Some Studies On The Commutativity Of Finite P-Groups

A finite group whose order is a power of a prime number p is called a finite pgroup,abbreviated as a p-group.It is not only an important research object in the field of finite groups,but also closely related to group theory and other branches of algebra,especially with the relationship between finite simple groups,ring theory,Lie algebras,and homological algebras.In recent years,with the completion of the classification of finite simple groups and the development of solvable group theory,the study of p-groups becomes more and more active.Many group theorists have begun to work on the study of p-groups.In the research field of p-groups,using the commutativity of subgroups to study the structure of large groups is one of the important research topics,and the commutativity of subgroups can be represented by the derived group,the center,or the centralizer of the subgroup.Starting from the commutativity of subgroups,this thesis studies five classes of finite p-groups,and gives their properties or classifications.This thesis is divided into seven chapters,which mainly include the following contents:In Chapter 1,we introduce the research background of this thesis and the main results obtained.In Chapter 2,we introduce the basic concepts,common conclusions and some important lemmas used in this thesis.In Chapter 3,we investigate the structure and properties of finite DC-groups.A finite group G is said to be DC-group if the set {H’|H≤G} is a chain under subgroup inclusion.This chapter proves that a finite DC-group is a semidirect product of a Sylow p-subgroup and an abelian p’-subgroup.Based on this result,we emphatically study the finite DC-p-group G(G is both a DC-group and a pgroup),and obtain the optimal upper bound of d(G’).Further,we prove that when p≤3,the finite DC-p-group is a metabelian group;when p≥5,there is a group example that is a non-metabelian finite DC-p-group.In particular,we characterize the DC-2-groups.In Chapter 4,we study the structure of finite P-groups.A finite p-group G is called a P-group if all A2-subgroups H of G satisfy H’=G’.This chapter gives the complete classification of finite P-groups,and using this classification,we solve a problem posed by Y.Berkovich.In Chapter 5,we are interested in the structure of finite CC-groups.The finite non-abelian p-group G is called a CC-group if all non-abelian proper subgroups of G have cyclic centers.This chapter gives the complete classification of finite CC-groups,and solves the problem posed by Y.Berkovich.Hence using this classification,we also determine the finite non-abelian p-groups all of whose proper subgroups being neither abelian nor minimal non-abelian have cyclic centers.In Chapter 6,we study the structure of finite OC-groups.A finite non-abelian p-group G is called an OC-group if the centers of all non-abelian proper subgroups of G have the same order.This chapter gives a complete classification of finite OC-groups,and using this classification,we obtain the classification of finite nonabelian p-groups whose non-abelian proper subgroups have the same center.In Chapter 7,we investigate the structure and properties of finite MC-groups.A finite non-abelian group G is called a MC-group if all non-abelian subgroups of G have minimum centralizers.In this chapter,we give some equivalent conditions for the MC-groups,and prove that MC-groups are just the finite groups with modular centralizer lattice of length 2 studied by R.Schmidt.According to R.Schmidt’s result,we get the classification of MC-groups.However,R.Schmidt did not delve into the MC-p-groups G(G is both a MC-group and a p-group).Therefore,we give a characterization of the MC-p-groups.In particular,we characterize the special MC-p-groups by means of commutator matrices,and provide a method to classify special MC-p-groups.Using this method,we obtain the isoclinism classification of the special MC-p-groups with an abelian maximal subgroup.