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

TL;DR

The authors extend Hall's solvability criterion by permitting maximal subgroups to be nilpotent in addition to having index $p$ or $p^2$, proving that such a group $G$ is solvable. They also show a related criterion: if every proper non-maximal subgroup of $G$ lies inside a subgroup of index $p$ or $p^2$, then $G$ is solvable. A key technical achievement is the classification of nonabelian simple groups with a squared-prime index subgroup, enabling a reduction to solvable groups and informing the limits of possible generalizations. The work blends elementary group-theoretic arguments with the Glauberman–Thompson $p$-nilpotency theory, Guralnick's simple-group classifications, and structural analyses of minimal normal subgroups to establish the solvability results.

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.

Theorems & Definitions (9)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Theorem A
• proof : Proof of Theorem C
• proof : Proof of Theorem B
• Remark 3.1