Finite 2-groups Whose Non-normal Subgroups Have Cyclic Maximal Subgroups

We say N is a normal subgroups of G if N9=N,for all g in G.As is well known,normal subgroups of a group play an important role in determining the structure of a group.A finite groups is called a Dedekind group if all its subgroups are normal.Such groups were classified by Dedekind in 1897.From then on,many scholars have studied finite groups which non-normal subgroups have some property.In this paper,we continue research work in this field,and finite 2-groups whose non-normal subgroups have cyclic maximal subgroups are classified completely by using methods of central extension,97 groups were obtained.For convenience,let this group be called P groupThis thesis consists of four chapters.Chapter ? is an introduction,which mainly introduction the research background,research methods and main results of this paper.Chapter ? gives a list of preliminaries,this chapter mainly introduces the definitions and lemmas to be used in this paper.Chapter ? gives the simple properties of P group and complete classification of P group,which is divided into six sections.The first section introductions some simple properties of P group.One of the key conclusions is that if G is P groups,d(G)?6.Section two to six are discussed according to the value of d(G).Finally,the isomorphic classification of P group was completed.Chapter ? is a summary and the prospect,exploring the problem that can be solved further.