Finite P-groups All Of Whose Non-normal Subgroups Have Cyclic Cores

Let G be a finite p-group,H?G.HG denote the normal core of H in G.The normality of a subgroup can be reflected by its normal core.In this paper,we mainly study the finite p-groups all of whose non-normal subgroups have cyclic normal cores.We call such groups P-groups.Obviously,a 2-group of maximal class is a P-group.Assume G is a P-group which is not a 2-group of maximal class.We prove that |G'|?p3.Moreover,if p>2,then |G'|?p2 and c(G)?3.Based on this,we give the structure of the P-groups with p>2 and do some research on the P-groups with p=2.This paper is divided into four chapters.The first chapter is the introduction,mainly introduces the research background,research methods and main results of this paper.The second chapter is the preliminaries,mainly introduces the concepts and lemmas to be used in this paper,and proves some basic properties of the P-groups.The third chapter gives the structures of the P-groups with p>2.Specifically,we shows that G is a P-group if and only if d(Z(G))?2 if |G'|=p.The structure of the P-group with |G'|=p is obtained by using the structure of the finite p-groups G with |G'|=p.If |G'|=p2,we firstly give two necessary and sufficient conditions for the P-groups.Secondly,we obtain the groups G with |G'|=p2 by the centra extension of the P-groups G with |G'|=p.Finally,the P-groups G with |G'|=p2 are obtained by checking the necessary and sufficient conditions.In the fourth chapter,we do some research on the P-groups for p=2.Concretely,we classify the P-groups satisfying one of following conditions:(1)|G'|=2;(2)|G'|=4,c(G)=3;(3)|G'|=8,c(G)=4.