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。     

阿贝尔群被超-(循环或有限)群的可裂扩张(Ⅰ)

设F是由f(p)所局部定义的可解群系,G∈F,A是ZG-模.我们称A的一个p-主因子U/V在G中是F-中心的,如果G/CG(U/V)∈f(p).否则称U/V在G中是非中心的.本文证明了:设G是超-(有限或循环)的局部可解群,A是Artinian ZG-模且所有的不可约ZG-因子都是有限的;F为由f(p)所局部定义的局部可解群系,且对任意的p∈π,f(p)≠φ,f(∞) f(p).如果G∈F,且A的所有不可约ZG-因子在G中均是F-非中心的,则A被G的扩张在A上共轭可裂..     

Locally nilpotent groups with a centralizer of finite rank

Locally nilpotent groups in which the centralizer of some finitely generated subgroup has finite rank are studied. It is shown that if G is such a group and F is a finitely generated subgroup with centralizer C_G(F) of finite rank, then the centralizer of the image of F in the factor group G/t(G) modulo the periodic part t(G) also has finite rank. It is also shown that G is hypercentral when F is cyclic and either G is torsion-free or all Sylow subgroups of the periodic part of C_G(F) are finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1511–1517, November, 1992.     

有限循环群上的Cayley有向图的哈密顿分解

本文给出了有限循环群上的Cayley有向图Cay(M,G)可哈密顿分解的一个充分条件,并证明了当|M|=2时此条件还是必要的.     

恰有两个子群共轭类长的有限群

设G是有限群,Ns(G)表示G的子群共轭类长构成的集合.本文研究Ns(G)中只有两个元素时有限群G的结构,在非幂零情形时给出了G的完全分类,在幂零情形时获得了G的一些性质.     

Generalized cyclic displacements : PMM vol. 37, n6, 1973, pp. 1135–1137

We consider the generalized cyclic displacements of holonomic mechanical systems with a finite number of degrees of freedom, and their application to integration of the equations of motion.N.G.Chetaev in [1] turned his attention to the formulation of problems dealing with general properties of mechanical systems and connected with the groups of transformations which leave the basic mechanical functions invariant. It was he who introduced [2] the concept of cyclic displacement of a mechanical system with smooth holonomic constraints. This concept was enlarged in [3] in the course of considering a particular case of motion of a mechanical system with three degrees of freedom.     

On finite groups with exactly seven element centralizers

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if $\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n$ . The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite groupG is primitive 7-centralizer if and only if $\frac{G}{{Z(G)}} \cong D_{10} $ orR, whereR is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute#Cent(G) for some finite groups, using the structure ofG modulu its center.     

有限群的特征标的零点和可解φ-群

有两个对偶的问题如下：问题Ⅰ：将满足下述条件的有限群G分类：G的特征标表中，除一行外其余各行最多有一个零．问题Ⅱ：将满足下述条件的有限群G分类：G的特征标表中，除一列外其余各列最多有一个零．在这篇文章中，我们对于有限可解群解答上述两个问题，并确定和这两个问题密切相关的一类有限可解群的结构（这类可解群在本文中称之为可解φ-群）．附带我们还完全回答了[4]中的问题1，并说明[6，定理]的条件可以极大地减弱．     

p-超循环嵌入子群的一个判别准则

令E是有限群G的一个正规子群,且U是所有有限超可解群的集合.E称为在G中是p-超循环嵌入的,如果E的每个pd-阶的G-主因子是循环的.G的子群H称为在G中是U-Φ-可补充的,如果存在G的一个次正规子群T,使得G=HT,且(H∩T)H_G/H_G≤Φ/(H/H_G)Z_U(G/H_G),其中Z_U(G/H_G)是商群G/H_G的U-超中心.作者证明,如果E的一些p-子群在G中是U-Φ-可补充的,那么E在G中是p-超循环嵌入的.作为应用,得到了有限群是p-超可解的若干判断准则,并且推广了一些已知的结果.     

Bi-Cayley图的一些代数性质

设G是一个有限群,S是G的一个子集,Bi-Cayley图BC(G,S)是一个二部图:其顶点集为G×{0,1},而边集为{{(g,0),(sg,1)}:g∈G,s∈S}.本文研究了有限阿贝尔群G上的Cayley图D(G,S)和Bi-Calyley图BC(G,S)之间特征值的关系,并由此得到循环群上的Bi-Cayley图的特征值.继而得到生成树数的一些渐进性定理.     

二次极大子群中2阶及4阶循环子群拟正规的有限群

本文讨论２阶及４阶循环子群对群结构的影响．主要结果是下述定理：如果有限群Ｇ满足标题的条件，那么下列情形之一成立：（１）Ｇ有正规Ｓｙｌｏｗ ２－子群；（２） Ｇ为 ２－幂零；（３） Ｇ ≌ Ｓ４；（４） Ｇ＝ＰＱ，其中 Ｐ为阶 ２４广义四元数群， Ｑ为 ３阶循环群；（５） Ｇ ≌ Ａ５或 ＳＬ（２，５）．     

ESSENTIALLY FINITELY INDECOMPOSABLE ABELIAN p-GROUPS

Abstract

An abelian p-group C is said to be essentially finitely indecomposable (efi) if given any decomposition of G as the direct sum of a family of subgroups, there exists a positive integer n such that all but at moat a finite number of subgroups of this family are bounded by n. We look at examples and related questions. We prove that a reduced abelian p-group G is efi if and only if G modulo its elements of infinite height is efi. In the proof of this we obtain the following result which is of independent interest: Let A be a reduced p-group with a summand K such that K is a direct sum of cyclic groups. Let B be a basic subgroup of A. Then B contains a subgroup C such that C is a summand of A and the final rank of C is equal to the final rank of K.     

Norm商群的Sylow子群皆循环的有限群

设$G$是有限群, $N(G)$为$G$的norm, 则$N(G)$是$G$的正规化G的每个子群的特征子群. 我们在下列条件之一下,研究了$G$的结构：1) Norm商群$G/N(G)$是循环群;2) Norm商群$G/N(G)$的所有Sylow子群都是循环群,特别地当$G/N(G)$的阶是无平方因子数时.     

Isomorphisms of finite semi-Cayley graphs

Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph(Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph(also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits(of equal size). In this paper, we introduce the concept of SCI-graph(semi-Cayley isomorphism)and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.     

On equality of central and class preserving automorphisms of finite p-groups

Let G be a finite non-abelian p-group, where p is a prime. Let Aut_c(G) and Aut_z(G) respectively denote the group of all class preserving and central automorphisms of G. We give a necessary and sufficient condition for G such that Aut_c(G) = Aut_z(G) and classify all finite non-abelian p-groups G with elementary abelian or cyclic center such that Aut_c(G) = Aut_z(G). We also characterize all finite p-groups G of order ≤ p ⁷ such that Aut_z(G) = Aut_z(G) and complete the classification of all finite p-groups of order ≤ p ⁵ for which there exist non-inner class preserving automorphisms.     

互素作用下的不变块(英文)

令A,G为有限群且(|A|,|G|)=1.设A通过自同构作用在G上.令B是G的A-不变块.若B的每一个指标都是A-不变的,且B的亏群是循环群,则A中心化B的某一亏群.     

Loops mit distributiven und geometrischen Unterloopverbänden

The following facts are shown: A loop with a finite distributive subloop lattice is finite, monogenic and all its subloops are monogenic. Therefore, power-associative loops having finite distributive subloop lattices are cyclic groups. A loop G with its subloop lattice L(G) being a finite n-dimensional projective geometry is generated by at most n+1 elements. For all n IN, n4, there are power-associative loops whose subloop lattices are projective lines with n points. Furthermore, for a given projective planeP _n (desarguesian or non-desarguesian) of order n there exists a power-associative loopG with L(G) -P _n.