The Description Of The Congruence Relation Of The Number Of Homomorphisms In Finite Groups

The characterization of congruence relation of group homomorphism number of finite groups is an older problem in group theory.However,it is also one of the hot issues in recent years.On the basis of previous studies,by using the relationship between the set of cocycles and the group homomorphism,a one-to-one correspondence between the equivalence class of the group action and the set of cocycles is established.Through this correspondence,we deeply discuss the congruence relation of finite group homomorphisms.In addition,we use the knowledge of number theory and the characteristics of specific group structure to calculate the number of homomorphisms between metacyclic groupG_n and quasi-dihedral groupQD_2α,the metacyclic group G_n is normal cyclic groups of the order n extended by cyclic groups of the order 4.The number of homomorphisms betweenG_n andQD_α2 satisfies the conjecture of T.Asai and T.Yoshida.