A Criterion for the Existence of a Solvable $\pi$-Hall Subgroup in a Finite Group

TL;DR

This paper establishes a precise criterion for the existence of solvable -Hall subgroups in finite groups, linking it to the existence of -Hall subgroups for all prime pairs in .

Contribution

It provides a necessary and sufficient condition for the existence of solvable -Hall subgroups based on pairwise prime subgroup existence.

Findings

A finite group has a solvable -Hall subgroup iff it has -Hall subgroups for all prime pairs in .

The criterion simplifies checking for solvable -Hall subgroups to examining prime pair subgroups.

The result advances understanding of subgroup structure in finite groups.

Abstract

Let $G$ be a finite group and let $\pi$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $\pi$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $\pi$-Hall subgroup if, and only if, $G$ has a $\{p,q\}$-Hall subgroup for any pair $p$, $q\in\pi$.