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, Ann. Math., 183:769--814, 2011) 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.


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