自同构群是循环群被交换群扩张的有限群

设Ｃ是有限群，ＡｕｔＧ＝ＡＢ，，Ａ是交换群且每Ｓｙｌｏｗ子群的秩≤２，Ｂ是循环群，本文得出了Ｇ的结构，特别地，证明了ＡｕｔＧ是秩≤２的交换群时，Ｇ循环。     

最高阶元素个数为2p~2的有限群是可解群

本文讨论了最高阶元素个数 ｜Ｍ（Ｇ）｜＝ ２Ｐ２ （为素数）的有限群，证明了群 Ｇ是可解群     

自同构群

设Ｇ是有限群，ＡｕｔＧ＝ＡＢ，Ｂ△ＡｕｔＧ，（｜Ａ｜，｜Ｂ｜）＝１，Ａ是交换群且每Ｓｙｏｗ子群的秩≤２，Ｂ是循环群。本文得出了Ｇ的结构，特别地，证明了ＡｕｔＧ是秩≤的交换群时，Ｇ循环。     

一类3元生成的有限可解群

设Ｇ是个有限可解解，若对Ｇ的每个商群Ｈ，Ｈ的正规Ａｂｅｌ子群都是可以由２元生成的，则称Ｇ为ＡＤ２－群，在本文里，我们证明了：如果Ｇ是个ＡＤ２－群，那么Ｇ是３元生定的，且Ｇ＾（６）＝１。另外，我们举了２个例子，说明这些界都是最好的。     

有限Abel群的一个特征

本文证明了有限群Ｇ是Ａｂｅｌ群当且仅当Ｇ_ｒ满足下列条件：（Ⅰ） Ｇ有一个幂自同构 ａ使得 ＣＧ（ａ）是一个初等 ＡｂｅｌＺ一群．（Ⅱ）Ｇ没有子群与２－群＜ａ,ｂ｜ａ~２~ｎ＝ｂ~２~ｍ＝１,ａ~ｂ＝ａ~（１＋２）~（ｎ－１）＞同构,其中ｎ≥３,ｎ≥ｍ．利用该结果,作者还证明若有限群Ｇ有一个幂自同构ａ使得Ｃ_Ｇ（ａ）是一个初等Ａｂｅｌ２－群,则Ｇ是幂零群     

一类不能作为自同构群的奇阶群

本文考虑如下问题：怎样的有限群可以作为另一个有限群的全自同构群？我们首先证明，若有限群Ｋ有一个正规Ｓｙｌｏｗｐ－子群使得｜Ｋ：Ｚ（Ｋ）｜ｐ＝ｐ２，那么Ｋ有２阶自同构．利用这个结果，我们证明了，若奇阶群Ｇ具有阶Ｐｓｍ（１≤ｓ≤３），ｐ为｜Ｇ｜的最小素因子，ｐｍ，ｍ无立方因子，则Ｇ不可能作为全自同构群．     

关于Chevalley群扩群的两点继续研究

构造出Ｄ２ｍ型Ｃｈｅｖａｌｌｅｙ群扩群Ｇ,证明了其唯一性,并讨论了文「１」中构造的Ａｎ型Ｇ的唯一性。     

无限的可解SD2—群（II）—有限剩余情形

如果对多重循环群Ｇ的每个有限剩余Ｇ，Ｇ的真子群都具有可以由二元生成的，那么我们就把Ｇ叫做ＲＤ２－群，在本文里，我们确定了无限的ＲＤ２－群的结构，证明了ＲＤ２－群是可以由二元生成的，这些结构推广了作者已经得到了关于无限的可解ＳＤ２群的全部结果，见（４）。     

非幂零极大子群指数为素数幂的有限群

本文证明了如下结果，１．设ｐ是一个素数，如果有限群Ｇ的每一非幂零的极大子群的指数都为ｐ的方幂，则Ｇ为可解群．２．如果有限群Ｇ的每一非幂零的极大子群的指数为素数幂，则Ｇ／Ｓ（Ｇ）１或ＰＳＬ（２，７），其中Ｓ（Ｇ）表示Ｇ的最大可解正规子群．     

关于自同构群方程｜Aut(G)｜=p~2q~2

本文给出满足｜Ａｕｔ（Ｇ）｜＝ｐ２ｑ２的有限群Ｇ的完全分类，ｐ和ｑ是不同的素数．     

On Some Groups with Finite Involution Saturated with Finite Simple Groups

In the paper, on the basis of the notion of saturation introduced in [1], we give a characterization of locally finite simple groups with an Abelian Sylow 2-subgroup in the classes of mixed and periodic groups. A part of the main results was announced in [2]. Theorem 2 of the paper generalizes results of [3, 4].     

On Subgroups and Quotient Groups of L-fuzzy Topological Groups

In paper [1], it gave the concept and basic properties of L-fuzzy topological groups. This paper will enter the concept of quotient groups of L-fuzzy topological groups and will prove some properties of subgroups and quotient groups. The persent work is the continuation of [1] and hence all the conventions in [1] still hold good in the present paper.     

Lexicographic Products of Half Linearly Ordered Groups

The notion of the half linearly ordered group (and, more generally, of the half lattice ordered group) was introduced by Giraudet and Lucas [2]. In the present paper we define the lexicographic product of half linearly ordered groups. This definition includes as a particular case the lexicographic product of linearly ordered groups. We investigate the problem of the existence of isomorphic refinements of two lexicographic product decompositions of a half linearly ordered group. The analogous problem for linearly ordered groups was dealt with by Maltsev [5]; his result was generalized by Fuchs [1] and the author [3]. The isomorphic refinements of small direct product decompositions of half lattice ordered groups were studied in [4].     

Suzuki-Ree群的自同构群的阶分量刻画

在《数学学报》2013年第56卷第4期中,"Suzuki-Ree群的自同构群的一个新刻画"一文证明了Aut(~2F_4(q)),q=2~f和Aut(~2G_2(q)),q=3~f,可由其阶分量刻画,其中f=3~s,s为正整数.本文证明了Aut(~2B_2(q)),q=2~f和Aut(2G2(q)),q=3~f,也可由其阶分量刻画,其中f为奇素数.结合二者得到结论:Suzuki-Ree单群的所有的素图不连通的自同构群皆可由其阶分量刻画.     

Cyclic Groups as Autocommutator Groups

This article classifies the groups X whose autocommutator subgroup [X, Aut(X)] is isomorphic to ? and the finite groups X for which [X, Aut(X)] ? C _p has a prime number p of elements.     

Extensions of Finite Quantum Groups by Finite Groups

We give a necessary and sufficient condition for two Hopf algebras presented as central extensions to be isomorphic, in a suitable setting. We then study the question of isomorphism between the Hopf algebras constructed in [AG] as quantum subgroups of quantum groups at roots of 1. Finally, we apply the first general result to show the existence of infinitely many non-isomorphic Hopf algebras of the same dimension, presented as extensions of finite quantum groups by finite groups. Partially supported by CONICET, ANPCyT, Secyt (UNC) and Ministerio de Ciencia y Tecnología de la Provincia de Córdoba.     

Expansive Convergence Groups are Relatively Hyperbolic

Let a discrete group G act by homeomorphisms of a compactum in a way that the action is properly discontinuous on triples and cocompact on pairs. We prove that such an action is geometrically finite. The converse statement was proved by P. Tukia [T3]. So, we have another topological characterisation of geometrically finite convergence groups and, by the result of A. Yaman [Y2], of relatively hyperbolic groups. Further, if G is finitely generated then the parabolic subgroups are finitely generated and undistorted. This answer to a question of B. Bowditch and eliminates restrictions in some known theorems about relatively hyperbolic groups. Received: April 2007, Revision: May 2008, Accepted: August 2008     

Finite Groups with only S-seminormal or Selfnormal Subgroups

Ｆｉｎｉｔｅ　Ｇｒｏｕｐｓ　ｗｉｔｈ　ｏｎｌｙ　Ｓ－ｓｅｍｉｎｏｒｍａｌ　ｏｒ　Ｓｅｌｆｎｏｒｍａｌ　ＳｕｂｇｒｏｕｐｓＦｉｎｉｔｅＧｒｏｕｐｓｗｉｔｈｏｎｌｙＳ－ｓｅｍｉｎｏｒｍａｌｏｒＳｅｌｆｎｏｒｍａｌＳｕｂｇｒｏｕｐｓ￥ＺｈａｎｇＱｉｎｈａｉ...     

Orthogonal Cayley-Klein Groups

A. Cayley [5] and F. Klein [12] were the first mathematicians who introduced the notion of a metric in a real projective space, by specializing a set of quadrics (called the absolute). We show in this paper, that from an algebraic point of view these projective spaces can be described by vector spaces, in which a metric structure is given by one or more symmetric bilinear forms. We study these Cayley- Klein vector spaces and their groups of automorphisms (orthogonal Cayley- Klein groups) over arbitrary commutative fields of characteristic ≠ 2.