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.     

On F-Abnormal Maximal Subgroups of Finite Groups

For any saturated formation F of finite groups containing all supersolvable groups, the groups in F are characterized by F-abnormal maximal subgroups.     

On ℳ p -Supplemented Subgroups of Finite Groups

A subgroup H of G is called ? _p -supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H with |H: T| =p ^α. In this paper, we investigate the influence of ? _p -supplemented subgroup and some conditions for p-nilpotency and p-supersolvability of finite groups are obtained.     

On Weakly ℋ-embedded Subgroups of Finite Groups

Let G be a finite group and H a subgroup of G. We say that H is an ?-subgroup in G if N_G(H) ∩ H^g ≤ H for all g ∈ G; H is called weakly ?-subgroup in G if G has a normal subgroup K such that G = HK and H ∩ K is an ?-subgroup in G. We say that H is weakly ? -embedded in G if G has a normal subgroup K such that H^G = HK and H ∩ K is an ?-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that some subgroups of prime power order are weakly ?-embedded in G. Our results improve and generalize several recent results in the literature.     

Subgroups of Finite Symmetric Groups

Recent results about primitive permutation groups are surveyed.     

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 Π-Permutable Subgroups in Finite Groups

Ukrainian Mathematical Journal - Let σ = {σi | i ∈ I} be a partition of the set of all primes ? and let Π be a nonempty subset of the set σ. A set ? of...     

On Subgroups and Quotient Groups of L-fuzzy Topological Groups

In paper [1], it gave the concept and basic properties of L-fuzzy topological groups. This paper will enter the concept of quotient groups of L-fuzzy topological groups and will prove some properties of subgroups and quotient groups. The persent work is the continuation of [1] and hence all the conventions in [1] still hold good in the present paper.     

On Finite Groups with Some Semi Cover-Avoiding Subgroups

A subgroup H of a finite group G is said to have the semi-cover-avoiding property in G if there is a chief series of G such that H covers or avoids every chief factor of the series. In this paper, some new results are obtained based on the assumption that some subgroups have the semi-cover-avoiding property in the group.     

Finite Groups with Only s-quasinormal and Abnormal Subgroups

The concept of an abnormal subgroup of a group G is, in a sense, opposite of thatof a normal subgroup of G. In [J. Algebra, 28(1974), 15-19) Fattahi studied those finitegroups in which each subgroup is either normal or abnormal and characterized all suchgroups. In this paper, as a generalization of Fattahi's result, we study more general groups,i. e., groups with only s-quasinormal and abnormal subgroups and characterize all suchgroups. Morever, some interesting conclusions are also obtained.     

On Π-Supplemented Subgroups of a Finite Group

A subgroup H of a finite group G is said to satisfy Π-property in G if for every chief factor L/K of G, |G/K: N_G/K(HK/K ∩ L/K)| is a π(HK/K ∩ L/K)-number. A subgroup H of G is called Π-supplemented in G if there exists a subgroup T of G such that G = HT and H ∩ T ≤ I ≤ H, where I satisfies Π-property in G. In this article, we investigate the structure of a finite group G under the assumption that some primary subgroups of G are Π-supplemented in G. The main result we proved improves a large number of earlier results.     

On Two Theorems of Finite Solvable Groups

For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.     

On Weakly ℋ-Subgroups of Finite Groups

Let G be a finite group. A subgroup H of G is called an ?-subgroup in G if N _G (H) ∩ H ^x  ≤ H for all x ∈ G. A subgroup H of G is called weakly ?-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ?-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ?-subgroups in G. Some recent results are extended and generalized.     

On the Syskin Problem of Finite Groups

In 1980, at the Seventh All-Union Symposium on Group Theory, ProfessorS.A .Syskin raised the following well-known problem: In a finite group G, supposeany two elemeuts of the same order are conjugate. Is |G|≤6? It is quite easy to