Finite simple groups with two maximal subgroups of coprime orders

TL;DR

This paper classifies finite simple groups with two maximal subgroups of coprime orders, extending classical results by identifying possible subgroup configurations in these groups.

Contribution

It determines the possible triples (G, H, M) where G is a finite simple group and H, M are maximal subgroups with coprime orders, building on prior work by Liebeck and Saxl.

Findings

Identifies all possible configurations of (G, H, M) with coprime maximal subgroup orders in finite simple groups.

Extends classical results by providing a detailed classification of such subgroup triples.

Clarifies the structure of finite simple groups with specific maximal subgroup properties.

Abstract

In 1962, V.A. Belonogov proved that if a finite group $G$ contains two maximal subgroups of coprime orders, then either $G$ is one of known solvable groups or $G$ is simple. In this short note based on results by M. Liebeck and J. Saxl on odd order maximal subgroups in finite simple groups we determine possibilities for triples $(G,H,M)$, where $G$ is a finite nonabelian simple group, $H$ and $M$ are maximal subgroups of $G$ with $(|H|,|M|)=1$.