On the converse of Hall's theorem

TL;DR

This paper explores the conditions under which the existence of certain Hall subgroups in a finite group implies the group is p-soluble, providing detailed characterizations and extending Hall's theorem.

Contribution

It offers new characterizations of finite groups with specific Hall subgroups, particularly relating to the converse of Hall's theorem under various assumptions.

Findings

If p ≠ 3 and G has a Hall {p,q}-subgroup for every q ≠ p, then G is p-soluble.

Provides detailed conditions under which the existence of Hall subgroups implies group solubility.

Extends the classical Hall's theorem by exploring its converse in finite groups.

Abstract

In this paper, we mainly investigate the converse of a well-known theorem proved by P. Hall, and present detailed characterizations under the various assumptions of the existence of some families of Hall subgroups. In particular, we prove that if $p\neq 3$ and a finite group $G$ has a Hall $\{p,q\}$-subgroup for every prime $q\neq p$, then $G$ is $p$-soluble.