具有较少顶点个数的共轭类长图
发布时间：2018-08-18 19:56
【摘要】：共轭类与有限群的结构在群论的研究中有着重要的地位,在过去的几十年中比较活跃,并取得了很多成果。本文主要利用共轭类长图来研究有限群的结构。共轭类长图T(G)是满足下面两个条件的无向图：(1)以群的非中心共轭类长的集合Cl(G)中的元素为顶点；(2)如果两个顶点|Gi|和|Cj|之间有一条边相连,当且仅当(|Ci|,|Cj|)1。利用共轭类长图的顶点和边的个数可以对有限群进行分类。 在本文首先给出了顶点个数最多为4的图,共有18个图。根据群的共轭类长图的定义以及共轭类长具有的一些性质,得到18个图中只有8个图可以作为有限群的共轭类长图。对于这8个图,得到了一些结果。若图为一孤立点,则群只有一个非中心共轭类长,此时群为其Sylow p子群与交换群的直积；若图为两个孤立点,则群为拟Frobenius群。 通过GAP程序计算出群阶在100以内的群的共轭类长以及对应群的结构,利用共轭类长图给出了100阶以内的有限群的一个分类。 共轭类长图是以一个群所有的共轭类长为研究对象来进行考察的,而在研究过程中发现当共轭类长满足一定算数条件时也能得到一些有意义的结果。文中第四章定义了共轭类长的平方整除群阶的群,即SCLD群,得到结论：(1)有限交换群为SCLD群；(2)单群、几乎单群及Frobenius群不是SCLD群；(3)幂零群为SCLD群当且仅当它的Sylow p子群均为SCLD群。
[Abstract]:The structure of conjugate classes and finite groups plays an important role in the study of group theory and has been active in the past few decades and many achievements have been made. In this paper, the structure of finite groups is studied by using conjugate class graph. The conjugate class length graph T (G) is an undirected graph satisfying the following two conditions: (1) taking the elements in the set Cl (G) of the group's noncentral conjugate class length as vertices; (2) if there is an edge connected between the two vertices, if and only if (ci, Cj) 1. Finite groups can be classified by the number of vertices and edges of conjugate class graphs. In this paper, we first give a graph with the number of vertices up to 4, a total of 18 graphs. According to the definition of conjugate class length graph of group and some properties of conjugate class length, only 8 of 18 graphs can be regarded as conjugate class length graph of finite group. For these eight graphs, some results are obtained. If the graph is an isolated point, then the group has only one noncentral conjugate class length, where the group is the direct product of its Sylow p subgroup and abelian group, and if the graph is two isolated points, then the group is quasi Frobenius group. The conjugate class length and the structure of corresponding group are calculated by GAP program, and a classification of finite groups of order 100 is given by using conjugate class length graph. The conjugate class length graph is investigated by taking the conjugate class length of a group as the object of study, and it is found that some meaningful results can be obtained when the conjugate class length satisfies certain arithmetic conditions. In the fourth chapter, we define the group of square division group of conjugate class length, that is, SCLD group. The conclusions are as follows: (1) finite abelian group is SCLD group, (2) simple group, almost simple group and Frobenius group are not SCLD group; (3) Nilpotent groups are SCLD groups if and only if their Sylow p subgroups are SCLD groups.
【学位授予单位】：沈阳工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O152.1;O157.5
本文编号：2190520
[Abstract]:The structure of conjugate classes and finite groups plays an important role in the study of group theory and has been active in the past few decades and many achievements have been made. In this paper, the structure of finite groups is studied by using conjugate class graph. The conjugate class length graph T (G) is an undirected graph satisfying the following two conditions: (1) taking the elements in the set Cl (G) of the group's noncentral conjugate class length as vertices; (2) if there is an edge connected between the two vertices, if and only if (ci, Cj) 1. Finite groups can be classified by the number of vertices and edges of conjugate class graphs. In this paper, we first give a graph with the number of vertices up to 4, a total of 18 graphs. According to the definition of conjugate class length graph of group and some properties of conjugate class length, only 8 of 18 graphs can be regarded as conjugate class length graph of finite group. For these eight graphs, some results are obtained. If the graph is an isolated point, then the group has only one noncentral conjugate class length, where the group is the direct product of its Sylow p subgroup and abelian group, and if the graph is two isolated points, then the group is quasi Frobenius group. The conjugate class length and the structure of corresponding group are calculated by GAP program, and a classification of finite groups of order 100 is given by using conjugate class length graph. The conjugate class length graph is investigated by taking the conjugate class length of a group as the object of study, and it is found that some meaningful results can be obtained when the conjugate class length satisfies certain arithmetic conditions. In the fourth chapter, we define the group of square division group of conjugate class length, that is, SCLD group. The conclusions are as follows: (1) finite abelian group is SCLD group, (2) simple group, almost simple group and Frobenius group are not SCLD group; (3) Nilpotent groups are SCLD groups if and only if their Sylow p subgroups are SCLD groups.
【学位授予单位】：沈阳工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O152.1;O157.5
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