Title of article :
The number of maximal subgroups and probabilistic generation of finite groups
Author/Authors :
Bolinches, Adolfo Ballester Departament de Matematiques - Universitat de Valencia - Spain , Esteban-Romero, Ramón Departament de Matematiques - Universitat de Valencia - Spain , Jiménez-Seral, Paz Departamento de Matematicas - Universidad de Zaragoza - Pedro Cerbuna - Spain , Meng, Hangyang Departament de Matematiques - Universitat de Valencia - Spain
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, /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.
Keywords :
Finite group , maximal subgroup , probabilistic generation , primitive group
Journal title :
International Journal of Group Theory
