The Congruence Of The Number Of Homomorphism Between The Dihedral Group And A Class Of Metacyclic Groups

Studying the properties and structures of different groups is an important task of group theory.Calculating the number of homomorphisms between two groups is one of the basic problems in group theory.Frobenius proved that the number of homomorphisms from order cyclic groups to finite groups is a multiple of the maximum common factor in 1903.This paper is based on group theory,structure of metacyclic groups and dihedral groups,and the characteristics of the group elements.It uses the basic method of algebra and number theory and specifically calculates the the number of homomorphisms between dihedral groups and a class of metacyclic groups.As an application,it proves that the conjecture of Asai T and Yoshdia T satisfies the number of homomorphisms of such groups.