Isomorphism between $Hom$ and $\mathbb{Z}_n$

Question

What is the isomorphism between $$\mathrm{Hom}(Z_n,\mathbb Q/\mathbb Z) \ \ \text{and} \ \ Z_n?$$ I guess its not very hard, but I can't find anyone.

Answer 1

Here's an approach you can take: any homomorphism $\Bbb Z_n\to\Bbb Q/\Bbb Z$ is uniquely determined by the image of $\bar 1\in\Bbb Z_n$. Define a homomorphism $f:\Bbb Z\to Hom(\Bbb Z_n,\Bbb{Q/Z}$) by sending $1\mapsto \varphi$ where $\varphi(\bar 1)=\overline{1/n}$. You need to check $(1)$ that this map $\varphi$ is actually a homomorphism, $(2)$ that $f$ is surjective (which amounts to showing $\varphi$ generates $Hom(\Bbb Z_n,\Bbb{Q/Z})$), and $(3)$ that $f$ has kernel $n\Bbb Z$.