On the product of two $\pi$-decomposable groups

Abstract

The aim of this paper is to prove the following result: let $\pi$ be a set of odd primes. If the finite group $G=AB$ is a product of two $\pi$-decomposable subgroups $A={\mathrm{O}}_{\pi }(A)\times {\mathrm{O}}_{{\pi }^{\prime }}(A)$ and $B={\mathrm{O}}_{\pi }(B)\times {\mathrm{O}}_{{\pi }^{\prime }}(B)$, then ${\mathrm{O}}_{\pi }(A){\mathrm{O}}_{\pi }(B)={\mathrm{O}}_{\pi }(B){\mathrm{O}}_{\pi }(A)$ and this is a Hall $\pi$-subgroup of $G$.