Full Automorphism Groups Of Frobenius Groups And Related Normal Edge-transitive Cayley Graphs

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.