On Finite Groups with $\mathbb{P}_{\pi}$-Subnormal Subgroups

Let $\pi$ be a set of primes. A subgroup $H$ of a group $G$ is said to be $\mathbb{P}_{\pi}$-subnormal in $G$ if either $H=G$ or there exists a chain of subgroups beginning with $H$ and ending with $G$ such that the index of each subgroup in the chain is either a prime in $\pi$ or a $\pi'$-number. Properties of $\mathbb{P}_{\pi}$-subnormal subgroups are studied. In particular, it is proved that the class of all $\pi$-closed groups in which all Sylow subgroups are $\mathbb{P}_{\pi}$-subnormal is a hereditary saturated formation. Criteria for the $\pi$-supersolvability of a $\pi$-closed group with given systems of $\mathbb{P}_{\pi}$-subnormal subgroups are obtained.