Finite groups with some SS-supplemented subgroups

Czechoslovak Mathematical Journal - A subgroup H of a finite group G is said to be SS-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormal in K. We...     

Finite groups with some pronormal subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H ^x in 〈H, H ^x 〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.     

The embedding of CAP-subgroups in finite groups

This paper features new and unpublished results about the cover-avoidance property presented in the context of previously established ones. The goal of this work is to contribute to the breadth of the current body of knowledge and to fill in some gaps in the literature. On one hand, we summarize and investigate further the relationships between the cover-avoidance property and normal, conjugate, Sylow, and maximal subgroups. On the other hand, we define a subgroup U to be a subCAP-subgroup of the finite group G if there exists a chain of subgroups \({U = U_{0}\leq U_{1}\leq \dotsm \leq U_{r} = G}\) such that U _i-1 is a CAP-subgroup of U _i for all \({i \in \{1, \dots, r\}}\), and show how this property can be used to characterize solvable groups.     

A question on partial CAP-subgroups of finite groups

A subgroup H of a finite group G is a partial CAP-subgroup of G if there is a chief series of G such that H either covers or avoids every chief factor of the series.The structural impact of the partial cover and avoidance property of some distinguished subgroups of a group has been studied by many authors.However,there are still some open questions which deserve an answer.The purpose of the present paper is to give a complete answer to one of these questions.     

Finite groups with some ?-quasinormal minimal subgroups

We use ?-quasinormal condition on minimal subgroups to Characterize the structure of a finite group through the theory of formation. We give some equivalent conditions of a nilpotent group or a saturated formation containing the nilpotent groups. Our results generalize earlier theorems of Yokoyama, Ballester-Bolinches and Pedraza Aguilera.     

Finite groups with some ss-quasinormal and c-normal subgroups

Let ℱ be a saturated formation containing the class of supersolvable groups. In this paper, we use c-normal subgroups and ss-quasinormal subgroups to give criteria for a group to belong to ℱ.     

Finite groups with some non-Abelian subgroups of non-prime-power order

Let G be a finite group and NA(G) denote the number of conjugacy classes of all nonabelian subgroups of non-prime-power order of G. The Symbol π(G) denote the set of the prime divisors of |G|. In this paper we establish lower bounds on NA(G). In fact, we show that if G is a finite solvable group, then NA(G) = 0 or NA(G) ≥ 2^|π(G)|?2, and if G is non-solvable, then NA(G) ≥ |π(G)| + 1. Both lower bounds are best possible.     

Finite groups with three conjugacy class sizes of some elements

Let G be a finite group. We prove as follows: Let G be a p-solvable group for a fixed prime p. If the conjugacy class sizes of all elements of primary and biprimary orders of G are {1,p ^a , n} with a and n two positive integers and (p,n)?=?1, then G is p-nilpotent or G has abelian Sylow p-subgroups.     

一类分式线性变换群的有限子群

证明PGL(2,Q)中仅有17个系数形如m n p的分式线性变换有限子群并且决定了它们的构造,其中m,n是整数,p是素数.     

Finite groups with some given systems of Xm‐semipermutable subgroups

Let X be a nonempty subset of a group G. We call a subgroup A of G an X_m‐semipermutable subgroup of G if A has a minimal supplement T in G such that for every maximal subgroup M of any Hall subgroup T₁ of T there exists an element x ∈ X such that AM^x = M^xA. In this paper, we study the structure of finite groups with some given systems of X_m‐semipermutable subgroups (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite groups with some NR-subgroups or {\mathcal{H}}-subgroups

Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR-subgroup (Normal Restriction) if whenever ${K \trianglelefteq H}$ , then ${K^G \cap H = K}$ , where K ^G is the normal closure of K in G. Bianchi, Gillio Berta Mauri, Herzog and Verardi proved a characterization of soluble T-groups by means of ${\mathcal{H}}$ -subgroups: a subgroup H of G is said to be an ${\mathcal{H}}$ -subgroup of G if ${H^g \cap N_G(H) \leq H}$ for all ${g \in G}$ . In this article we give new characterizations of finite soluble PST-groups in terms of NR-subgroups or ${\mathcal{H}}$ -subgroups. We will show that they are different from the ones given by Ballester-Bolinches, Esteban-Romero and Pedraza-Aguilera. Robinson established the structure of minimal non-PST-groups. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PST-groups.     

n-极大子群为共轭可换的有限群

群G的子群H称为G的共轭可换子群,若HHg=HgH,对任意g∈G都成立,本文考查了n-极大子群为共轭可换时对有限群构造的影响.     

Finite groups with transitive semipermutability

A group G is said to be a T-group (resp. PT-group, PST-group), if normality (resp. permutability, S-permutability) is a transitive relation. In this paper, we get the characterization of finite solvable PST-groups. We also give a new characterization of finite solvable PT-groups.      

Finite groups with Frobenius subgroup

Suppose the normalizer N of a subgroup A of a simple group G is a Frobenius group with kernel A, and the intersection of A with any other conjugate subgroup of G is trivial, and suppose, if A is elementary Abelian, that ¦a¦> 2n+1, where n=¦N:A¦. It is proved that if A has a complement B in G, then G acts doubly transitively on the set of right cosets of G modulo B, the subgroup B is maximal in G, and ¦B¦ is divisible by ¦a¦–1. The proof makes essential use of the coherence of a certain set of irreducible characters of N.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 177–186, August, 1976.The author would like to thank V. D. Mazurov for helpful discussions concerning the theorem proved in this note.     

Finite graphs of groups with isomorphic fundamental groups

Translated from Algebra i Logika, Vol. 30, No. 5, pp. 595–623, September–October, 1991.     

Finite groups

The present survey is formed primarily on the basis of works reviewed in the Referativnyi Zhurnal Matematika during 1976–1983 and is a continuation of surveys of the same name published in 1966, 1971, 1976 in the series Albegra, Topologiya, Geometriya (INT). Principal attention is devoted to finite simple groups and their classification.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 24, pp. 3–120, 1986.