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

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.