Classes of finite groups with generalized subnormal cyclic primary subgroups

Abstract

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.

A generalization of Hall's theorem on hypercenter

“…Let F be a hereditary formation. In [19] and [27] the classes of groups wF and v * F all whose Sylow and cyclic primary subgroups respectively are K-F-subnormal were studied. From the results of these papers follows Proposition 1.…”

“…Then N σ has the lattice property for K-F-subnormal subgroups (see [26,Lemma 2.6(3)] or [5,Chapter 3]). According to [19,Theorem B and…”

On some applications of Fitting like subgroups of finite groups

“…Let F be a hereditary formation. In [16,25] the classes of groups wF and v * F all whose Sylow and cyclic primary subgroups respectively are K-F-subnormal were studied. In these papers the following results were proved.…”

“…If F = N is the formation of all nilpotent groups, then the notions of K-F-subnormal and subnormal subgroups coincide. Groups with different systems of K-F-subnormal are the main object of many papers (for example, see [16,24,25,28]). The main idea of this paper is to consider K-F-subnormality of a subgroup not in the whole group but in some subgroup related to some generalization of the Fitting subgroup in the sense of the following definition:…”

On generalizations of Baer's theorems about the hypercenter of a finite group

“…Vasil'ev and T.I. Vasil'eva [14] studied a class of groups wF whose all Sylow subgroups are F-subnormal for a given hereditary saturated formation F. Let us note that in this case Z wF (G) lies in the intersection of all F-subnormalizers of all Sylow subgroups of a group G. Author [15] studied a class of groups vF whose all cyclic primary subgroups are F-subnormal for a given hereditary saturated formation F. Again Z vF (G) lies in the intersection of all Fsubnormalizers of all cyclic primary subgroup of a group G.…”