On weakly $\mathbb{P}$-subnormal subgroups of finite groups

A subgroup $H$ of a finite group $G$ is called a weakly $\mathbb{P}$-subnormal subgroup if $H$ is generated by two subgroups, one of which is subnormal in $G$, and the other one can be connected to $G$ by a subgroup chain with prime indexes. We establish the properties of weakly $\mathbb{P}$-subnormal subgroups and one makes possible to extend the known results on finite groups with sets of $\mathbb{P}$-subnormal subgroups to finite groups with weakly $\mathbb{P}$-subnormal subgroups. In particular, we prove that a finite group with weakly $\mathbb{P}$-subnormal normalizers of Sylow subgroups is supersolvable and a group with weakly $\mathbb{P}$-subnormal $B$-subgroups is metanilpotent.