一类可解的（q）－群

有限群Ｇ叫（ｑ）－群，如果Ｇ中每个次正规子群均为拟正规子群，群Ｇ叫Ｅｑ－群，若Ｇ中每个子群在Ｇ中拟正规或自正规，有限群Ｇ叫内Ｅｑ－群，如果Ｇ本身不是Ｅｑ－群，但Ｇ的每个真子群是Ｅｑ－群，本文确定了Ｅｑ－群的结构与内Ｅｑ－群的分类．     

一类可解的（q）－群

有限群Ｇ叫（ｑ）－群，如果Ｇ中每个次正规子群均为拟正规子群，群Ｇ叫Ｅｑ－群，若Ｇ中每个子群在Ｇ中拟正规或自正规，有限群Ｇ叫内Ｅｑ－群，如果Ｇ本身不是Ｅｑ－群，但Ｇ的每个真子群是Ｅｑ－群，本文确定了Ｅｑ－群的结构与内Ｅｑ－群的分类．     

有限CN-群与有限c-可补群*

有限群G的子群H称为G的c-可补子群(c-正规子群),如果存在G的子群(正规子群)N, 使得 G = NH 且 N\cap H \leq H_G,这里 H_G =\bigcap\limits_g\in G H^g 是 H 在 G 中的核.每个子群都c-可补(c-正规)的有限群称为有限c-可补群(CN-群).本文研究有限CN-群与有限c-可补群, 获得了CN- 群与c-可补群的一些新的结果.特别地, 在方法上有一定的创新, 完善近期关于CN-群的研究.     

内PN*-群的结构(英文)

若有限群G的每个极小子群及4阶循环子群在G中正规,则称G为PN~*-群.本文给出了有限群的每个真子群都是PN~*-群但其本身不是PN~*-群的完全分类.     

一般投影线性群PGL(2,q)和4-(q+1,5,λ)设计

本文主要考虑了一般投影线性群PGL(2,q)区传递作用下的4-(q+1,5,λ)设计的存在性问题。经讨论知λ的可能值是4。     

有限群的Fuzzy拟正规子群和Fuzzy次正规子群

讨论有限群的Fuzzy拟正规子群和Fuzzy次正规子群的一些性质。     

弱s-拟正规子群对有限群的p-幂零性的影响

群G的子群H称为G的弱s-拟正规子群,若G有次正规子群T,使得G=HT且H ∩T≤HsG,其中HsG是包含在H中的G的最大的s-拟正规子群.本文利用Sylow p-子群的极大子群的弱s-拟正规性得到有限群为p-幂零群的一些充分条件,并给出Schur-Zassenhaus定理的一种推广.     

每个子群都c-正规的有限群

在本文中我们研究有限CN-群, 即每个子群都c-正规的有限群. 我们得到以下结果:群G是CN-群当且仅当G^∈的每个子群都在G中正规.群G是CN-群当且仅当G可解且c-正规性是传递的.设p是一个奇素数, P是一个p-群, 则P是一个CN-群当且仅当Φ(P)≤Z(P).我们也得到了一些CN-群的直积为CN-群的判别条件.     

一类有限群的超可解性

<正> 不少人对每个极小子群均是正规子群(简称为PN-群)的群进行了研究.Gaschutz和Ito[1]证明了这种群的导群是P-幂零的,其中P是任意奇素数.Buckley[2]证明了奇阶PN-群是超可解的.近年来,人们放宽了对极小子群正规性的限制,亦得到了一些结果.本文主要是研究部分极小子群具有某种正规性(比如,S-拟正规,予正规等等)的群G,获得了G为超可解群的充要条件是G没有截断D_(2q)(截断即是子群的商群),其中D_(2q)是如     

二次极大子群中极小子群和4阶循环子群拟正规的有限单群

如果有限群 G 的各个极小子群和4阶循环子群在 G 中是拟正规的,我们就称 G 是强 PQN-群.本文主要目的是:(一)讨论极大子群是强 PQN-群的有限群的结构,证明它们除三种群之外都是超可解的,而对这三种例外的群,我们给出了详尽的结构描述;(二)确定2-极大子群是强 PQN-群的有限非 Abel 单群,证实这种群恰是 A_5.(注:在正文中我们将强 PQN-群一律简称为 PQN-群.)     

ON PRODUCTS OF GROUPS FOR WHICH NORMALITY IS A TRANSITIVE RELATION ON THEIR FRATTINI FACTOR GROUPS

Abstract

This paper is devoted to the study of groups with the property that the Frattini factor group is a T-group, i.e. a group in which every subnormal subgroup is normal. We give necessary and suffucient conditions for a direct product G = H x K of finite groups H and K to have such a property. Some structure theorems are also discussed.     

On <Emphasis Type="Italic">T</Emphasis>-groups,supersolvable groups,and maximal subgroups

A group is called a T-group if all its subnormal subgroups are normal. Finite T-groups have been widely studied since the seminal paper of Gaschütz (J. Reine Angew. Math. 198 (1957), 87–92), in which he described the structure of finite solvable T-groups. We call a finite group G an NNM-group if each non-normal subgroup of G is contained in a non-normal maximal subgroup of G. Let G be a finite group. Using the concept of NNM-groups, we give a necessary and sufficient condition for G to be a solvable T-group (Theorem 1), and sufficient conditions for G to be supersolvable (Theorems 5, 7 and Corollary 6).     

Cuccia定理的一个推广

For a finite group G,let S(G)be the set of minimal subgroups of odd order of G which are complemented in G.It is proved that if every minimal subgroup X of odd order of G which does not belong to S(G),C_G(X)is either subnormal or abnormal in G.Then G solvable.     

Permutable subnormal subgroups of finite groups

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called -groups. In particular, it is shown that the finite solvable -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not -groups but all subnormal subgroups of defect two are permutable. Received: 13 August 2008     

Finite Soluble Groups with Permutable Subnormal Subgroups

A finite group G is said to be a PST-group if every subnormal subgroup of G permutes with every Sylow subgroup of G. We shall discuss the normal structure of soluble PST-groups, mainly defining a local version of this concept. A deep study of the local structure turns out to be crucial for obtaining information about the global property. Moreover, a new approach to soluble PT-groups, i.e., soluble groups in which permutability is a transitive relation, follows naturally from our vision of PST-groups. Our techniques and results provide a unified point of view for T-groups, PT-groups, and PST-groups in the soluble universe, showing that the difference between these classes is quite simply their Sylow structure.     

On Locally Nilpotent Subgroups of GL 1(D)

In this article we study locally nilpotent subgroups of D*: = GL ₁(D), where D is a division ring. It is proved that every locally nilpotent subnormal subgroup of D* is central. If D is algebraic over its centre then every locally solvable subnormal subgroup of D* is central. Also, in this case, it is shown that every locally nilpotent maximal subgroup of D* can occur as the multiplicative group of some maximal subfield of D.     

Every 2-group with all subgroups normal-by-finite is locally finite

A group G has all of its subgroups normal-by-finite if H/H _G is finite for all subgroups H of G. The Tarski-groups provide examples of p-groups (p a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2-group with every subgroup normal-by-finite is locally finite. We also prove that if |H/H _G | 6 2 for every subgroup H of G, then G contains an Abelian subgroup of index at most 8.     

关于几乎可解群

本文推广了关于局部有限群的Asar定理及p.Hall—Kulatilaka,Kargapolov定理.