A generalization of the Arad--Ward theorem on Hall subgroups

TL;DR

This paper generalizes the Arad--Ward theorem by exploring the intersection properties of classes of finite groups with Hall subgroups, leading to new conditions for the existence of solvable Hall subgroups.

Contribution

It extends the understanding of Hall subgroup structures by proving intersection properties and conditions for solvable Hall subgroups in finite groups.

Findings

Proves that the intersection of classes with Hall $ ho$-subgroups is contained in the class with Hall $ ho$-subgroups.

Shows that groups with certain Hall subgroup conditions contain solvable Hall $ ext{pi}$-subgroups.

Establishes a generalized framework for Hall subgroup existence based on prime set intersections.

Abstract

For a set of primes $\pi$, denote by $E_\pi$ the class of finite groups containing a Hall $\pi$-subgroup. We establish that $E_{\pi_1}\cap E_{\pi_2}$ is contained in $E_{\pi_1\cap\pi_2}$. As a corollary, we prove that if $\pi$ is a set of primes, $l$ is an integer such that $2\leqslant l<|\pi|$ and $G$ is a finite group that contains a Hall $\rho$-subgroup for every subset $\rho$ of $\pi$ of size $l$, then $G$ contains a solvable Hall $\pi$-subgroup.