【作者】 王磊；

【导师】 李才恒；

【作者基本信息】 云南大学 ， 基础数学， 2015， 博士

【摘要】 本文主要研究了几类Frobenius群的全自同构群的结构,刻画了两类相关正规边传递Cayley图.Frobenius群是一类极为重要的群,其本身具有很强的性质,在有限群的特征标理论与群的结构理论中均扮演重要的角色.第一章是绪论部分,主要介绍Frobenius群的相关背景知识和现状,以及本文将要研究的问题.第二章主要介绍了本文所要用到的一些有关群,表示论及图的基本概念及相关定理,性质.为了更好地研究一类正规边传递Cayley图,第三章提出了相对初等交换群(简称为REA群)的概念,并给出了REA群的相关性质.应用这些性质,给出了完全多部图为正规边传递Cayley图的一个充分条件；同时,分析了幂零群,Frobenius群与REA群的关系.第四章继续对REA群的性质进行了研究：将对REA群可解性的研究转化为对REA群为几乎单群的研究,分析了几乎单群的无不动点的自同构,进而得到REA群一定是可解群的结论.群的全自同构群的结构是随着代数学的发展所提出的课题之一,其研究在有限群论中占有至关重要的地位.第五章主要刻画了Frobenius群(Πik=1 Cpidi):Cn的全自同构群,研究发现,k=1和k≥2时Frobenius群的全自同构群有些许不同.进而,我们刻画了一类Frobenius REA群.在第六章,我们给出了Frobenius群为REA群的充分必要条件,在此基础上,分别对Frobenius补为Cn:C2f,Cn:C3f,Cn:Q2f的Frobenius REA群进行了研究,这在某种程度上是对第三章结论的补充与完善.Frobenius补作为Frobenius群的一个重要组成部分,具有深刻的研究意义.有关学者已经得到了Frobenius补的一些比较好的性质A. I. Starostin把Frobenius补分成了六类.在此基础上,本文第七章对其中的四类可解Frobenius补的结构进行了细致分析,从而得到一些方便我们使用的群类.作为此结论的应用,我们构造了Frobenius核为初等交换群的本原Frobenius群,推广了已有的结果.此外,把群与图结合起来,利用群来研究图的结构也是本文的研究重点之一.基于前几章对Frobenius群的全自同构群的研究,本文第八章对Frobe-nius群上的4度边传递Cayley图进行了刻画.更多还原

【Abstract】 This paper mainly investigates the full automorphism groups of sev-eral Frobenius groups and characterizes two kinds of related normal edge-transitive Cay ley graphs.Frobenius groups are essential in group theory and have strong proper-ties, which play an important role in the character theory of finite groups and have great influences on the structure of finite groups. Chapter Ⅰ is an intro-duction part, it involves the background and current situations of Frobenius groups, and the problems which we will study.Chapter Ⅱ are preliminaries for this dissertation, which introduce some basic definitions, related theorems and properties of groups, representations and graphs.In Chapter Ⅲ, we present the notion of relative elementary abelian group (or REA group for short), and reach the relevant properties of the REA groups. By the definition of the REA groups, we obtain a sufficient condition on which the complete multipartite graphs are normal edge-transitive Cay-ley graphs. Meanwhile, we also analyze the relationship between nilpotent groups, Frobenius groups and REA groups.Chapter Ⅳ continues to study the properties of the REA groups. The research on the solvability of the REA groups can be transformed into the study of whether the almost simple groups are the REA groups, we analyze the fixed-point-free automorphisms of the almost simple groups, and conclude that every REA group is solvable.As is known to all, the study of the full automorphism group of a group is one of the topics presented along with the development of the algebra. Usually, the value of existence of a group is reflected by its action and full automorphism group, which, therefore, plays an important role in finite group theory.Chapter V mainly characterizes the full automorphism group of Frobe-nius group (Πik=1 Cpidi):Cn. In the research, we find that the full automorphism groups are a slightly different between k= 1 and k≥ 2. Moreover, we also characterise a class of Frobenius REA groups.In Chapter VI, we obtain a necessary and sufficient condition on which Frobenius groups are REA groups. Based on the results we have obtained, we study a kind of Frobenius REA groups, which the Frobenius complement is Cn:C2f, Cn:C3f or Cn:Q2f, respectively. This is, to some extent, a supplement and perfection of the conclusions of Chapter Ⅲ.As an important part of Frobenius groups, Frobenius complement also has profound research significance, and scholars have acquired some nice properties about Frobenius complement. For example, A. I. Starostin has divided Frobenius complement into six kinds of groups. Then, Chapter Ⅶ of this thesis analyzes its four types of solvable Frobenius complements in detail, and obtains eight class of groups which can be used conveniently later on. As an application of these conclusions, we also construct several classes of primitive Frobenius groups, which generalized the existing results.In addition, combining groups with graphs together, using group theory to study the structure of graphs is one of the key research in this paper. Based on the study of full automorphism groups of Frobenius groups of the previ-ous chapters, Chapter Ⅷ characterizes tetravalent edge-transitive Cayley graphs of Frobenius group Cp:Cn.更多还原