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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]

We determine finite simple groups which have a subgroup of index with exactly two distinct prime divisors. Then from this we derive a classification of primitive permutation groups of degree a product of two prime-powers. [ABSTRACT FROM AUTHOR]

Under study is the influence of the index of H in 〈 H,H〉 for g ∈ G on the structure of a group G, where H is either a second maximal subgroup of G or a Sylow subgroup of G. [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]

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 The classification of all block-transitive 2 − (v , k , 1) designs has been known as a long-standing and still open problem. This work is a contribution to the general program to classify the pairs (D , G) in which D is a 2 − (v , k , 1) design and G act as a point-primitive block-transitive automorphism group of D with a Suzuki group Sz (q) socle. We determine the point-stabilizer G α ∩ Sz (q) and the block-stabilizer G B ∩ Sz (q), and the parameter of D is given. [ABSTRACT FROM AUTHOR]

Thirty years ago, a six-person team classified the pairs ( D , G ) where D is a 2 − ( v , k , 1 ) design and G is a flag-transitive automorphism group of D , with the exception of those in which G is a one-dimensional affine group. Since then the effort has been to classify those D , designs which are block-transitive but not flag-transitive. This paper contributes to the program for determining the pairs 2 − ( v , k , 1 ) in which ( D , G ) has a block-transitive group G of automorphisms. It is clear that if one wishes to study the structure of a finite group acting on a D design then describing the socle is an important first step. Here we prove that if G is a block-transitive group of automorphisms of 2 − ( v , k , 1 ) which has D as its socle then T is also transitive on the blocks of T = 2 F 4 ( q ) . [ABSTRACT FROM AUTHOR]