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

Abstract

Let the group $G=AB$ be a product of two $\pi$-decomposable subgroups $A=O_\pi(A) \times O_{\pi'}(A)$ and $B=O_\pi(B) \times O_{\pi'}(B)$ where $\pi$ is a set of primes. The authors conjecture that $O_\pi(A)O_\pi(B)=O_\pi(B)O_\pi(A)$ if $\pi$ is a set of odd primes. In this paper it is proved that the conjecture is true if $A$ and $B$ are soluble. A similar result with certain additional restrictions holds in the case $2 \in \pi$. Moreover, it is shown that the conjecture holds if $O_{\pi'}(A)$ and $O_{\pi'}(B)$ have coprime orders.