有限群的Fuzzy次正规子群与Fuzzy极大子群

本文研究了有限群的Ｆ次正规子群，得出了一个Ｆ子群是Ｆ次正规子群的充要条件，讨论了Ｆ次正规子群的一些重要性质。另外，本文还引入了有限群的Ｆ极大子群的概念，给出了Ｆ子群是Ｆ极大群的充要条件。最后，给出了三个定理，讨论了有限群Ｇ可解、超可解、幂零与Ｇ的Ｆ次正规子群、Ｆ极大子群之间的联系。     

S-拟正规嵌入子群

群G的一个子群H称为在G中S-拟正规嵌入的,如果对H阶中的每一个素因子p,H的Sylowp-子群也是G的某个S-拟正规子群的Sylowp-子群.利用子群的S-拟正规嵌入性给出了群为p-幂零群及超可解群的一些特征.     

非幂零真子群同阶类类数给定的有限群

作为Schmidt定理的推广,证明了:(1)非幂零真子群同阶类类数<3的有限群可解;(2)G为非幂零真子群同阶类类数=3的非可解群当且仅当G≌A_5或G≌SL_2(5).此外,完全分类了非平凡幂零子群同阶类类数≤5的非可解群和非平凡子群同阶类类数≤9的非可解群.     

有限群的弱正规子群与p-幂零性

研究了p2阶子群以及一般的pk阶子群为弱正规子群时有限群G的结构.给出了有限群为p-幂零群以及超可解群的一些条件.     

两个正规p-超可解子群的积

本文研究了两个正规的p-超可解子群的积构成的极小非P-超可解群的结构的问题.利用有限群论和群类论的一些基本方法,获得了两个正规的p-超可解子群的积仍为P-超可解群的一些充分条件,并推广了文[1]中关于超可解群的情况.     

子群次正规性对有限群可解性的影响

考查了次正规子群对有限群结构的影响,得到有限群可解的若干充分条件和超可解的一个充分条件.     

共轭置换、半正规与Sylow子群的极大子群

结合共轭置换与半正规的概念,研究了群G的Sylow子群的极大子群或共轭置换或半正规这一条件与群G的超可解性之间的关系.     

次正规子群与有限群的超可解性

通过Sylow子群的极大子群和次正规性,利用极小阶反例的方法,得出群p-幂零性和超可解性的结论.本文的创新改进之处在于结合Sylow子群的极大子群和次正规性,研究p-幂零性和超可解性的相关结论.     

无限对称群的正规子群

有限对称群Ｓｎ（ｎ≠４）非平凡的正规子群仅有一个交代群Ａｎ。无限集合Ｍ上的对称群ＳＭ则不是这样。本文的主要结果是：（１）确定了ＳＭ的全部正规子群，它们形成一个整序集；（２）ＳＭ正规子群的正规子群仍是ＳＭ的正规子群；（３）ＳＭ的正规子群是特征子群；（４）ＳＭ的正规子群（≠１）的自同构群与ＳＭ同构，ＳＭ是完全群。     

非幂零极大子群共轭类类数给定的有限群

证明了非幂零极大子群共轭类类数等于2的有限群必可解,并给出了非幂零极大子群同阶类类数等于2的非可解群的等价刻画.     

Bounded Rank Subgroups of Coxeter Groups, Artin Groups and One-Relator Groups with Torsion

We obtain a number of results regarding the freeness of subgroupsof Coxeter groups, Artin groups and one-relator groups withtorsion. In the case of Coxeter groups, we also obtain resultson quasiconvexity and subgroup separability. 2000 MathematicsSubject Classification 20F65, 20F55, 20F36, 20F06.     

Countable Primitive Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product. Received: January 2006, Revision: May 2006, Accepted: May 2006     

Quotient Stable Uniform Groups

Abstract

In a subclass of rigid almost completely decomposable groups with global homocyclic regulator quotient a classification up to near-isomorphism, in terms of modified diagonal similarity, and a decomposition criterion is given. This subclass contains the local uniform groups, and generalizations of results of Dugas and Oxford are obtained, first to some global case and second to modified diagonal similarity instead of diagonal equivalence.     

A New Approach to Extracting Roots in Garside Groups

We study the problem of extracting roots in Garside groups by reducing it to the calculation of certain ultra summit sets. Several properties concerning the roots and an effective algorithm are derived. In particular, in the case of braid groups, a conjecture on the bound of super summit set implies that, for fixed number of strands and ordinal number of root, the algorithm is polynomial in the word length.     

Some finiteness conditions on normalizers or centralizers in groups

We consider the following two finiteness conditions on normalizers and centralizers in a group G: (i) |N_G(H) : H| < ∞ for every H ? G, and (ii) |C_G(x):?x?|<∞ for every ?x??G. We show that (i) and (ii) are equivalent in the classes of locally finite groups and locally nilpotent groups. In both cases, the groups satisfying these conditions are a special kind of cyclic extensions of Dedekind groups. We also study a variation of (i) and (ii), where the requirement of finiteness is replaced with a bound. In this setting, we extend our analysis to the classes of periodic locally graded groups and non-periodic groups. While the two conditions are still equivalent in the former case, in the latter the condition about normalizers is stronger than that about centralizers.     

Classification of connected commutative locally Nash groups

In this article, we classify connected commutative (locally) Nash groups, which is a continuation of our previous work on the classification of abelian Nash manifolds. Our results generalize the classification of the one-dimensional case by Madden-Stanton and the two-dimensional case by Baro-Vicente-Otero. Moreover, we determine the affineness and toroidal affineness of a connected commutative Nash group.     

非平凡正规子群只有两种阶数的有限群

In this paper, we investigate the structure of the groups whose nontrivial normal subgroups have order two. Some properties of this kind of groups are obtained.     

On the Direct Indecomposability of Infinite Irreducible Coxeter Groups and the Isomorphism Problem of Coxeter Groups

In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components.     

L—Fuzzy群的同志和同构

本文研究了Ｌ—Ｆｕｚｚｙ群的同态和同构，给出了它们的一些性质，并把群论中的同态定理和同构定理推广到Ｌ—Ｆｕｚｚｙ群上