Extensions of a theorem of P. Hall on indexes of maximal subgroups

TL;DR

This paper generalizes P. Hall's theorem by showing that a finite group remains solvable if all maximal subgroups are nilpotent or if all proper non-maximal subgroups are contained in subgroups of prime or squared-prime index.

Contribution

It extends classical results by relaxing conditions on maximal subgroups, demonstrating solvability under broader subgroup index assumptions.

Findings

Finite groups with nilpotent maximal subgroups are solvable.

Groups where all proper non-maximal subgroups lie in subgroups of prime or squared-prime index are solvable.

Generalization of P. Hall's theorem to broader subgroup conditions.

Abstract

We extend a classical theorem of P. Hall that claims that if the index of every maximal subgroup of a finite group $G$ is a prime or the square of a prime, then $G$ is solvable. Precisely, we prove that if one allows, in addition, the possibility that every maximal subgroup of $G$ is nilpotent instead of having prime or squared-prime index, then $G$ continues to be solvable. Likewise, we obtain the solvability of $G$ when we assume that every proper non-maximal subgroup of $G$ lies in some subgroup of index prime or squared prime.