Sub-cover-avoidance Properties and the Structure of Finite Groups

A subgroup H of a group G is said to have the sub-cover-avoidance property in G ffthereis a chief series 1 = G0 ≤ G1 ≤…≤ Gn - G, such that Gi-1（H ∩ Gi） G for every i = 1,2,... ,l. In this paper, we give some characteristic conditions for a group to be solvable under the assumptions that some subgroups of a group satisfy the sub-cover-avoidance property.     

关于有限群的$CAP$-嵌入子群

A subgroup H of a finite group G is said to be CAP-embedded subgroup of G if, for each prime p dividing the order of H, there exists a CAP-subgroup K of G such that a Sylow p-subgroup of H is also a Sylow p-subgroup of K. In this paper some new results are obtained based on the assumption that some subgroups of prime power order have the CAP-embedded property in the group.     

A question on partial CAP-subgroups of finite groups

A subgroup H of a finite group G is a partial CAP-subgroup of G if there is a chief series of G such that H either covers or avoids every chief factor of the series.The structural impact of the partial cover and avoidance property of some distinguished subgroups of a group has been studied by many authors.However,there are still some open questions which deserve an answer.The purpose of the present paper is to give a complete answer to one of these questions.     

On -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.     

On <Emphasis Type="Italic">c</Emphasis>*-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 ∩ K is an S-quasinormal embedded subgroup of G. In this paper, the structure of a finite group G with some c*-normal maximal subgroups of Sylow subgroups is characterized and some known related results are generalized.     

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.     

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) ∩Hg≤H for all g∈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∩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.     

s-Permutably embedded subgroups and p-supersolvable groups

Let G be a finite group,and H a subgroup of G.H is called s-permutably embedded in G if each Sylow subgroup of H is a Sylow subgroup of some s-permutable subgroup of G.In this paper,we use s-permutably embedding property of subgroups to characterize the p-supersolvability of finite groups,and obtain some interesting results which improve some recent results.     

Finite groups whose n-maximal subgroups are σ-subnormal

Let σ = {σ_i | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σ_i-subgroup of G, for some i ∈ I, and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HA~x= A~xH for all A ∈ H and x ∈ G:σ-subnormal in G if there is a subgroup chain A = A_0≤A_1≤···≤ A_t = G such that either A_(i-1)■A_i or A_i/(A_(i-1))A_i is a finite σ_i-group for some σ_i ∈σ for all i = 1,..., t.If M_n M_(n-1) ··· M_1 M_0 = G, where Mi is a maximal subgroup of M_(i-1), i = 1, 2,..., n, then M_n is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal(σ-quasinormal,respectively) in G but, in the case n 1, some(n-1)-maximal subgroup is not σ-subnormal(not σ-quasinormal,respectively) in G, we write m_σ(G) = n(m_(σq)(G) = n, respectively).In this paper, we show that the parameters m_σ(G) and m_(σq)(G) make possible to bound the σ-nilpotent length l_σ(G)(see below the definitions of the terms employed), the rank r(G) and the number |π(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give the conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when m_σ(G) = |π(G)|. Some known results are generalized.     

Finite p-Groups with a Class of Complemented Normal Subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified.     

A generalized CAP-subgroup of a finite group

Let A be a subgroup of a finite group G. We say that A is a generalized CAP-subgroup of G if for each chief factor H/K of G either A avoids H/K or the following holds:(1) If H/K is non-abelian, then|H :(A ∩H)K | is a p′-number for every p ∈π((A ∩H)K/K);(2) If H/K is a p-group, then |G : NG(K(A ∩H))| is a p-number. In this paper, we use the generalized CAP-subgroup to characterize the structure of finite groups.Some new characterizations of the hypercyclically embedded subgroups of a finite group are obtained and a series of known results are generalized.     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

On Nearly SS-Embedded Subgroups of Finite Groups

Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ HseG, where HseG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups.     

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 of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S4-free and every minimal subgroup of P n GN is c-supplemented in NG(P), and when p = 2 P is quaternion-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.     

On σ-semipermutable Subgroups of Finite Groups

Let σ = {σ_i|i ∈ I } be some partition of the set of all primes P, G a finite group andσ(G) = {σ_i |σ_i ∩π(G) = ?}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member = 1 of H is a Hall σ_i-subgroup of G for some σ_i ∈σ and H contains exactly one Hallσ_i-subgroup of G for every σ_i ∈σ(G). A subgroup H of G is said to be: σ-semipermutable in G with respect to H if H H_i~x= H_i~xH for all x ∈ G and all H_i ∈ H such that(|H|, |H_i|) = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.     

关于有限群某些子群的X-半置换性

Let A be a subgroup of a group G and X a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we try to use the X-semipermutability of some subgroups to characterize the structure of finite groups.     

半覆盖远离性质与有限群的结构

In this paper,the so-called π-cover-avoiding properties of subgroups are defined and investigated.In terms of this property,we characterize the π-solvability of finite groups.Some other new results are also obtained based on the assumption that some subgroups have the semi cover-avoiding properties in a finite group.     

On Φ-τ-Supplement Subgroups of Finite Groups

Let τ be a subgroup functor and H a p-subgroup of a finite group G. Let G= G/H_G and H= H/H_G. We say that H is Φ-τ-supplement in G if G has a subnormal subgroup T and a τ-subgroup S contained in H such that G=H T and H∩T≤SΦ(H). In this paper,some new characterizations of hypercyclically embedability and p-nilpotency of a finite group are obtained based on the assumption that some primary subgroups are Φ-τ-supplement in G.     

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 = P H 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 ∩ T≤H SE,where H SE denotes the subgroup of H generated by all those subgroups of ...     

The influence of s-conditionally permutable subgroups on finite groups

A subgroup H of a group G is called s-conditionally permutable in G if for every Sylow subgroup T of G there exists an element x ∈ G such that HTx = TxH. Using the concept of s-conditionally permutable subgroups, some new characterizations of finite groups are obtained and several interesting results are generalized.