Compute $\operatorname{Hom}(\mathbb{Z}/{n \mathbb{Z}}, \mathbb{Q}/\mathbb{Z}$)

Question

I need to calculate $$\operatorname{Hom}(\mathbb{Z}/{n \mathbb{Z}}, \ \mathbb{Q}/\mathbb{Z})$$ but I can't see a way to do it...any hints?

Answer 1

Every map $\phi : \mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Q}/\mathbb{Z}$ is determined by $\phi(1)$.

We know that $0=\phi(n)=n\phi(1)$. This means that $n\phi(1)$ is an integer, since all the integers are zero in $\mathbb{Q}/\mathbb{Z}$. There are therefore $n$ different values for $\phi(1)$:

$\frac{1}{n}, \frac{2}{n}, \dots, \frac{n}{n}=1$

So $\text{Hom}(\mathbb{Z}/n\mathbb{Z},\mathbb{Q}/\mathbb{Z})\cong\mathbb{Z}/n\mathbb{Z}$