二面体群上构作的结合方案（英文）

本文研究了二面体群的元素的等价划分问题.利用群在集合上的作用,在二面体群上构造了一类新的结合方案,并且计算了这类结合方案的所有参数.进一步,得到了一类强正则图.所得到的结果丰富了结合方案理论.     

Cayley图的同构分解及弱DCI-子集的充要条件

本文得到了Ｃａｙｌｅｙ图的同构分解定理及弱ＤＣＩ－子集的充要条件．证明了二面体群是弱２－ＤＣＩ－群，同时确定了二面体群上２度Ｃａｙｌｅｙ图的自同构群．     

具有2p阶二面体群次成份的本原群

本文确定了具有同构于2p 阶二面体群的极大子群的全部有限单群,进而得到了对于具有素数次二面体群次成份的本原置换群的一个完全分类.     

交错群A_5与二面体群D_6直积的整群环的挠单位

本文利用Luthar-Passi方法,研究了五次交错群A_5与六阶二面体群D_6直积的整群环的挠单位,得到了该群的Zassenhaus猜想成立.     

一类关于两个子群的相对差集

本文在一般有限群 G=G_1×G_2上,利用它的两个子群 G_1、G_2构作了一类新的相对差集,给出了一些简单性质(第一节),得到了由群差集构造这类相对差集的一般方法(第二节)。证明了由这类相对差集可以得到一类相应的,具有3个结合类的 PBIB 设计(第三节)。最后给出了二个构造性结果(第四节)。     

一类拟二面体群的四度Cayley图的正规性

群G的一个Cayley图X=Cay(G,S)称为正规的,如果右乘变换群R(G)在AutX中正规.研究了4m阶拟二面体群G=a,b|a~(2m)=b~2=1,a~b=a~(m+1)的4度Cayley图的正规性,其中m=2~r,且r2,并得到拟二面体群的Cayley图的同构类型.     

有限群的广义交换度

给出了有限群广义交换度的一些性质.某种意义下,广义交换度测度了所有子群正规化子阶的平均值.通过考虑一类内交换群,探索了有限群的幂零性(可解性)与其广义交换度之间的关系.对于二面体群及含有循环极大子群的P-群,给出了其广义交换度的详细公式.     

4p阶二面体群的弱3-CI性

决定了4p(p是奇素数)阶二面体群的连通3度Cayley图的完全分类,并证明4p阶二面体群不是弱3-CI群,从而否定了C.H.Li关于"所有有限群都是弱3-CI群"的猜想     

伪辛空间F_q~(2v+2+l)中一类2-维子空间的结合方案及其结构

该文利用伪辛空间Fq(2v+2+l)中一类2-维非迷向子空间构作了具有2q-1个结合类的交换 的但非对称的结合方案,并且讨论了它的结构,证明了它是其基础域上的加法群和乘法群上的 熟知的结合方案的扩张.     

伪辛空间F^(2v+2+l)_q中一类2维子空间的结合方案及其结构

该文利用伪辛空间F\-q\+\{(2v+2+l)中一类2 维非迷向子空间构作了具有2q-1个结合类的交换的但非对称的结合方案，并且讨论了它的结构，证明了它是其基础域上的加法群和乘法群上的熟知的结合方案的扩张。     

Linked systems of symmetric group divisible designs

We introduce the concept of linked systems of symmetric group divisible designs. The connection with association schemes is established, and as a consequence we obtain an upper bound on the number of symmetric group divisible designs which are linked. Several examples of linked systems of symmetric group divisible designs are provided.     

Paley type group schemes and planar Dembowski–Ostrom polynomials

In this paper we give some necessary and sufficient conditions for Dembowski–Ostrom polynomials to be planar. These conditions give a simple explanation of the Coulter–Matthews and Ding–Yin commutative semifields and enable us to obtain permutation polynomials from some of the Zha–Kyureghyan–Wang commutative semifields. We then give a generalization of Feng’s construction of Paley type group schemes in extra-special p-groups of exponent p and construct a family of Paley type group schemes in what we call the flag groups of finite fields. We also determine the strong multiplier groups of these group schemes. In the last section of this paper, we give a straightforward generalization of the twin prime power construction of difference sets to a construction of Hadamard designs from twin Paley type association schemes.     

Classification of association schemes with 18 and 19 vertices

The isomorphism classes of association schemes with 18 and 19 vertices are classified. We obtain 95 isomorphism classes of association schemes with 18 vertices and denote the representatives of the isomorphism classes as subschemes of 7 association schemes. We obtain 7 isomorphism classes of association schemes with 19 vertices and six of them are cyclotomic schemes.     

Schur rings and non-symmetric association schemes on 64 vertices

In this paper we enumerate essentially all non-symmetric association schemes with three classes, less than 96 vertices and with a regular group of automorphisms. The enumeration is based on a computer search in Schur rings. The most interesting cases have 64 vertices.In one primitive case and in one imprimitive case where no association scheme was previously known we find several new association schemes. In one other imprimitive case with 64 vertices we find association schemes with an automorphism group of rank 4, which was previously assumed not to be possible.     

On class dimension of flat association schemes in affine and affine-symplectic spaces

In this paper, we obtain upper bounds of the class dimension of flat association schemes in affine and affine-symplectic spaces and construct resolving sets for these schemes.     

Association schemes based on attenuated spaces

The subspaces of a given dimension in an attenuated space form the points of an association scheme. If the dimension is maximal, it is the bilinear forms graph, which has been thoroughly studied. In this paper, we discuss the general case, and obtain a family of symmetric association schemes which are a common generalization of the Grassmann schemes and the bilinear forms schemes. Moreover, all the parameters of the association scheme are computed.     

A Family of Partial Geometric Designs from Three‐Class Association Schemes

In this paper, we show that partial geometric designs can be constructed from certain three‐class association schemes and ternary linear codes with dual distance three. In particular, we obtain a family of partial geometric designs from the three‐class association schemes introduced by Kageyama, Saha, and Das in their article [“Reduction of the number of associate classes of hypercubic association schemes,” Ann Inst Statist Math 30 (1978)]. We also give a list of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.     

Characters of table algebras and applications to association schemes

Hanaki [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195-201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363-369] generalized many properties of characters of finite groups to characters of association schemes. In this paper we show that many of these properties also hold for table algebras. Our approach is not to generalize the proofs in [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195-201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363-369] to table algebras, but to prove many stronger properties, and then obtain results in [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195-201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363-369] as direct consequences.     

二面体群D_2上Hopf路余代数的结构分类

本文研究了Hopf代数的构造问题.利用模范畴和箭图,获得了当G是二面体群D2这一4阶交换群时的Hopf路余代数kQc的同构分类及其子Hopf代数kG[kQ1]的结构,推广了当G是2阶循环群时的相应结论.     

On a Class of Non-commutative Imprimitive Association Schemes of Rank 6

Many facts about group theory can be generalized to the context of the theory of association schemes. In particular, association schemes with fewer than 6 elements are all commutative. While there is a nonabelian group with 6 elements which is unique up to isomorphism, there are infinitely many isomorphism classes of non-commutative association schemes with 6 elements. All examples previously known to us are imprimitive, and fall into three classes which are reasonably well understood. In this paper, we construct a fourth class of noncommutative, imprimitive association schemes of rank 6.