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

TL;DR

The paper generalizes the Arad–Ward result by proving $E_{\pi_1}\cap E_{\pi_2}\subseteq E_{\pi_1\cap\pi_2}$ for sets of primes $\pi_1,\pi_2$, establishing a strong intersection property for Hall subgroups. As a corollary, for a prime set $\pi$ and $l$ with $2\le l<|\pi|$, a finite group $G$ that has Hall $\rho$-subgroups for every subset $\rho\subseteq\pi$ of size $l$ must contain a solvable Hall $\pi$-subgroup. The proof employs a minimal counterexample strategy and hinges on the classification of finite simple groups, leveraging Table 1 from $\text{VdRev11}$ and auxiliary results on Hall subgroups in $GL_2(q)$, $SL_2(q)$, and symmetric groups to rule out obstructions across all simple group families. This yields a lattice-theoretic view of the sets $\Pi(G)$ of prime sets for which $G\in E_\pi$, illustrating that $\Pi(G)$ forms a lattice under inclusion with meet given by intersection and a more nuanced join operation.

Abstract

For a set of primes $π$, denote by $E_π$ the class of finite groups containing a Hall $π$-subgroup. We establish that $E_{π_1}\cap E_{π_2}$ is contained in $E_{π_1\capπ_2}$. As a corollary, we prove that if $π$ is a set of primes, $l$ is an integer such that $2\leqslant l<|π|$ and $G$ is a finite group that contains a Hall $ρ$-subgroup for every subset $ρ$ of $π$ of size $l$, then $G$ contains a solvable Hall $π$-subgroup.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Corollary 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Proposition 1
• proof
• Corollary 2