Homomorphism from $\mathbb{Q}/ \mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$ Let $\mathbb{Q}$ be the group ($\mathbb{Q}$,+) and $\mathbb{Z}$ is a sub-group of $\mathbb{Q}$.
It is quite easy to find all homomorphism from $\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Q}/ \mathbb{Z}$.
However, I couldn't find what would be all  homomorphism from $\mathbb{Q}/ \mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$.
Please help me.

Because there is only the trivial one!
Any morphism $\Bbb Q/\Bbb Z\to\Bbb Z/n\Bbb Z$ is induced by a morphism $\phi:\Bbb Q\to \Bbb Z/n\Bbb Z$ such that $\phi(1)=0$.
Now let $u\in\Bbb Q$ arbitrary, then we must have
$$\phi(u)=\phi\left(n\cdot\frac un\right)=n\cdot \phi\left(\frac un\right)=0$$
in $\Bbb Z/n\Bbb Z$.
