Finite Groups With Maximal Normalizers

In the case of finite groups,the properties of the normalizers of subgroups are closely related to the properties of the group itself.Group G is a finite group.A finite group G is called a NP-group,if there is a prime number p such that?G:NGg(A)??p for every cyclic subgroup A of G.In this paper,it studies the struc-tures and properties of NP-groups.In the third chapter.Firstly,we prove that the non-nilpotent NP-group has Sylow tower.Secondly,The NP-group by discussing the Sylow subgroup is only one prime factor p and its Sylow p-subgroup is not Dedekind group.Thirdly,Let G be a finite group with ?G? = P1?1…ps?s,p1<…<Ps are primes and GPi is Sylow pi-subgroup of G,GPm is the first non-normal subgroup of G,and?G:NG(GPm)?= pk.Then,G =(GP1x…x Gpm-1x GPk×…× GPs)x(GPm×…×Gpk-1),the derived length is at most 2.In particular,if Gpm is not Dedekind group,G =(GP1×…× GPm-1 x GPm+1x…×GPs)(?)Gpm];If Gpm is Dedekind group,All Sylow subgroups of G are Dedekind group.Finally,Pm?Pk-1 and(1)if GPm is not Dedekind group,?G:Z(G)??2(pk-1)pk.(2)if GPm is not Dedekind group,?G:Z(G)??4pm?mPk.