Classes of finite groups with generalized subnormal cyclic primary subgroups

We study the properties of the classes nu (pi)a"OE (nu (pi) (*) a"OE) of finite groups whose all cyclic primary pi-subgroups are a"OE-subnormal (respectively, K-a"OE-subnormal) for a set of primes pi and a hereditary homomorph a"OE. It is established that is a hereditary saturated formation if is a hereditary saturated formation. We in particular obtain some new criteria for the p-nilpotency and I center dot-dispersivity of finite groups. A characterization of formations with Shemetkov property is obtained in the class of all finite soluble groups.