On weakly <Emphasis Type="Italic">S</Emphasis>Φ-supplemented subgroups of finite groups

Let G be a finite group. We say that a subgroup H of G is weakly SΦ-supplemented in G if G has a subgroup T such that G = HT and H∩T ≤ Φ(H)H_sG, where H_sG is the subgroup of H generated by all those subgroups of H that are s-permutable in G. In this paper, we investigate the influence of weakly SΦ-supplemented subgroups on the structure of finite groups. Some new characterizations of p-nilpotency and supersolubility of finite groups are obtained.     

On weakly s-permutable subgroups of finite groups

A subgroup H of a group G is said to be weakly s-permutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ H _sG where H _sG is the largest s-quasinormal subgroup of G contained in H. In this paper, we investigate the influence of weak s-permutability of some primary subgroups in finite groups. Some new results about p-supersolvability and p-nilpotency of finite groups are obtained.     

The Influence of Weakly s-Supplemented Subgroups on the Structure of Finite Groups

Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H _{s G} , where H _{s G} is the subgroup of H generated by all those subgroups of H which are s-quasinormal in G. In this article, we investigate the structure of G under the assumption that some families of subgroups of G are weakly s-supplemented in G. Some recent results are generalized.     

Finite groups with given nearly s-embedded subgroups

Let G be a finite group and H a subgroup of G. We say that H is s-permutable in G if HP = PH for all Sylow subgroups P of G; H is s-semipermutable in G if HP = PH for all Sylow subgroups P of G with (|P|, |H|) = 1. Let H _s G be the subgroup of H generated by all those subgroups of G which are s-permutable in G and H ^sG the intersection of all such s-permutable subgroups of G contain H. We say that H is nearly s-embedded in G if G has an s-permutable subgroup T such that H ^sG = HT and \({H \cap T \leqq H_{ssG}}\) , where H _ssG is an s-semipermutable subgroup of G contained in H. In this paper, we study the structure of a finite group G under the assumption that some subgroups of prime power order are nearly s-embedded in G. A series of known results are improved and extended.     

A note on a result of Skiba

A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H _sG , where H _sG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following

Theorem. Let ? be a saturated formation containing the class of all supersoluble groups

Open image in new window

and let G be a group with E a normal subgroup of G such that G/E ∈ ?. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ?.

     

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 ⩽ H _sG , where H _sG 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.     

Nearly s-Normality of Groups and Its Properties

We call a subgroup H of a group G nearly s-normal in G if there exists N ? G such that HN ? G and H ∩ N ≤ H _sG , where H _sG is the largest s-permutable subgroup of G contained in H. In this article, we obtain some results about the nearly s-normal subgroups and use them to characterize the structure of finite groups.     

Finite groups with <Emphasis Type="Italic">S</Emphasis>-supplemented <Emphasis Type="Italic">p</Emphasis>-subgroups

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by B _sG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ B _sG . A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem.     

Finite groups with some maximal subgroups of Sylow subgroups ℳ-supplemented

A subgroup H of a group G is said to be ?-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. In this paper, we obtain the following statement: Let ? be a saturated formation containing all supersolvable groups and H be a normal subgroup of G such that G/H ε ?. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of F*(H), having no supersolvable supplement in G, is ?-supplemented in G. Then G ε ?.     

On Weakly s-permutably Embedded Subgroups of Finite Groups

Suppose G is a finite group and H is subgroup of G. H is said to be s-permutably embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup H _se of G contained in H such that G = HT and H ∩ T ≤ H _se . We investigate the influence of weakly s-permutably embedded subgroups on the structure of finite groups. Some recent results are generalized.     

New Characterizations of p-Supersolubility of Finite Groups

Let H be a subgroup of a finite group G. H is said to be λ-supplemented in G if G has a subgroup T such that G = HT and H ∩ T ≤ H _SE , where H _SE denotes the subgroup of H generated by all those subgroups of H, which are S-quasinormally embedded in G. In this article, some results about the λ-supplemented subgroups are obtained, by which we determine the structure of some classes of finite groups. In particular, some new characterizations of p-supersolubility of finite groups are given under the assumption that some primary subgroups are λ-supplemented. As applications, a number of previous known results are generalized.     

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.     

The influence of <Emphasis Type="Italic">M</Emphasis>-supplemented subgroups on the structure of finite groups

A subgroup H of a group G is said to be M-supplemented in G if there exists a subgroup B of G such that G = HB and T B < G for every maximal sub-group T of H. Moreover, a subgroup H is called c-supplemented in G if there exists a subgroup K such that G = HK and H ∩ K ≤ H _G where H _G is the largest normal subgroup of G contained in H. In this paper we give some conditions of supersolv-ability of finite group under assumption that some primary subgroups have some kinds of supplements, which are generalizations of some recent results.     

<Emphasis Type="Italic">S</Emphasis>-embedded subgroups of finite groups

A subgroup H of G is said to be S-embedded in G if G has a normal subgroup N such that HN is s-permutable in G and H ∩ N ⩽ H _sG , where H _sG is the largest s-permutable subgroup of G contained in H. S-embedded subgroups are used to give novel characterizations for some classes of groups. New results are obtained and a number of previously known ones 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...     

c*-Quasinormally embedded subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be s-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G; H is called c*-quasinormally embedded in G if there is a subgroup T of G such that G = HT and H??T is s-quasinormally embedded in G. We investigate the influence of c*-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.     

On weakly S-embedded and weakly τ-embedded subgroups

Let G be a finite group. A subgroup H of G is said to be weakly S-embedded in G if there exists a normal subgroup K of G such that HK is S-quasinormal in G and H ∩ K ≤ H _seG , where H _seG is the subgroup generated by all those subgroups of H which are S-quasinormally embedded in G. We say that a subgroup H of G is weakly τ-embedded in G if there exists a normal subgroup K of G such that HK is S-quasinormal in G and H ∩ K ≤ H _seG , where H _seG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. In this paper, we study the properties of weakly S-embedded and weakly τ-embedded subgroups, and use them to determine the structure of finite groups.     

<Emphasis Type="Italic">M</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript>-supplemented subgroups of finite groups

A subgroup K of G is M_p-supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p^α. In this paper we prove the following: Let p be a prime divisor of |G| and let H be ap-nilpotent subgroup having a Sylow p-subgroup of G. Suppose that H has a subgroup D with D_p ≠ 1 and |H: D| = p_α. Then G is p-nilpotent if and only if every subgroup T of H with |T| = |D| is M_p-supplemented in G and N_G(T_p)/C_G(T_p) is a p-group.     

A condition for the solvability of finite groups

A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H ₁ B is a proper subgroup of G for every maximal subgroup H ₁ of H. We investigate the influence of ?-supplementation of Sylow subgroups and obtain a condition for solvability and p-supersolvability of finite groups.     

Finite Groups Whose Minimal Subgroups are c-Supplemented

Let G be a finite group. A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H _G , where H _G  = Core _G (H) is the largest normal subgroup of G contained in H. In this article, we investigate the structure of a finite group G under the assumption that subgroups of prime order are c-supplemented in G. Moreover, we analyze the structure of a group G when the minimal subgroups of the generalized Fitting subgroup F?(G) of G are c-supplemented in G through the theory of formations.