The major aim of the present paper is to strengthen a nice result of Shemetkov and Skiba which gives some conditions under which every non-Frattini $G$-chief factor of a normal …
The major aim of the present paper is to strengthen a nice result of Shemetkov and Skiba which gives some conditions under which every non-Frattini $G$-chief factor of a normal subgroup $E$ of a finite group $G$ is cyclic. As applications, some recent kno
Consider a finite group G. A subgroup H of G is said to be E-supplemented in G if there is a subgroup T of G such that G = HT …
Consider a finite group G. A subgroup H of G is said to be E-supplemented in G if there is a subgroup T of G such that G = HT and H ∩ T ≤ H eG , where H eG denotes the subgroup of H generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, we investigate the influence of E-supplemented subgroups on the structure of finite groups.
Let [Formula: see text] be a finite group. If [Formula: see text] is a subgroup of [Formula: see text] and [Formula: see text] a subgroup of [Formula: see text], we …
Let [Formula: see text] be a finite group. If [Formula: see text] is a subgroup of [Formula: see text] and [Formula: see text] a subgroup of [Formula: see text], we say that [Formula: see text] is strongly closed in [Formula: see text] with respect to [Formula: see text] if [Formula: see text] for any [Formula: see text]. We say that a subgroup [Formula: see text] of [Formula: see text] is strongly closed in [Formula: see text] if [Formula: see text] is strongly closed in [Formula: see text] with respect to [Formula: see text]. A subgroup [Formula: see text] of a group [Formula: see text] is said to be weakly [Formula: see text]-supplemented in [Formula: see text] if [Formula: see text] has a subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is strongly closed in [Formula: see text]. In this paper, we study the structure of a group [Formula: see text] under the assumption, that some subgroups of prime power order are weakly [Formula: see text]-supplemented in [Formula: see text]. Our results extend and generalize several recent results in the literature.
A subgroup H of a group G is said to be weakly M-supplemented in G if there exists a subgroup B of G provided that (1) G = HB, and …
A subgroup H of a group G is said to be weakly M-supplemented in G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B=BH1<G, where HG is the largest normal subgroup of G contained in H. In this paper, we study the influence of weakly M-supplemented subgroups with given order on the structure of finite groups.
Let H be a subgroup of a finite group G. We say that H is weakly ss-supplemented in G if there exists a subgroup T of G and an s-semipermutable …
Let H be a subgroup of a finite group G. We say that H is weakly ss-supplemented in G if there exists a subgroup T of G and an s-semipermutable subgroup Hss of G contained in H such that G = HT and H∩T ≤ Hss. In this paper, we get some new characterizations of supersolvability and nilpotentcy of G by assuming some minimal subgroups of G are weakly ss-supplemented. Some recent results are extended and generalized.
A subgroup H of a group G is said to be weakly ss-supplemented in G if there exists a subgroup K of G such that HK is a subnormal subgroup …
A subgroup H of a group G is said to be weakly ss-supplemented in G if there exists a subgroup K of G such that HK is a subnormal subgroup of G and H∩K≤HuG,where HuG is the largest subnormal subgroup of G contained in H.In this paper,the solvability of a finite group G with some subgroups of Sylow subgroups weakly ss-supplemented in G is characterized.
Let H be a subgroup of a finite group G.H is called weakly S-supplemented if there exists a subgroup T of G such that G=TH and H∩T≤HG,where HG is the …
Let H be a subgroup of a finite group G.H is called weakly S-supplemented if there exists a subgroup T of G such that G=TH and H∩T≤HG,where HG is the subgroup generalized by all subgroups of H which are s-semipermutable in G.Let G be a finite group and p∈π(|G|).If one of the following ① and ② holds,then G is p-nilpotent: ①(|G|,p-1)=1,G has a Sylow p-subgroup P such that every minimal subgroup of P is weakly -supplemented in G,moreover if p=2,then P is A4-free;② G is A4-free,p=min π(G),N■G such that G/N is p-nilpotent,every subgroup of order p2 in every Sylow p-subgroup P of N is weakly -supplemented in G.
A subgroup H is called ℳ-supplemented in a finite group G if there exists a subgroup B of G such that G = HB and such that H1B is a …
A subgroup H is called ℳ-supplemented in a finite group G if there exists a subgroup B of G such that G = HB and such that H1B is a proper subgroup of G for any maximal subgroup H1 of H. In this paper we fix a subgroup D in every non-cyclic Sylow subgroup P of G satisfying 1 < D < P and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are ℳ-supplemented in G or have a supersolvable supplement in G.
A subgroup of a group is said to be supplemented in if there exists a subgroup of such that is a permutable subgroup of and,where is the largest permutable subgroup …
A subgroup of a group is said to be supplemented in if there exists a subgroup of such that is a permutable subgroup of and,where is the largest permutable subgroup of contained in In this paper,the solvability of a finite group with some subgroups of Sylow subgroups supplemented in is characterized.
Background and Objective: A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G.A subgroup, where H G is the core …
Background and Objective: A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G.A subgroup, where H G is the core of H in G and Z T (G/H G ) is the hypercenter of G/H G .The main objective of this study is to study the structure of a finite group under the assumption that some subgroups of prime power order are T hp -supplemented in the group.Methodology: This study can improve previous results by studying the properties of the concept of T hq -supplemented and using some lemmas on these concept.Results: Results clearly reveal the influence the concept of T hq -supplemented of some subgroups of prime power order on the group.Conclusion: This study improves and extends some results of super solvability of the group by using the concept of T hq -supplemented.
A subgroup H of a group G is said to be weakly M -supplemented in G if there exists a subgroup B of G provided that (1) G = HB …
A subgroup H of a group G is said to be weakly M -supplemented in G if there exists a subgroup B of G provided that (1) G = HB , and (2) if H1/HG is a maximal subgroup of H/HG , then H1B = BH1 < G.where HG is the largest normal subgroup of G contained in H .In this paper we will prove that: Let F be a saturated formation containing all supersolvable groups and G be a group with a normal subgroup H such that G/H ∈ F .If every maximal subgroup of every noncyclic Sylow subgroup of F * (H) having no supersolvable supplement in G , is weakly M -supplemented in G , then G ∈ F .
Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HG p = G p H for any Sylow …
Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HG p = G p H for any Sylow p-subgroup G p of G; H is called weakly s-supplemented subgroup of G if there is a subgroup T of G such that G = HT and H ∩ T ≤ H sG , where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of minimal weakly s-supplemented subgroups on the structure of finite groups and generalize some recent results. Furthermore, we give a positive answer in the minimal subgroup case for Skiba's Open Questions in [On weakly s-permutable subgroups of finite groups, J. Algebra315 (2007) 192–209].
The major aim of the note is to strengthen the results of Asaad in Communications in Algebra, 42 (2014), 2319–2330.
The major aim of the note is to strengthen the results of Asaad in Communications in Algebra, 42 (2014), 2319–2330.
A subgroup H of a group G is said to be c-supplemented inG if there exists a subgroup K of G such thatHK=G and H \cap K is contained in …
A subgroup H of a group G is said to be c-supplemented inG if there exists a subgroup K of G such thatHK=G and H \cap K is contained in Core_G (H). We follow Hall's ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some 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 …
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.
Let G be a finite group. A subgroup A of G is said to be S-quasinormal in G if AP = PA for all Sylow subgroups P of G. The …
Let G be a finite group. A subgroup A of G is said to be S-quasinormal in G if AP = PA for all Sylow subgroups P of G. The symbol HsG denotes the subgroup generated by all those subgroups of H which are S-quasinormal in G. A subgroup H is said to be S-supplemented in G if G has a subgroup T such that T ∩ H ⩽ HsG and HT = G; see [Skiba, J. Algebra 315: 192–209, 2007].
Let G be a group and H be a subgroup of G. We say that H is weakly Φ-supplemented in G if G has a subgroup T such that HT …
Let G be a group and H be a subgroup of G. We say that H is weakly Φ-supplemented in G if G has a subgroup T such that HT = G and , where denotes the Frattini subgroup of H. In this paper, properties of this new kind of inequalities of subgroups are investigated and new characterizations of nilpotency and supersolubility of finite groups in terms of the new inequalities are obtained. MSC:20D10, 20D15, 20D20.
Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HG p = G p H for any Sylow …
Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HG p = G p H for any Sylow p-subgroup G p of G; H is called weakly s-supplemented subgroup of G if there is a subgroup T of G such that G = HT and H ∩ T ≤ H sG , where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of minimal weakly s-supplemented subgroups on the structure of finite groups and generalize some recent results. Furthermore, we give a positive answer in the minimal subgroup case for Skiba's Open Questions in [On weakly s-permutable subgroups of finite groups, J. Algebra315 (2007) 192–209].