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In this paper, we study how certain arithmetical conditions on the lengths of conjugacy classes of primary elements and π-elements of a finite group G influence the structure of G, some new results are obtained and several well-known results in the literature are developed. In particular, the structure of a kind of outer p-supersolvable groups is described, some new conditions implying π-supersolvability are found, and the upper bound for the order of Sylow subgroups is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the influence of weakly s-semipermutable subgroups on the p F -hypercenter of a finite group and obtained some new criteria for a normal subgroup to be contained in the p F -hypercenter of a finite group. Many known results are generalized and uniformed. [ABSTRACT FROM AUTHOR]

Let G be a finite group and H a subgroup of G. H is said to be N H -embedded in G if there exists a normal subgroup T of G such that H T is a Hall subgroup of G and H ∩ T ≤ H s ̄ G , where H s ̄ G is the largest s -semipermutable subgroup of G contained in H. In this paper, we give some new characterizations of p -nilpotent and supersolvable groups by using N H -embedded subgroups. Some known results are generalized. [ABSTRACT FROM AUTHOR]

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 1 , 2 , 3 be classes of all p -groups, p -nilpotent groups and p -supersolvable groups, respectively, G be the -residual of G. Let X ∈ { (G p ∗) 1 , G 2 , G 3 }. A subgroup H of a finite group G is said to have Π -property in G , if for any G -chief factor L / K , | G / K : N G / K ((H ∩ L) K / K) | is a π ((H ∩ L) K / K) -number. A normal subgroup E of G is said to be p -hypercyclically embedded in G if every p - G -chief factor of E is cyclic, where p is a fixed prime. In this paper, we prove that E is p -hypercyclically embedded in G if and only if for some p -subgroups H of E , H ∩ X have Π -property in G. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called an ICΦs -subgroup of G if (H ∩ [ H , G ]) H G / H G ≤ Φ (H / H G) H s G / H G , where H G = ∩ g ∈ G H g and Φ (H / H G) is the Frattini subgroup of H / H G and HsG is the largest s-permutable subgroup of G contained in H. In this paper, we investigate the structure of finite group G under the assumption that some subgroups of G are ICΦs -subgroups of G, some results are obtained. [ABSTRACT FROM AUTHOR]

A subgroup S of G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. In this paper, we give some new characterizations of p-solvable and p-supersolvable groups by the p-cover-avoidance properties of the p-sylowizers of some p-subgroups and the p-sylowizers in some classes of groups. [ABSTRACT FROM AUTHOR]

In this paper, we investigated influence of M p -supplemented p-subgroups contained in a normal subgroup on structure of finite groups. We obtained some new criteria for p-supersolvability of finite groups, decided all possible non-Abelian pd-chief factors under our assumption and gave a Frobenius type theorem. Some earlier results on M p -supplemented p-subgroups appear as particular cases. [ABSTRACT FROM AUTHOR]

Let Φ a (G) and Φ n (G) denote the intersection of all abnormal and normal maximal subgroups of G, respectively. S is called Φ a (or Φ n )-supplemented in G if there exists a proper normal subgroup N of G such that G = SN and S ∩ N ≤ Φ a (S) (or Φ n (S) ). In this paper we characterize many properties of groups by using Φ a and Φ n -supplementation. Communicated by Sudarshan Kumar Sehgal [ABSTRACT FROM AUTHOR]

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]

Let be a finite group. A subgroup of is said to be a Hall -semiembedded subgroup of if is a Hall subgroup of for any , where . In this paper, we investigate the influence of Hall -semiembedded subgroups on the structure of the finite group . Some new results about to be a -group are obtained, where is a saturated formation. [ABSTRACT FROM AUTHOR]

LetGbe a finite group, andT(G) be the sum of all complex irreducible character degrees ofG. In this paper, we get the exact lower bound of |G|∕T(G) for a non-r-solvable groupG. [ABSTRACT FROM AUTHOR]

Let G be a finite group and $$H\le G$$ . In this paper, we investigate the influence of the index of H in $$\langle H, H^{g}\rangle $$ on the structure of G. [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 p-group. If |CG(x): ⟨ x ⟩ | ≤pkfor every noncentral element x of G, then G is called a k − 𝒩𝒞 −p-group. In this article, we give a classification of k − 𝒩𝒞 −p-groups, where k = 1, 2 and p > 2. [ABSTRACT FROM PUBLISHER]

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]

In this paper, we classify the -groups all of whose proper subgroups have cyclic Frattini subgroups, for an odd prime . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the concept of weakly -supplemently embedded subgroup and characterize -nilpotency and supersolvability of a finite group under the assumption that some maximal subgroups of Sylow subgroup are weakly -supplemently embedded in the normalizer of the Sylow subgroup. [Copyright &y& Elsevier]

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]

In this article, we classify p-groups whose non-normal subgroups have index p in their normalizers. This answers a problem posed by Berkovich and Janko. [ABSTRACT FROM AUTHOR]

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 study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or As if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s. [ABSTRACT FROM AUTHOR]

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]

Let N be a finite group and G be a finite simple group and let πt(N) and πt(G) be the set of indices of maximal subgroups of groups N and G respectively. If πt(N) ⊆ πt(G), we give the structure of N. [ABSTRACT FROM AUTHOR]

Let G be a finite group. In this paper, we study how certain arithmetical conditions on the conjugacy class lengths of real elements of G influence the structure of G. In particular, a new type of prime graph is introduced and studied. We obtain a series of theorems which generalize some existed results. [ABSTRACT FROM AUTHOR]

A subgroup S of a group G is called a p -sylowizer of a p -subgroup R in G if S is maximal in G with respect to having R as its Sylow p -subgroup. In this paper, we investigate the structure of the group G in which the sylowizers of each non-trivial primary subgroups in G are conjugate and give the non-solvable chief factors of G. [ABSTRACT FROM AUTHOR]