| The non-abelian subgroups of class two are important subgroups of p-groups.Every non-abelian p-group has a subgroup of class two and can be generated by its subgroups of class two.Considering the importance of subgroups of class two in p-groups,scholars at home and abroad have studied the influence of subgroups of class two with certain properties on the structure of p-groups.Scholars have made a lot of progress in studying the influence of minimal non-abelian subgroups to the structures of finite p-groups,thus they have turned to study the influence of the general subgroups of class two to the structures of finite p-groups.In particular,we already know that if subgroups of class two of G are minimal non-abelian,then subgroups of class two of G are of the same order or subgroups of class two of G are of the same order except one.The classification of finite p-groups all of whose non-abelian subgroups of class two are of the same order and finite p-groups all of whose non-abelian subgroups of class two are minimal non-abelian are given.Naturally,the study of finite p-groups all of whose non-abelian subgroups of class two are of the same order except one are important.In this thesis,we mainly study the finite p-groups all of whose non-abelian subgroups of class two are of the same order except one.We call such groups P1-groups,and we call finite p-groups all of whose non-abelian subgroups of class two are two-generators as P2-groups.In this thesis,the classification of Pi-groups are given.There are ten classes of groups which are not isomorphic to each other are obtained.This thesis is divided into four chapters.The first chapter is the introduction,and mainly introduces the research background,research methods and main results of this paper.The second chapter is the preliminaries,and mainly introduces the concepts and results to be used in this pape.r.The third chapter is the classification of P1 ∩P2 groups.In four chapter,we completely classify P1-P2 groups by using methods of cyclic extension. |
