2pq阶Cayley图是Hamilton图

一、引言对Cayley图的Hamilton性的研究近几年有所突破[1]现最好的结果是[2]的主要定理:若群G上的换位子群C′是p~n(p是素数,n是正整数)阶循环群时,G上的每个Cayley图皆为Hamilton图。1987年D.Marusic还证明了2p~2(p是素数)阶Cayley图为Hamilton图[4]。本文用群的构造理论证明:2pq(p,q是素数)阶Cayley图是Hamilton图。本文中所提到的群G皆指有限群;群的有关术语和记号同于文献[3];图的有关术     

奇数阶亚循环p-群分类的一个证明

一个有限p群p称为亚循环的,如果P有一个循环的正规子群A,使得PA是循环的.1973年King在文[2]中对亚循环p群进行了分类;在文[4,5]中M.F.NewmanandMingYaoXu发现了这些群的新的表示,给出了新的分类方法,这种方法是由p群生成算法(见文[3])得到的.本文的目的是给这些结果的另一种证明,与p群生成算法是不相关的.     

能表示成四个真子群并的有限群

本文研究了子群覆盖问题.利用计算子群阶的方法,给出了能表示成p+1或p+2个真子群并的有限群G的结构,这里p是群G阶的最小素因子,从而推广了文献[1]中的结果.     

多元算子群的根

多元算子群（简称Ω－群）的概念见[1]或[2]。本文在Ω－群中引进R－根的概念，并对R（G）－半单Ω－群作了刻划，从而把[4]与[5]中关于环和模的结果推广到多元算子群中。     

基于MSP秘密共享的（t，n）门限群签名方案

门限群签名是群签名中重要的—类，它是秘钥共享与群签名的有机结合．本文通过文献[5]中的MSP方案（Monotone Span Program），提出了一种新的门限群签名方案．在本签名方案建立后，只有达到门限的群成员的联合才能生成—个有效的群签名，并且可以方便的加入或删除成员．一旦发生争议，只有群管理员才能确定签名人的身份．该方案能够抵抗合谋攻击：即群中任意一组成员合谋都无法恢复群秘钥k．本方案的安全性基于Gap Diffie-Hellman群上的计算Diffie-Hellmanl可题难解上，因此在计算上是最安全的．     

半正规n-极大子群对有限群结构的影响

设△↓n(G）为有限群G的n次极大子群的全体。1．若△↓4（G）中的子群均在G中半正规,则下述结论之一成立：（1）G是可解群;（2）G／φ（G）＝A5,（3）G/φ(G)=PSL(2,13);(4)G／φ（G）＝PSL（2,p),满足p=4p1 1=6p2-1,这里p1≥43,p2≥29;(5)G/φ(G)=PSL(2,p),满足p=6p1 1=4p2-1,这里p1≥7,p2≥11.2。2．设3不属于π（G）,若△↓（G）中的子群均在G中半正规,则G是可解群,或G／φ(G)=Sz(2^3).     

Steenrod运算和同伦群(Ⅰ)

<正> 设 n 为一大于1的整数,p 为一奇素数,当 n相似文献   

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．     

最高阶元个数是4p的有限群

本文证明了如果有限群G恰有4p个最高阶元,p为素数,则群G为可解群,除非G同构于S5.     

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.     

OD-CHARACTERIZATION OF ALMOST SIMPLE GROUPS RELATED TO U6（2）

Let G be a finite group and π(G) = { p 1 , p 2 , ··· , p k } be the set of the primes dividing the order of G. We define its prime graph Γ(G) as follows. The vertex set of this graph is π(G), and two distinct vertices p, q are joined by an edge if and only if pq ∈π e (G). In this case, we write p ～ q. For p ∈π(G), put deg(p) := |{ q ∈π(G) | p ～ q }| , which is called the degree of p. We also define D(G) := (deg(p 1 ), deg(p 2 ), ··· , deg(p k )), where p 1 < p 2 < ··· < p k , which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L := U 6 (2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S 3 is 5-fold OD-characterizable.     

Distance Integral Complete Multipartite Graphs with s=5, 6

Let D(G) = (dij )n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vertices vi and vj in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a su?cient and necessary condition for complete r-partite graphs Kp1,p2,··· ,pr =Ka1·p1,a2·p2,··· ,as···ps to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs Ka1·p1,a2·p2,··· ,as·ps with s>4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with s = 5, 6. The problem of the existence of such distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with arbitrarily large number s remains open.     

有限秩的幂零群的自同构(Ⅱ)

设幂零群G=KP=PK,其中P是有限秩的幂零p-群,K是G的有限秩的p-自由的正规子群,p不属于K的谱Sp(K).设1=ζ0Gζ1G···ζcG=G是G的上中心列,α和β是G的两个p-自同构,把α,β在每个ζiG/ζi-1G上的诱导自同构分别记为αi和βi,又记Ii:=Im(αiβi-βiαi),则(i)如果每个Ii都是有限循环群,并且I:=(αβ(g))(βα(g))-1|g∈G是G的有限子群,那么α和β生成一个有限p-群;(ii)如果Ii或为有限循环群,或为拟循环p-群,或为Zpn⊕Zp∞对某自然数n,那么α和β生成一个可解的剩余有限p-群,它是有限生成的无挠幂零群被有限p-群的扩张;(iii)如果Ii或为有限循环群,或为拟循环p-群,或为Zpn⊕Zp∞,或为无挠的局部幂零群,或Ii有正规列1JiIi,其商因子分别为有限循环群、无挠的局部幂零群,或Ii=Zp∞⊕Ji,Ji为无挠的局部幂零群,或Ii有正规列1KiJiIi,其商因子分别为有限循环群、拟循环p-群、无挠的局部循环群,那么α和β生成一个可解的剩余有限p-群,它的幂零长度至多是3.特别地,当K是一个FC-群时,在情形(iii),α和β生成的群也是有限生成的无挠幂零群被有限p-群的扩张.此外,如果G=KP里,K是一个FC-群,对G的下中心列考虑了类似的问题,得到了"对偶"的结果.     

Covering the Hecke Triangle Surfaces

In the mid-1980s an equivalence was established between the simple closed geodesics on the Riemann surfaces obtained as quotients of the upper half plane H by any of the following subgroups of the modular group (1) : , (3), and ³. An axis of a hyperbolic element of (1) projects to a simple closed geodesic on one of these surfaces if and only if it does so on the other two.This equivalence was used to obtain a variety of Diophantine and geometric results. In subsequent related investigations, the role of (1) was assumed by the Hecke triangle group G_q for q 3. (For q = 3, we have (1) = G₃.) These works employed the analog of ³, denoted _q.In the context of the G_q, the present paper gives the analog of , which we denote _q. As in the case q = 3, we have [_q:_q] = 2. A rather full discussion of geometry of _q\ H is given. In particular, we demonstrate that the equivalence of simple closed geodesics on _q\ H and _q\ H does not hold for q 7.As of this writing, we have not been able to obtain an appropriate analog of (3).     

Locally soluble infinite-dimensional linear groups with restrictions on nonAbelian subgroups of infinite ranks

We are concerned with locally soluble linear groups of infinite central dimension and infinite sectional p-rank, p ⩾ 0, in which every proper non-Abelian subgroup of infinite sectional p-rank has finite central dimension. It is proved that such groups are soluble. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 601–616, September–October, 2008.     

Wreath Hurwitz numbers, colored cut-and-join equations, and 2-Toda hierarchy

Let G be arbitrary finite group,define H G· (t;p +,p) to be the generating function of G-wreath double Hurwitz numbers.We prove that H G· (t;p +,p) satisfies a differential equation called the colored cutand-join equation.Furthermore,H G·(t;p +,p) is a product of several copies of tau functions of the 2-Toda hierarchy,in independent variables.These generalize the corresponding results for ordinary Hurwitz numbers.     

关于内7-闭单群的结构

研究内ｐ－闭群的结构是一个很活跃的课题．对于ｐ＝２，３，５的内ｐ－闭群的结构已经被确定（见［１，２，３］）．本文确定内７－闭单群的结构     

A Poincar′e Inequality in a Sobolev Space with a Variable Exponent

Let Ω be a domain in RN. It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space W1,p(·)(Ω), where p(·) :(-Ω)→ [1,∞[ is a variable exponent. This inequality is itself a corollary to a more general result about equivalent norms over such cones. The approach in this paper avoids the difficulty arising from the possible lack of density of the space D(Ω) in the space {v ∈ W1,p(·)(Ω);tr v= 0 on aΩ}. Two applications are also discussed.     

Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let C be an Abelian group. An Abelian group A in some class of Abelian groups is said to be _C H-definable in the class if, for any group B\in , it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _C H-definable in , then the class is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.     

6p阶可解对称图的分类

设ｘ是简单无向图，Ｇ是Ａｕｔ（Ｘ）的一个于群，Ｘ称为Ｇ－对称的，如果Ｇ在ｘ的１－孤（即两相邻顶点构成的有序偶）集合上的作用是传递的；ｘ称为对称图，如果Ｘ是Ａｕｔ（ｘ）－对称的；ｘ称为可解对称的，如果Ａｕｔ（Ｘ）包含可解子群Ｇ，使Ｘ是Ｇ－对称的．本文给出了具有６Ｐ个顶点的可解对称图的一个分类，这里ｐ≥５是素数．