Abstract: Firstly, by using group theory and congruence theory, the author calculated the number of endomorphisms and automorphisms between a class of non commutative inner cyclic groups, and gave the quantity relation of the number of homomorphisms from such inner cyclic groups to finite groups. Secondly, the author verified that the conjectures of Asai and Yoshida were true for the above cases.

Key words: inner cyclic groups, homomorphism, endomorphism, automorphism