On supersolvability of fatorized finite groups

In this paper, we investigate the structure of finite groups that are products of two supersolvable groups and gain a sufficient condition for a group to be supersolvable. Our main theorem is the following: Let the group $G=HK$ be the product of the subgroups $H$ and $K$ . Assume that $H$ permutes with every maximal subgroup of $K$ and $K$ permutes with every maximal subgroup of $H$ . If $H$ is supersolvable, and $K$ is nilpotent and $K$ is $\delta $ -permutable in $H$ , where $\delta $ is a complete set of Sylow subgroups of $H$ , then $G$ is supersolvable. Some known results are generalized.