Simple groups, generation and probabilistic methods

TL;DR

This survey explores the generation properties of finite simple groups, emphasizing probabilistic methods, random generation, and subgroup generation, highlighting recent advances and applications in group theory.

Contribution

It provides a comprehensive overview of generation properties of simple groups, emphasizing probabilistic techniques and recent results on subgroup generation.

Findings

Finite simple groups are typically 2-generated.

Probabilistic methods are effective in proving generation properties.

Recent work links subgroup generation to primitive permutation groups.

Abstract

It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the extraordinary generation properties of simple groups, focussing on topics such as random generation, $(a,b)$-generation and spread, as well as highlighting the application of probabilistic methods in the proofs of many of the main results. We also present some recent work on the minimal generation of maximal and second maximal subgroups of simple groups, which has applications to the study of subgroup growth and the generation of primitive permutation groups.