有限二元生成平衡p-群

Blackburn, Deaconescu和 Mann 的文章[1], 群G叫做平衡群，如果对G的满足HK=KH的任意子群H,K, 或者H 属于K的正规化子，或者K 属于H的正规化子.继续Blackburn, Deaconescu和 Mann 的工作，并基于 Newman和本文第四作者关于亚循环p-群的分类（见文[4,5],也见文[8]）,我们在本文中给出了有限二元生成平衡p-群的完全分类。     

奇阶亚循环p~-群的完全分类

亚循环p~-群的完全分类是由Bruce W.King在1973年首先给出的,但他的计算太繁.据[2]Ⅲ,10.2c),奇阶亚循环p~-群是正则的,于是可应用正则p~-群的理论大大简化计算.关于正则p~-群的知识可见[2]Ⅲ§10.此外,还需要一个简单的数论结果,不加证明地陈述于下 引理 设p是奇素数,n是正整数.则与p互素的正整数模p~n组成一个乘法群(依通常乘法),记作M(p~n),且|M(p~n)|=φ(p~n)=p~n—p~(n-1).M(p~n)的Sylow p~-子群S(M)={x∈ M(p~n)|x≡1(modp)},是循环群.S(M)的唯一的p~i阶子群     

r-循环阵的逆阵或广义逆阵

在应用数学和其他学科(如数理统计、固态物理等)中,都将遇到求循环阵的逆阵或广义逆阵的问题.如何求非奇异循环阵的逆阵?文[1]提出了一种算法而无证明,文[2]则给出了这种算法的一个证明,文[3]又提出一种新算法,但上述两种算法的计算量大,实际使用时是很繁的.针对这一情况,文[4]除了对[1]中提出的方法重新给了一个初等证明外,还导出了一些特殊循环阵的逆阵公式.关于求奇异循环阵的广义逆阵的问题,则除了[3]中给出了某类特殊的奇异循环阵的 Moor-Penrose 逆阵外,还未见到有文章论述求奇异循环阵的广义逆阵的一般方法.本文给出了 r-循环阵的逆阵或一个反射 g 逆阵的公式和具体算法.特别,这个公式可用来求通常的循环阵及反循环阵的逆阵和 Moor-Penrose 逆阵.文[3]、[4]中的各个公式可用本文的统一方法推广到 r-循环阵的情形.     

关于群布尔矩阵的收敛和周期

在文[1]和[2]中,各自得到了如下结果：一个循环布尔矩阵A是本原的当且仅当gcd（i2-i1,…,i1-i1,n）＝1,其中A＝Pi1十Pi2十…＋Pi1,0≤i1＜i2＜…＜i1≤n-1,P是对应于n阶循环置换（123…n）的置换矩阵．在本文中,先把此结果推广到群矩阵（一种循环矩阵的推广）．其次,讨论群布尔矩阵的周期．给出了计算周期的算法,最后,探讨循环布尔矩阵A的使Am p＝Am的最小正整数m.     

图上Z~2可扩作用的非存在性

文[9]中作者考虑连续统上可扩群作用的存在性问题,证明了单位闭区间上不存在自由交换群Z×Z的可扩作用,并且给出一个例子表明闭区间上存在自由积Z*Z的可扩作用.换句话说,由两个交换同胚生成的群是不能可扩作用在闭区间上的,但还是存在由两个非交换同胚生成的群能够可扩作用在闭区间上.本文证明了图上不存在Z×Z的可扩作用,解决了文[9]所提的一个问题.     

非交换Hadamard差集存在的一个必要条件

本文把文[1]的想法推广到非交换的情形,得到非交换Hadamard差集存在的一个必要条件.作为它的推论,一是解决了文[2]遗留下的一个未决情形,简化了其相应结果的证明;二是在自共轭条件满足时,对著名的交换Hadamard差集的Turyn指数界条件作出了改进.最后,提出了一个4p4阶群中交换Hadamard差集不存在的一个猜想.     

非交换Hadamard差集存在的一个必要条件

本文把文[1]的想法推广到非交换的情形,得到非交换Hadamard差集存在的一个必要条件.作为它的推论,一是解决了文[2]遗留下的一个未决情形,简化了其相应结果的证明;二是在自共轭条件满足时,对著名的交换Hadamard差集的Turyn指数界条件作出了改进.最后,提出了一个4p4阶群中交换Hadamard差集不存在的一个猜想.     

任意两个非交换元均生成 p3阶子群的有限 p -群

该文分类了任意两个非交换元均生成 p³阶子群的有限 p -群.作为推论, 完全解决了文献[1]中提出的第237个问题: 对于所有的 x,y∈ G, 研究满足条件( x, y) |≤p³的 p -群 G.     

四次循环域的明显刻划

<正> 一、引言本文的目的是对于四次循环域给出明显的刻划,将这类域用 Conductor 加以分类,计算出具有同一 Conductor 的四次循环域的个数,给出这些域的明显形式(即域的生成元和它所满足的方程),同时还讨论了域中整基、素理想分解等一系列算术问题.关于这方面的文献已有[1],[3],[4],其中 Albert 在计算整基过程中有许多错误,而文献[3]和[4]中采用了初等方法,但是未能反映出 Galois 群所起的本质作用.本文则是运用[2]中的处理风格.本文所给出的结果,今后将用来研究四次循环域的进一步性质.事实上,张贤科利用这里所给出的结果研究了循环四次数域的相对整基问题,推广了前人的结果,得到了完善的答案.     

字典序地生成有序根树的一个算法

§1.引言由于树的生成在计算机科学中有着重要应用,近年来许多文章研究了树的生成,其中大多数文章是讨论2分树及 k 分树的生成.研究一般有序根树的文章尚少.文献[1]给出了有序根树的一个序列表示法,并描述了一个生成有序根树的算法.文献[2]及[3]讨论了生成2分树及 k 分树的算法.本文用0,1序列表示有序根树,并给出了一个字典序地生成具有 n 个顶点的所有有序根树的算法.本文的表示法及算法与文献[1]中所提方法不同.本算法亦可用来生成具有 n 个叶子的所有2分树.它比[2]中的算法更简单.本文中未加说明的术语皆见[1].     

Finite metacyclic groups as active sums of cyclic subgroups

The notion of active sum provides an analogue for groups of what the direct sum is for abelian groups. One natural question then is which groups are the active sum of a family of cyclic subgroups. Many groups have been found to give a positive answer to this question, while the case of finite metacyclic groups remained unknown. In this note we show that every finite metacyclic group can be recovered as the active sum of a discrete family of cyclic subgroups.     

Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some metabelian groups

We prove a conjecture of Zassenhaus that every normalized torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a group element for some metabelian groups including metacyclic groups G containing a normal cyclic group A such that G/A is cyclic of prime power order. The relative prime case was done in [11]. Received: 21 April 2005     

On the problem of classification of finite groups associated to multiplicative <Emphasis Type="Italic">η</Emphasis>-products

In this article, we study finite groups such that the cusp forms associated to all elements of these groups by means of some faithful representation are modular forms with multiplicative Fourier coefficients from a special class. The Sylow subgroups of such groups of odd order are found. We consider such metacyclic groups. The groups of order 16 and the groups of order 32 that are metacyclic or are direct products of a group of order 16 and the cyclic group of order 2 are considered in detail. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 43–64, 2004.     

具有循环极大子群的亚循环p-群的自同构群

利用半直积和群扩张的理论,给出了具有极大循环子群的亚循环p-群的自同构群.     

The Farrell–Jones Conjecture for the solvable Baumslag–Solitar groups

In this paper, we prove the Farrell–Jones Conjecture for the solvable Baumslag–Solitar groups with coefficients in an additive category. We also extend our results to groups of the form, Z[1/p] semidirect product with any virtually cyclic group, where p is a prime number.     

Finite separable metacyclic 2-groups

A finite separable metacyclic 2-group G can be written as the semidirect product of a cyclic group with another cyclic group. Necessary and sufficient conditions are given for when all other split decompositions of G result in, up to isomorphism, this same semidirect product representation for G.     

Finite groups embeddable in division rings

In a tour de force in 1955, S. A. Amitsur classified all finite groups that are embeddable in division rings. In particular, he disproved a conjecture of Herstein which stated that odd-order emdeddable groups were cyclic. The smallest counterexample turned out to be a group of order 63. In this note, we prove a non-embedding result for a class of metacyclic groups, and present an alternative approach to a part of Amitsur's results, with an eye to ``de-mystifying" the order 63 counterexample.

     

Towards Practical Non-Interactive Public-Key Cryptosystems Using Non-Maximal Imaginary Quadratic Orders

We present a new non-interactive public-key distribution system based on the class group of a non-maximal imaginary quadratic order Cl( _p ). The main advantage of our system over earlier proposals based on (Z/nZ)* [25,27] is that embedding id information into group elements in a cyclic subgroup of the class group is easy (straight-forward embedding into prime ideals suffices) and secure, since the entire class group is cyclic with very high probability. Computational results demonstrate that a key generation center (KGC) with modest computational resources can set up a key distribution system using reasonably secure public system parameters. In order to compute discrete logarithms in the class group, the KGC needs to know the prime factorization of _p =₁ p ². We present an algorithm for computing discrete logarithms in Cl( _p ) by reducing the problem to computing discrete logarithms in Cl(₁) and either F* _p or F* _p ². Our algorithm is a specific case of the more general algorithm used in the setting of ray class groups [5]. We prove—for arbitrary non-maximal orders—that this reduction to discrete logarithms in the maximal order and a small number of finite fields has polynomial complexity if the factorization of the conductor is known.     

Ring Isomorphism of Cyclic Cyclotomic Algebras

It is shown that ring isomorphisms between cyclic cyclotomic algebras over cyclotomic number fields are essentially determined by the list of local Schur indices at all rational primes. As a consequence, ring isomorphisms between simple components of the rational group algebras of finite metacyclic groups are determined by the center, the dimension over ?, and the list of local Schur indices at rational primes. An example is given to show that this does not hold for finite groups in general.     

具有指数为2的循环子群的亚循环群

我们给出了具有指数为2的循环子群的亚循环群的同构分类.这个结果可应用于正则地图的研究中.