Normalizers Of Nilpotent Residuals Of Subgroups Of Finite Groups

It is a very important topic in finite group to investigate the relationshipbetween the normality and the structure of finite groups. The famous Dedekindgroup is a group that every subgroup is normal. In the classification of infiniteDedekind group, a characteristic subgroup named norm plays an impotent role.Later, Wielandt introduced a subgroup related with norm, named Wielandt sub-group. Since then, many experts in group theory try their best to investigatenorm or Wielandt subgroups and their influence toward the structure of finitegroups, and some outstanding results were obtained. Not only that, many ex-perts in group theory also proposed many meaningful questions concerned.Meanwhile, it is also very important and interesting to investigate the re-lationship between the automorphism and the structure of finite groups. In thestudy of the relationship between the power automorphism and the structure offinite groups, norm also plays an impotent role. Here, we study norm, Wielandtsubgroup and the structure of finite groups, which starting from power automor-phism induced on some special subgroup (i.e. nilpotent residual).In chapter III, we investigate the intersection of the normalizers of nilpotentresiduals of all the subgroups in finite groupG:N~N(G). Firstly, we obtain somebasic propositions onN~N(G). Secondly, the structures and propositions of agroup G thatN~N(G) contains some minimal subgroup were obtained. Thirdly,we give a new character on meta-nilpotent group and the intersection of maximalmeta-nilpotent subgroup in finite groupG byN~N(G). Finally, we introduce somegroups similar toN~N(G).In chapter IV, we investigateN~N-groups.N~N-groups is characterized by the complement of nilpotent residual, the simple and soluble minimal non--groups are characterized. Also, we consider the number of conjugacy classes ofnon-normal subgroups in-groups and the groups expressed by the product oftwo special-groups.In chapter V, we obtain a note on the size of nilpotent residual in finitegroup. It is proved that|G: Z(G)|<|G′|·|G~N|if is a finite non-abelian groupwith Φ(G)=1.