The $D_{\pi }$-property on products of $\pi$-decomposable groups

Abstract

The aim of this paper is to prove the following result: Let $\pi$ be a set of odd primes. If the group $G=AB$ is the product of two $\pi$-decomposable subgroups $A=A_{\pi }\times A_{{\pi }^{\prime }}$ and $B=B_{\pi }\times B_{{\pi }^{\prime }}$, then $G$ has a unique conjugacy class of Hall $\pi$-subgroups, and any $\pi$-subgroup is contained in a Hall $\pi$-subgroup (i.e. $G$ satisfies property $D_{\pi }$).