt.ui.ac.ir/article_23260_a1e34939c5076bd89068d8b7b5d04b45.pdf