Some Results Of Conjectur Of T.Asai & T.Yoshida

The number of homomorphisms of groups is a basic quantity to study the relationship between groups,that can describe some structures and properties of groups.It has been becoming one of the most popular problems in group theory to study the related properties of homomorphisms of groups.Conjecture of T.Asai & T.Yoshida plays an important role in exploring the congruence relationship between the number of homomorphisms of finite groups and the order of groups.Based on the structure and element properties of central-bihedral group,bihedral group,quasi-bihedral group and quaternion group in group theory,using the knowledge of algebra and number theory,this paper concretely constructs all homomorphisms between central-bihedral group and bihedral group,central-bihedral group and quasi-bihedral group,central-bihedral group and quaternion group,and it is proved that conjecture of T.Asai & T.Yoshida is valid for these groups.In addition,by using the properties of homomorphism and anti-homomorphism of groups,the relationship between the number of homomorphisms and the number of anti-homomorphisms between the two groups is studied,and an equivalent proposition of conjecture of T.Asai & T.Yoshida is obtained.