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Let G be a finite group, p a prime, P a Sylow p-subgroup of G and d a power of p such that 1 < d < | P |. Let G p * denote the unique smallest normal subgroup of G for which the corresponding factor group is abelian of exponent dividing p – 1. Let F 1 , F 2 , F 3 be classes of all p-groups, p-nilpotent groups and p-supersolvable groups, respectively, G F be the F -residual of G. Let E ∈ { G F 1 , (G p *) F 1 , G F 2 , G F 3 }. We prove that if H ∩ E is weakly s-semipermutable in G for all normal subgroups H of P with | H | = d , then either G ∈ F 3 and G ∈ F 2 if p is the least prime factor of | G | , or else | P ∩ E | > d ; and for all subgroups H of P with | H | = d and 4 (if p = 2), then G ∈ F 3 and G ∈ F 2 if p is the least prime factor of | G |. Many known theorems become Corollaries of our results. Communicated by Peter Sin [ABSTRACT FROM AUTHOR]

LetGbe a finite group,pa prime andPa Sylowp-subgroup ofG. The main goal of the paper is to determine sufficient conditions for thep-nilpotence of a finite group withp-nilpotent Sylow normalizers. [ABSTRACT FROM AUTHOR]

We prove that a finite group G is p-supersolvable or p-nilpotent if some subgroups of G are weakly s-semipermutable in G. Several earlier results are generalized. [ABSTRACT FROM AUTHOR]

Let G be a finite group and p a prime. In this article, we give a criterion of p-nilpotency of G by means of the normalizer and a generalization of normality of subgroups. [ABSTRACT FROM PUBLISHER]

We call H an SΦ-supplemented subgroup of a finite group G if there exists a subnormal subgroup T of G such that G = HT and H ∩ T ≤ Φ( H), where Φ( H) is the Frattini subgroup of H. In this paper, we characterize the p-nilpotency and supersolubility of a finite group G under the assumption that every subgroup of a Sylow p-subgroup of G with given order is SΦ-supplemented in G: Some results about formations are also obtained. [ABSTRACT FROM AUTHOR]

Suppose that G is a finite group and H is a subgroup of G. We call H a weakly s-supplementally embedded subgroup of G if there exist a subgroup T of G and an s-quasinormally embedded subgroup Hse of G contained in H such that G = HT and H ∩ T ≤ Hse. We investigate the influence of the weakly s-supplementally embedded property of some minimal subgroups on the structure of finite groups. As an application of our results, some earlier results are generalized. [ABSTRACT FROM AUTHOR]

Abstract: A subgroup is said to be an -subgroup of a finite group if for all . For every prime dividing the order of , let be a Sylow -subgroup of and a subgroup of with . We investigate the structure of under the assumption that each subgroup of with is an -subgroup of . Some earlier results are generalized. Some results about formation are obtained. [Copyright &y& Elsevier]

A subgroup H of a finite group G is said to be SS-quasinormal in G if there exists a supplement B of H in G such that H is permutable with every Sylow subgroup of B. In this paper, we get some new characterizations of supersolvability and p-nilpotency of G by assuming some subgroups of prime power order of G are SS-quasinormal. [ABSTRACT FROM AUTHOR]

Abstract: We extended the normal index from maximal subgroups to proper subgroups. We give a quantitative version of all results obtained by using -normal subgroups and obtain some new characterizations of solvable, supersolvable and nilpotent groups by the normal indices of proper subgroups. [Copyright &y& Elsevier]

Abstract: Let be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, contains one and only one Sylow p-subgroup of G. A subgroup H of G is said to be -permutable in G if H permutes with every member of . In this paper we characterize p-nilpotency of finite groups G with assumption that some maximal subgroups or some 2-maximal subgroups of Sylow subgroups of G are -permutable. Some recent results are extended. [Copyright &y& Elsevier]

We introduce the notion of θ-pair of a proper subgroup of a finite group and give necessary and sufficient conditions for supersolvability and nilpotence of a finite group. As application, we give a proof to a recent open problem of Skiba in the unsolved problems in group theory. [ABSTRACT FROM AUTHOR]