On some products of finite groups

A classical result of Baer states that a finite group $G$ which is the product of two normal supersoluble subgroups is supersoluble if and only if $G^{\mathrm{\prime }}$ is nilpotent. In this article we show that if $G=AB$ is the product of supersoluble (respectively, $w$-supersoluble) subgroups $A$ and $B$, $A$ is normal in $G$, $B$ permutes with every maximal subgroup of each Sylow subgroup of $A$, then $G$ is supersoluble (respectively, $w$-supersoluble) provided that $G^{\mathrm{\prime }}$ is nilpotent. We also investigate products of subgroups defined above when $A\cap B=1$ and obtain more general results.