Study On The Congruence Problem Of Number Of Homomorphisms From Several Non-Commutative Groups Into Dihedral Groups

It is a basic problem in group theory to study the homomorphism between groups based on the know groups.In this paper,we introduce the basic concepts of a class of non-commutative group G of order 10pⁿ,dihedral group D_2m and finite 2 group G₂.The images of the generators of these groups are constructed in the sense of homomorphism by using the proper-ties of element orders and subgroups.According to the uncertainty of m in dihedral group,the number of homomorphisms from D_2m into G and G₂ are classified and discussed.So far the homomorphism problem between binary generating groups have been improved,in which a group can be tenary generating group.As an application of the above results,the famous co-njecture of T.Asai and T.Yoshida is verified.Finally,we make a prospect for the research of this paper.