The P-nilpotent Residuals Of Subgroups And The Structure Of Finite Groups

It is very important to investigate the relationship between automorphism groups of finite groups and the structure of finite groups in finite group theory.On the one hand, one wants to decide automorphism groups of finite groups;on the other hand, one hope to study the structure of finite groups through automorphism groups or an automorphism which has some special properties. In this field power automorphisms is one of the most interesting objects. In 1934,Baer introduced a concept of in a finite group. The importance lies in that every element of of a finite group can induce a power automorphism in the group and many interesting results has been given. Recently, “generalized norms” become popular and some new problems are raised.In chapter III, we first introduce a characteristic subgroup () of a finite group , that is the intersection of the normalizers of -nilpotent residuals of all subgroups of a finite group , where is the class of all -nilpotent groups.Next we study some basic properties of (). Finally we give the relationship between () and the centralizer of , and find out the relationship between () and (), where () is the intersection of the normalizers of nilpotent residuals of all subgroups of a finite group .In chapter IV, we continue to investigate the intersection ?() of the normalizers of ?-residuals of all subgroups of a finite group under any formation ?.By using the relationship between ?() and the centralizer of the ?-residual of, we give the relationship between ?() and the ?-hypercentre of . Then the structure of finite groups is given by “generalized power automorphism groups”.In chapter V, we investigate the intersection of the normalizers of some subgroups, that is, the intersection N() of the normalizers of all non -nilpotent subgroups of a finite group . We give not only some properties of N(), but also the relationship between N() and the structure of .