The Normal Closure Of Non-normal Subgroups Of Finite 2-groups

A Group G is called a Dedekind group if each subgroup of G is normal in G.The structure of Dedekind groups are given by Dedekind in 1897.Since then,Dedekind groups are generalized from different aspects.One of the methods commonly used by many group theorists is adding restricted condition on non-normal subgroups.Many interesting results are obtained.In this paper,we investigate the structure of finite p-groups by the normal closures of non-normal subgroups.Specifically,we study a finite 2-groups whose order of the normal closure of non-normal subgroups takes only two values on the basis of predecessors.For convenience of description,this group is called ?2-group.The structure of this paper is as follows:the first chapter is the introduction,which mainly introduces the research background,research methods and main results of this paper.The second chapter is the preparatory knowledge,mainly introduces the definitions and lemmas used in this paper.The third chapter is the main content.Firstly,some basic properties of ?2-groups are given;Secondly,we classify the ?2-group with d(G)=2 and 3;Finally,this paper gives some discriptions of ?2-group for d(G)? 4.The fourth chapter is a summary and prospect,exploring the problems that can be solved further.