Some Generalized Normal Subgroups and p-nilpotency of Finite Groups

In this paper, the influence of s-semipermutable, c~#-normal, subnormally embedded and ss-quasinormal subgroups on the p-nilpotency of finite groups is investigated and some recent results are generalized.

S-seminormality and P-nilpotency

In this paper, we deal mainly with the following problem: if every 2-maximal subgroup of a Sylow p-subgroup of a finite group G is S-seminormal in G, what conditions force G to be p-nilpotent? As an application of main results, some sufficient conditions for finite nilpotent groups and finite supersolvable groups are obtained.

C_p-Groups and Some Results on the p-nilpotency

Let P∈SylpG,ch(P) be any characteristic subgroup of P. A group C is called a Cp-group if G satisfies the condition 'NG(ch(P)) is p-nilpotent implies that G is p-nilpotent'. In this paper some properties on Gp-groups and some suffcient conditions for a group to be p-nilpotent are obtained.

s-Conditionally Permutable Subgroups and p-Nilpotency of Finite Groups

A subgroup H of G is called s-conditionally permutable in G if for every Sylow subgroup T of G, there exists an element x ∈ G such that HTK = T^KH. In this paper, we investigate further the influence of s-conditionally permutability of some 2-maximal subgroups of the Sylow subgroup of G, on the structure of finite groups. New criteria for a group G being p-nilpotent are obtained.

ON F-z-SUPPLEMENTED SUBGROUPS OF FINITE GROUPS

A subgroup H of a group G is called F-z-supplemented in G if there exists a subgroup K of G, such that G = HK and H∩K≤ Z∞F（G）, where Z∞F（G） is the F-hypercenter of G. We obtain some results about the F-z-supplemented subgroups and use them to determine the structure of some groups.

A Note on s-quasinormally Embedded and c-supplemented Subgroups of Finite Groups

In this paper the influence of s-quasinormally embedded and c-supplemented subgroups on the p-nilpotency of finite groups is investigate and some recent results are generalized.

FINITE GROUPS WHOSE MINIMAL SUBGROUPS ARE WEAKLY H-SUBGROUPS

Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG（H）∩ H^g ≤ H for all g C G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H N K is an H-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G. Our results improve and generalize several recent results in the literature.

A Note on Zorn's Theorem

In this note, we prove that if G is a finite group, then G is p-nilpotent if and only if there exists a positive integer n such that [x,y,…y（n）]∈Op′（G） for any x,y∈G.

On Restricted Lie Superalgebras with Semisimple Elements

In the present paper, we give some sufficient conditions for the commutativity of restricted Lie superalgebras and characterize some properties of restricted Lie superalgebras with semisimple elements.

On p-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K ofG such that G = HK and H ∩K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotentif G is S4-free and every minimal subgroup of P ∩ GN is c-supplemented in NG(P), and when p = 2 P isquaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G.As some applications of this result, some known results are generalized.

Engel Condition and p-nilpotency of Finite Groups

Let G be a finite group,and let P be a Sylow p-subgroup of G.Under the hypothesis that NG(P)is p-nilpotent,we provide some conditions to give a p-nilpotency criterion of finite groups by Engel condition,which improves some recent results.

On the Product of Two Nilpotent Subgroups of a Finite Group

It is known that the product of two nilpotent subgroups of a finite group is not necessarily nilpotent.In this paper, we study the influence of the Engel condition on the product of two nilpotent subgroups. Our results generalize some well-known results.

On s-Quasinormal and c-Normal Subgroups of a Finite Group

Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.

Finite groups with systems of Σ-embedded subgroups

In recent years,a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group.However,about the cover-avoiding property of subgroups for non-chief factor,there is no study up to now.The purpose of this paper is to build the theory.Let A be a subgroup of a finite group G and Σ:G0≤G1≤…≤Gn some subgroup series of G.Suppose that for each pair(K,H) such that K is a maximal subgroup of H and G i 1 K < H G i for some i,either A ∩ H = A ∩ K or AH = AK.Then we say that A is Σ-embedded in G.In this paper,we study the finite groups with given systems of Σ-embedded subgroups.The basic properties of Σ-embedded subgroups are established and some new characterizations of some classes of finite groups are given and some known results are generalized.

Some open questions in the theory of generalized permutable subgroups

A subgroup H of a group G is said to be weakly s-supplemented in G if H has a supplement T in G such that H ∩ T HsG, where HsG is the largest s-permutable subgroup of G contained in H. This paper constructs an example to show that the open questions 6.3 and 6.4 in J Algebra, 315: 192–209 (2007) have negative solutions, and shows that in many cases Question 6.4 is positive. A series of known results are unified and generalized.

On SS-quasinormal subgroups and the structure of finite groups

A subgroup H of a finite group G is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B. In this paper, we investigate the structure of a group under the assumption that every subgroup with order pm of a Sylow p-subgroup P of G is SS-quasinormal in G for a fixed positive integer m. Some interesting results related to the p-nilpotency and supersolvability of a finite group are obtained. For example, we prove that G is p-nilpotent if there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| or 2|D| whenever p = 2 and |D| = 2 is SS-quasinormal in G, where p is the smallest prime dividing the order of G and P is a Sylow p-subgroup of G.

On s-semipermutable Subgroups of Finite Groups

Suppose that G is a finite group and H is a subgroup of G. We say that H is ssemipermutable in G if HGv = GpH for any Sylow p-subgroup Gp of G with （p, ｜H｜） = 1. We investigate the influence of s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized and unified.

On weakly S-embedded subgroups of finite groups

Let H be a subgroup of a group G. Then H is said to be S-quasinormal in G if HP = PH for every Sylow subgroup P of G; H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G for each prime p dividing the order of H. In this paper, we say that H is weakly S-embedded in G if G has a normal subgroup T such that HT is an S-quasinormal subgroup of G and H VIT ≤ HSE, where HSE denotes the subgroup of H generated by all those subgroups of H which are S-quasinormally embedded in G. Some results about the influence of weakly S-embedded subgroups on the structure of finite groups are given.

On c#-normal subgroups in finite groups

A subgroup H of a finite group G is called a c#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H n K is a CAP-subgroup of G. In this paper, we investigate the influence of fewer c#- normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.

S-semiembedded subgroups of finite groups

A subgroup H of a finite group G is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） = 1. We say that a subgroup H of a finite group G is S-semiembedded in G if there exists an s-permutable subgroup T of G such that TH is s-permutable in G and T ∩ H ≤ H-sG, where HsG is an s-semipermutable subgroup of G contained in H. In this paper, we investigate the influence of S-semiembedded subgroups on the structure of finite groups.