How to calculate number of onto and one homomorphism?

Question

How to calculate the number of one one onto homomorphism from one finite cyclic group to another?

I tried using first theoram of isomorphism but could not find the result. Any help would be greatly appreciated.

Answer 1

Suppose $C_1=\langle g\rangle$ and $C_2$ are cyclic groups. A homomorphism $C_1\to C_2$ is completely determined by the image of $g$. If $h\in C_2$, then:

1. There is a homomorphism taking $g$ to $h$ if and only if the order of $h$ divides the order of $g$ (or the order of $g$ is infinite).
2. This homomorphism is 1-1 if and only if the orders of $h$ and $g$ are equal.
3. This homomorphism is onto if and only if the order of $h$ equals the order of $C_2$.

These observations reduce the problem to elementary number theory.