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

In this paper, the classes of groups with given systems of [Formula: see text]-subnormal subgroups are studied. In particular, it is showed that if [Formula: see text] and [Formula: see text] are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose [Formula: see text]-subgroups are all [Formula: see text]-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with [Formula: see text]-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong [Formula: see text], where [Formula: see text] is a hereditary saturated formation, is obtained.

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