应用群论研究无平方因子阶2-弧传递图的对称性

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups and the two theories have influenced each other. This project is a contribution to the application of group theory to graph theory. The application mainly investigates the graph symmetries, which can be described by the action of automorphism group on the different objects such as vertices, edges, arcs and so on. The study of arc-transitive graphs has recently been an active topic in algebraic theory. There are two major branches to investigate the arc-transitive graphs. One branch is to investigate graphs with a higher degree of symmetry such as distance transitive graphs and s-arc transitive graphs with s no less than 2. The other is to investigate symmetric graphs on various restrictions such as small valency or (and) certain order. For the branch to investigate graphs with a higher degree of symmetry, the almost perfect achievement is the classification of the finite distance transitive graphs. In this subject, we will investigate s-arc transitive graphs with s no less than 2. The study of this kind of graphs was initiated by Tutte in 1947, and since then, the study of finite s-arc transitive graphs has received considerable attention in lots of literatures. In particular, the classification and charcterization of such graphs has been closely watched, which is recognized hard but of great theoretical meaning. It is obvious that every finite G-vertex-transitive, (G, s)-arc transitive graphs of valency at least 3 with s no less than 2 are also (G, 2)-arc transitive. Thus we will focus on 2-arc transitive graphs in this project. Further, the research on graphs with a higher degree of symmetry was mainly analyzing the local properties of graphs, as an example of the structure of stabilizer sequence. Therefore, we also study the local properties of 2-arc transitive graphs. In the literature, the problem of characterizing and classifying edge-transitive graphs of square-free order has received considerable attentions, and so does the vertex-transitive case. The aim of this project is to give a complete classification of 2-arc transitive graphs of square-free order and analyze their local properties. We will also investigate the tructure of their automorphism groups and their stabilizers. The research results of this project are meaning for the study of symmetries of graphs and the permutation group theory, also they can provide effective model in the application.

用代数的理论和方法对组合结构以及图的对称性质进行研究是代数组合论中最重要的分支之一。 近年来，群与图已经成为国际学术界一个非常活跃的研究领域，其主要研究对象是点、边传递图和对称图等具有较高传递性质的图类。本项目旨在结合群论和组合方法来考察无平方因子阶的2-弧传递图，侧重于该类图的刻画与分类以及其自同构群和点稳定子群的结构。本项目将首先分析2-传递置换群的非可解合成因子的分布情况，然后构造稳定子群和自同构群之间的一个关于合成因子的子群链，进而递归地得到它们的精确结构，从而构造出2－弧传递图并进行刻画和分类。本项目预期将给出无平方因子阶2-弧传递图一个完全分类。本项目对图的对称性研究及置换群理论的完善都有重要的意义。

群与图的主要研究对象是具有某种传递性的图类，给出其完全分类或者刻画其全自同构群的结构。该项目首先利用GAP软件和组合中的整数分拆得到无平方因子数的一些特征，然后通过对子群在集合上的作用是否传递和在集合是否有不动点等情况分别讨论，得到了基柱为交错群的几乎单群的群论刻画。在上述群论刻画的基础上，将其划分为三类即特殊群类、 子群在集合上传递、子群在集合上至少有两个轨道(此时通过分析子群的不可解合成因子)，分别来讨论其所可能对应的2-弧传递图，得到了无平方因子阶的基柱为交错群的几乎单群所对应的2-弧传递图的分类。此外，还给出了几乎单群对应的无平方因子阶2-弧正则图的分类和无平方因子阶三度点传递图的分类。 本项目的研究基本上按照预期的计划进行，基本上解决了项目申请书中提出的两个问题。除此之外，还考察了具有更高对称性的图，即无平方因子阶的2-弧正则图和高弧正则图。 但是由于时间关系，对于高弧正则图还没有得到很好的结果。