Bounds on the number of maximal subgroups of finite groups

TL;DR

This paper establishes bounds on the number of maximal subgroups of finite groups, which in turn inform the probabilistic generation of such groups by random elements.

Contribution

It provides new bounds on the number of maximal subgroups of finite groups and applies these to improve estimates for random generation.

Findings

Bounds on the number of maximal subgroups of finite groups

Improved estimates for the number of generators needed

Enhanced understanding of group generation probabilities

Abstract

In this paper we obtain significant bounds for the number of maximal subgroups of a given index of a finite group. These results allow us to give new bounds for the number of random generators needed to generate a finite $d$-generated group with high probability.