On Finite Groups with Pπ-Subnormal Subgroups

Let pi be a set of primes. A subgroup H of a group G is said to be 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 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 P-pi-subnormal is a hereditary saturated formation. Criteria for the P-pi-supersolvability of a P-pi-closed group with given systems of Pr-subnormal subgroups are obtained.