On the products of $\mathbb P$-subnormal subgroups of finite groups

A subgroup $H$ of a finite group $G$ is called $\mathbb P$-subnormal in $G$ whenever $H$ either coincides with $G$ or is connected to $G$ by a chain of subgroups of prime indices. If every Sylow subgroup of $G$ is $\mathbb P$-subnormal in $G$ then $G$ is called a w-supersoluble group. We obtain some properties of $\mathbb P$-subnormal subgroups and the groups that are products of two $\mathbb P$-subnormal subgroups, in particular, of $\mathbb P$-subnormal w-supersoluble subgroups.