On The Norm And Wielandt Subgroups Of Finite Groups

It is very important and interesting to investigate the relationship betweenthe automorphisms in finite groups and the structure of finite groups. In thestudy, a characteristic subgroup called norm plays an impotent role. The normN (G) of a group G is the intersection of the normalizers of all the subgroups ofG. A related concept is the Wielandt subgroup of a group G, denoted w(G),is the intersection of the normalizers of all the subnormal subgroups of G. TheWielandt subgroup of a nilpotent group coincides with the norm of the group.Many authors have investigated both norm and Wielandt subgroups of finitegroups. In this paper, we continue to discuss this topic.In chapter III, we investigate the norm and Wielandt series of finite groups.Firstly, we give a necessary and sufcient condition for a capable group G tobe satisfied N (G)=Z(G), and then some sufcient conditions for a capablegroup with N (G)=Z(G) are obtained. Meanwhile, a necessary and sufcientcondition for a nilpotent group of odd order to be satisfied ωi(G)=Zi(G) isgiven. Secondly, we discuss the norm of a nilpotent group with cyclic derivedsubgroup. The relationship between the Wielandt series and the upper centralseries of the groups is gained. Finally, we completely classify two generated finitep-groups with cyclic derived subgroups and N (G)=Z(G) when p>2.In chapter IV, we are interested in p-groups of maximal class. We discuss therelationship between the Wielandt series and the upper central series of p-groupsof maximal class. A necessary and sufcient condition for a regular p-group ofmaximal class G to be satisfied ω(G)=Z2(G) is given.In chapter V, we discuss finite p-groups whose all non-normal cyclic sub-groups have small index in their normalizers. For convenience, we introduce Npm groups. We prove that the order of non-Dedekind Npmgroups are bounded byp(2m+1)(m+1)when p>2, we completely classify non-Dedekind Np2groups forp=2.