Normality Of Subgroups And Its Dual In Finite Groups

There has been much interest in investigating the relationships between special normality of subgroups and the structures of finite groups in the past decades. Not only have many kinds of concepts of generalized normality been given, but also a large amount of research fruit is gained. This strongly drives the development of the theory of finite groups. Among these concepts, c-normality and cover-avoidance property attract more attention, and the research about them is being deepened. However, these two important concepts have no certain connection. Recently, Fan, Guo and Shum introduce the concept of semi cover-avoiding property. Their research shows that the semi cover-avoiding property covers both cover-avoidance property and c-normality, and so it is a unified generalization of cover-avoidance property and c-normality. In the first part of this thesis, we continue to study the finite groups with some subgroups having semi cover-avoiding property. Many sufficient and necessary conditions for a finite group to be solvable are given, which generalize some known results. Moreover, several sufficient conditions for a finite group to be p-nilpotent or supersolvable are obtained, some of which are extended to formation.As a dual of the above idea, we try to study the influence of power automorphism groups on the structures of finite groups. First, we investigate the properties of the norm of a group, which are necessary for our subsequent work. The norm N(G) of a group G is a characteristic subgroup of G consisting of elements which induce power automorphisms on G. Then, in Chapter 4, we generalize the famous It(?) Theorem and Burnside Theorem about p-nilpotence. In Chapter 5, we classify several classes of finite groups whose power automorphism groups are of large order. For this purpose, the structure of a finite group G with |G: N(G)|=p or pq is determined. On the basis of that, a finite group G satisfying |Aut(G): Paut(G)|=1, p or pq is successfully classified, where p, q are distinct primes, Aut(G) is the automorphism group of G and Paut(G) is the power automorphism group of G. Finally, in Chapter 6, we determine the structures of so-called N-groups. A group G is called a N-group if for any x∈G, either (?)G or x∈N(G). The reason why we investigate N-groups is that: on one hand, a group G is Dedekind if and only if (?)G for any x∈G; on the other hand, a group G is Dedekind if and only if x∈N(G) for any x∈G. But there are examples showing that (?)G and x∈N(G) have no relation between them. So our investigation has more general meaning. The result shows that the class of N-groups properly contains the class of Dedekind groups.