The Influence Of Subnormal Subgroups And Their Deficits On The Group Structure

In the group theory,the structure of the group is often studied by it’s subgroups,and it is a very interesting subject to study the structure of finite groups by using subnormality of their subgroups.In this paper,we focus on two major problems,one is the study of the subnormal subgroups of finite groups,the other is the shortest length of successive normal closure series or the defect of subnormal subgroup in finite group.The main task of this paper shows as follows: The first chapter is to summarize some basic properties of subgroups and introduce some readiness knowledge.The second chapter is mainly to introduce the concept of the defect of subnormal subgroup in finite group.At the same time,some interesting properties about the defects are proved.Furthermore,in the third chapter,the finiteness and solubility of the groups which generated by two subgroups is discussed.The purpose is to investigate the influence of subnormal subgroups on finite groups.In the last chapter,we made a comparison of the defect and the class of nilpotent group,as well as the fitting height.For application,we give a example of equality between the defect of the subnormal subgroup and the class of the nilpotent group.