Large subgroups of simple groups

TL;DR

This paper classifies large maximal subgroups of finite simple groups and simple algebraic groups, and explores their role in triple factorizations, providing a comprehensive understanding of subgroup structures in these groups.

Contribution

It determines all large maximal subgroups of finite simple groups and extends the concept to simple algebraic groups, including applications to triple factorizations.

Findings

All large maximal subgroups of finite simple groups identified.

Analogous results established for simple algebraic groups.

Application to triple factorizations of simple groups discussed.

Abstract

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an analogous result for simple algebraic groups (in this context, largeness is defined in terms of dimension). An application to triple factorisations of simple groups (both finite and algebraic) is discussed.