关于有限特殊类群对合交换图的研究

发布时间：2018-06-11 13:36

本文选题：对合交换图 + k-正则图　；参考：《湖北民族学院》2017年硕士论文

【摘要】：对合交换图是以群的一个二阶元共轭类为顶点,两顶点有边当且仅当它们交换的图.本文首先研究了对合交换图为0-正则和1-正则图时群结构的相关问题.其次,本文讨论了一些特殊类群对合交换图的结构.全文由四章组成.第1章介绍了对合交换图的研究背景、相关的定义、全文涉及的基本理论及基本符号.第2章讨论了0-正则和1-正则对合交换图一些相关的结构、性质,并给出了对合交换图为0-正则图的一个充要条件.主要证明了若有限群G的对合交换图)(IG(38)为0-正则图,并且G(28)?I?,则G有一个指数为2的奇阶子群.最后,我们给出了对合交换图为1-正则图的若干群例及两个猜想.第3章研讨了亚循环2-群对合交换图的结构,主要证明了亚循环2-群的对合交换图为0-正则或1-正则图.第4章讨论了亚交换群对合交换图的结构,得出对于交换群A和B,若G(28)AfBExt),;,(?,?),(a为G中任意二阶元,|{??)((10)-(28)bbH Ab?}为A的子群,则G关于?),(a的对合交换图是r)1(--正则图,其中r为所有陪集?)(),(),(??????-(10)-(10)aaffH,??B?中二阶元的个数.最后,我们给出了一个具体计算亚交换群对合交换图的例子,即28?(28)ZZA,22?(28)ZZB,且G(28)AfBExt),;,(?,计算出所有的对合交换图的度,并得出G的对合交换图为0-正则、1-正则或2-正则图.
[Abstract]:Involutive commutative graph is a class of second order conjugate of group as vertex, two vertices have edge if and only if they commutate. In this paper, we first study the problem of group structure when involutive commutative graphs are 0-regular and 1-regular. Secondly, we discuss the structure of some special class groups involutive commutative graphs. The full text consists of four chapters. Chapter 1 introduces the background, definition, basic theory and symbol of involutive commutative graph. In chapter 2, we discuss some related structures and properties of 0-regular and 1-regular involutive commutative graphs, and give a necessary and sufficient condition for involutive commutative graphs to be 0-regular graphs. It is proved that G has an odd order subgroup if the involutive commutative graph of finite group G is a 0-regular graph and G ~ (2) is an odd order subgroup. Finally, we give some group examples and two conjectures that involutive commutative graphs are 1-regular graphs. In chapter 3, we discuss the structure of subcyclic 2-group involutive commutative graphs, and prove that the involutive commutative graphs of subcyclic 2-groups are 0-regular or 1-regular. In chapter 4, we discuss the structure of the involutive commutative graphs of subabelian groups A and B, and obtain that for the abelian groups A and B, if G is an arbitrary second order element in G, and G is a subgroup of A, then G's involutive commutative graph about A is a rtl ~ (1) -regular graph, where r is all the guest set ~ (-10) -10 ~ (aaffHB) ~ (?) _ B _ (?) _ _ _ The number of middle and second order elements. Finally, we give an example of calculating the involutive commutative graph of subabelian group, that is, 28AZA22 / 28ZZB, and GX28 / AfBExtl, calculate the degree of all involutive commutative graphs, and obtain that the involutive commutative graph of G is 0-regular 1-regular or 2-regular graph, and that the involutive commutative graph of G is 0-regular 1-regular or 2-regular graph, and the involutive commutative graph of G is 0-regular 1-regular or 2-regular.
【学位授予单位】：湖北民族学院
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O152.1

【参考文献】