## Abstract

This paper gives a characterization of finite groups *G* in which each cyclic subgroup either is normal in *G* or normalizes all subgroups of *G*.

## Abstract

Two sufficient conditions for a finite group *G* to be *p*-supersolvable have been obtained. For example (Theorem 1.1), let *N* be a normal subgroup of *G* such that *G*/*N* is *p*-supersolvable for a fixed odd prime *p* and let *N*
_{p} be a Sylow *p*-subgroup of *N*. Suppose that *N* is *p*-solvable and Ω_{1}(*N*
_{p}) is generated by the subgroups of order *p* of *N*
_{p} which are normal in *N*
_{G}(*N*
_{p}). Then *G* is *p*-supersolvable.

A subgroup *H* of *G* is called *M*
_{p}-embedded in *G*, if there exists a *p*-nilpotent subgroup *B* of *G* such that *H*
_{p} ∈ Syl* _{p}*(

*B*) and

*B*is

*M*

_{p}-supplemented in

*G*. In this paper, we use

*M*

_{p}-embedded subgroups to study the structure of finite groups.

## Abstract

Let *G* be a finite group. A PT-group is a group *G* whose subnormal subgroups are all permutable in *G*. A PST-group is a group *G* whose subnormal subgroups are all S-permutable in *G*. We say that *G* is a PT_{o}-group (respectively, a PST_{o}-group) if its Frattini quotient group *G*/Φ(*G*) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PT_{o}-groups and minimal non-PST_{o}-groups.

## Abstract

This paper represents an attempt to extend and improve the following result of Berkovich: Let *G* be a group of odd order. Let *G*=*G*
_{1}
*G*
_{2} such that *G*
_{1} and *G*
_{2} are subgroups of *G*. If the Sylow *p*-subgroups of *G*
_{1} and of *G*
_{2} are cyclic, then *G* is *p*-supersolvable.

## Abstract

A subgroup *H* of *G* is *c*-*permutable* in *G* if there exists a permutable subgroup *P* of *G* such that *HP*=*G* and *H*∩*P*≦*H*
_{pG}, where *H*
_{pG} is the largest permutable subgroup of *G* contained in *H*. A group *G* is called CPT-group if *c*-permutability is transitive in *G*. A number of new characterizations of finite solvable CPT-groups are given.

## Abstract

A number of authors have studied the structure of a finite group *G* under the assumption that some subgroups of *G* are well located in *G*. We will generalize the notion of *s*-permutable and *s*-permutably embedded subgroups and we will obtain new criterions of *p*-nilpotency and supersolvability of groups. We also generalize some known results.

Let *G* be a finite group. A subgroup *H* of *G* is called an

*G*if

*N*

_{G}(

*H*) ∩

*H*

^{g}≤

*H*for all

*g*∈

*G*. A subgroup

*H*of

*G*is called a weakly

*G*if there exists a normal subgroup

*K*of

*G*such that

*G*=

*HK*and

*H*∩

*K*is an

*G*. In this article, we investigate the structure of a group

*G*in which every subgroup with order

*p*

^{m}of a Sylow

*p*-subgroup

*P*of

*G*is a weakly

*G*, where

*m*is a fixed positive integer. Our results improve and extend the main results of Skiba [13], Jaraden and Skiba [11], Guo and Wei [8], Tong-Veit [15] and Li et al. [12].

## Abstract

Let be a class of groups and *G* a finite group. We call a set Σ of subgroups of *G* a *G-covering subgroup system for *
if whenever . Let *p* be any prime dividing |*G*| and *P* a Sylow *p*-subgroup of *G*. Then we write Σ_{p} to denote the set of subgroups of *G* which contains at least one supplement to *G* of each maximal subgroup of *P*. We prove that the sets Σ_{p} and Σ_{p}∪Σ_{q}, where *q*≠*p*, are *G*-covering subgroup systems for many classes of finite groups.