Finite Groups In Which The Normalizer Of Every Non-normal Cyclic Subgroup Is Maximal

It is well-known that the essential task of finite group is determining the structure of finite groups. There is a close relationship between the normality of subgroups and the structure of finite groups, and normalizer of subgroups is a measure of the normality of subgroups. So it is reasonable to investigate the structure of a group by using normalizers of some kind of subgroups. Since the p-nilpotent group is a basic and important class of group, many experts in group theory are focused on studying this kind of group. For example:Frobenius gives us a famous p-nilpotent criterion. Burnside and Thompson are also get. some important theorems on p-nilpotent group. On the other hand, the structure of maximal subgroup is very close to the structure of finite group. So it is reasonable to study the relationship between finite group and its maximal subgroups. For all the above reasons, in this paper, we study finite groups in which the normalizer of every non-normal cyclic subgroup is a maximal subgroup, we call this group a NCM-group.In chapter III, we study finite solvable NCM-group. Firstly, we investigate the structure of the normalizer of non-normal cyclic subgroup. We find that solvable NCM-group is either q-closed or q-nilpotent, with q a smallest prime dividing the order of G. Finally, we investigate solvable non-nilpotent NCM-groups when G is either q-closed or q-nilpotent and give the structure of this kind of group.In chapter IV, we investigate finite non-solvable NCM-groups. In section two. we study semisimlple NCM-groups. We proved that if G is a semisimple ACM-group, then there is a unique minimal normal subgroup N with N a non-abelian simple group such that N≤G≤Aut (N). We also find that the unique minimal normal subgroup N is PSL(2,p) with p an odd prime. Finally, we give the structure of semisimple NCM-group. In section three, we investigate non-semisimple NCM-group. and got. the structure of G.