All group homomorphism from $ \mathbb{Z} _m $ to $\mathbb{Z}_n $ [duplicate]

All group homomorphism from $ \mathbb{Z} _m $ to $ \mathbb{Z}_n $

How could I find every group homomorphism?

A homomorphism from the cyclic group $\mathbb{Z}_m$ into any other group is determined by where it sends a generator. The generator must be sent to an element whose order divides $m$.

In the case of this problem, let $d=\operatorname{gcd}(m,n)$. For every $d'|d$, there exists a unique subgroup $\mathbb{Z}_{d'}\le\mathbb{Z}_n$. Map a generator of $\mathbb{Z}_m$ to each generator of $\mathbb{Z}_{d'}$ for all $d'|d$ to obtain all possible homomorphisms. There are $\varphi(d')$ generators for $\mathbb{Z}_{d'}$, so the total number group homomorphisms is $$\sum_{d'\mid d}\varphi(d')=d$$ Here $\varphi$ is Euler's totient function. The above formula is a standard property of Euler's function.

The map $\phi : Z_m \to Z_n$ given by $\phi(r\mod m) = k r\mod n $ is a well-defined homomorphism if and only if $n$ divides $km$. Every homomorphism from $Z_m \to Z_n$ is of this form.

A homomorphism is determined by $\phi(1)$ in this case. (Of course, if the group has more generators, you should check all the generators.)

Hint: each homomorpism $f$ is uniquely determined by $f(1)$.