Factorization Numbers Of Two Classes Of Finite P-groups

In the study of finite groups,using order of group,properties of subgroup and element to research the combination of finite groups is an important direction.Computing the factorization numbers of finite p-groups is so interesting and meaningful in the study of combination of finite groups,and the study of this field is closely linked with the probability of finite groups.In particular,the subgroup commutativity degree of finite groups can be determined by the factorization numbers of finite groups,vice versa.Let G be a finite group,A and B be two subgroups of G.If G=AB,then G is said to be factorized by A and B.In this paper,we apply the order of group and element,m?bius inversion formula,the numbers of the subgroups of Abel p-group to investigate the factorization numbers of finite p-groups.The main contents of this thesis consists of the following three chapters.In the first chapter,we mainly introduce the current background and significance of the subgroup commutativity and factorization of finite groups,then introduce some necessary propaedeutics.In the second chapter,we compute the factorization numbers of a class of generalized extraspecial p-groups,as (?) where m?1.In the last chapter we compute the factorization numbers of a class of inner Abel p-groups,as (?) where n?m?1.The method: At first,three cases are analyzed,calculated and discussed according to whether A and B contained the derived group G'.In the first case,both A and B contain the derived group G'.In the second case,both A and B do not contain the derived group G'.In the third case,A contain the derived group G',but B do not contain the derived group G',vice versa.Then,we compute the numbers of A and B in all cases,which satisfyG=AB.Finally,we compute the factorization numbers of the group G.