Factorization Numbers Of Some Finite P-groups

In finite groups,the study of group combination problems is a meaningful thing.Furthermore,the research on factorization numbers of groups has a close relationship with the probability problems of finite groups,especially it has a direct relationship with the calculation of subgroup commutativity degree.In particular,to calculate the subgroup commutativity degree of a finite group,we only need to know the factorization numbers and subgroup lattice of the subgroup.Many theorists have done in-depth research on the factorization numbers of finite groups,which greatly promoted the development of finite group theory.Let G be a finite p-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 study the factorization numbers of four kinds of finite p-groups by the orders of groups,the numbers of subgroups and the properties of elements.In the second chapter,we further perfect the results of the factorization numbers of a class of inner Abel p-group.From Chapter three to Chapter five,we calculate the factorization numbers of three kinds of finite 2-groups with a cyclic Frattini subgroups,and we generalize the corresponding results of the finite 2-groups with a cyclic maximal subgroup.In this paper,three cases are analyzed according to whether the derived group G' or center Z(G)is included in subgroups A and B.Firstly,the derived group G'(or the center Z(G))is contained within the subgroups A and B,and the corresponding quotient group is a group with a known result or a group with a recurrence relationship.Secondly,the derived group G'(or the center Z(G))is not included in the subgroups A and B,and the corresponding factorization numbers does not exist or the situation is relatively simple.Thirdly,the derived group G'(or the center Z(G))is included in subgroup A,but not in subgroup B,and vice versa.Through the discussion of these three cases,we finally get the factorization numbers of the group G.