On The Calculation Of The Number Of Homomorphisms Between Certain Subcyclic Groups

Counting the number of homomorphisms between two groups is one of the basic problem in group theory.In this thesis,applying the number theory,generators of group and the corresponding relations between generator,we calculate the group homomorphisms from a class of metacyclic group of normal cyclic group of the order n exended by cyclic groups of the order 4 into the modular group,and from a class of metacyclic group of normal cyclic group of the order n exended by cyclic groups of the order 2p(p is odd prime)into the dihedral group,from the metacyclic group into the quasi-dihedral group,from the metacyclic group into the quaternion group.from the metacyclic group into the modular group.As an application,we prove those groups satisfying the conjecture of Asai and Yoshida.