五次自由阶拟本原置换群及相关三度对称图

发布时间：2018-04-24 05:24

本文选题：五次自由阶 + 单群　；参考：《云南财经大学》2017年硕士论文

【摘要】：这篇论文旨在研究五次自由阶拟本原置换群及其关联的三度对称图(注:一个置换群G称为d次自由阶的置换群,其中d为正整数,如果不存在素数P,使得pd整除|G|).著名的Cayley定理告诉我们:任何一个有限群都与一个置换群(其右正则表示)同构.这一定理奠定了置换群在现代数学中的重要地位.近年来,基于O'Nan-Scott-Praeger定理的发现(见[30])和Liebeck, Praeger, Saxl得到的有限单群的极大因子分解(见[24]),置换群论的研究得到了快速的发展,并且在许多领域(尤其在代数图论领域)发现了很好的应用.2005年,Dietrich和Erick [8]研究了立方自由阶群的性质,利用其结果,Li和Qiao[35]完全确定了立方自由阶群的结构,进而在后续文章[21]得到了四次自由阶群的一些好的结构性质.特别地,他们确定了所有立方自由阶和四次自由阶的非交换单群.本文的主要目的之一是研究五次自由阶群的性质,特别地,得到五次自由阶拟本原置换群的分类,并确定所有五次自由阶非交换单群.注:一个置换群G ≤ Sym(Ω))称为拟本原置换群如果其每个极小正规子群在腕上都是传递的.1938年,Fruchet证明了任何一个置换群都是一个图的自同构群,从而建立了置换群论和图论间的密切联系.本文的第二个工作是利用所得的五次自由阶拟本原置换群的分类结果去分类具有顶点本原五次自由阶自同构群的所有三度弧传递图r.具体而言,我们证明了下述之一成立:r≌K_4,K_4 - 4K_2或O_2 ( Petersen图),或者r为PSL(2,p)的陪集图,其中p≡±1 (mod 16)为素数.
[Abstract]:This paper aims to study the three degree Symmetric Graphs of the five free order quasi primitive permutation groups and their associations (Note: a replacement group G is called the replacement group of the D order of freedom, where D is a positive integer, if there is no prime number P, so that PD is divided into |G|). The famous Cayley theorem tells us what a finite group is all with a replacement group (its right regular representation). Isomorphism, which establishes the important position of displacement groups in modern mathematics. In recent years, the research on the maximal factorization of finite groups (see [24]) based on the discovery of the O'Nan-Scott-Praeger theorem (see [30]) and Liebeck, Praeger and Saxl (see [24]) has developed rapidly, and in many fields (especially in algebraic graph theory) Domain) found a good application in.2005 years. Dietrich and Erick [8] studied the properties of cubic free order groups. Using the results, Li and Qiao[35] completely determined the structure of the cubic free order group. Then, in the subsequent article [21], some good structural properties of the four order free order groups were obtained. In particular, they determined all cubic free order and four. One of the main purposes of this paper is to study the properties of the five order free order group. In particular, we obtain the classification of the five order quasi primitive permutation groups and determine all five free order noncommutative groups. Note: a replacement group G < Sym (omega)) is called the quasi primitive permutation group if each minimal normal subgroup is on the wrist. In all the.1938 years of transfer, Fruchet proves that any substitution group is an automorphism group of a graph, and thus establishes the close relation between the permutation group theory and the graph theory. The second work of this paper is to classify the five free order primitive permutation groups by using the results to classify the groups with the vertex primitive five order free order automorphism groups With three degrees of arc transitive graph R., we prove that one of the following is established: R K_4, K_4 - 4K_2 or O_2 (Petersen graph), or R is a coset graph of PSL (2, P), in which p + 1 (MOD 16) is a prime.

【学位授予单位】：云南财经大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5

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