Finite groups whose maximal subgroups have only soluble proper subgroups

TL;DR

This paper characterizes finite groups where all maximal subgroups have only soluble proper subgroups, addressing a key question about the composition factors of such groups with soluble second maximal subgroups.

Contribution

It provides a detailed description of finite groups with maximal subgroups having only soluble proper subgroups, advancing understanding of their composition factors.

Findings

Finite groups with maximal subgroups having only soluble proper subgroups are characterized.

The results answer a longstanding question about the composition factors of groups with soluble second maximal subgroups.

Abstract

We give a description of a finite group whose maximal subgroups possess only soluble proper subgroups, which implies the answer to the well-known question on composition factors of finite groups, whose second maximal subgroups are soluble.