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Let $A$ be a $\mathbb{Z}$-module, let $a$ be an element of $A$ and let $n$ be a positive integer. Prove that the map $\phi_a:\mathbb{Z}/n \mathbb{Z} \rightarrow A$ given by $\phi(\bar{k})=ka$ is a well-defined $\mathbb{Z}$-module homomorphism iff $na=0$. Prove that Hom$_{\mathbb{Z}}(\mathbb{Z}/n \mathbb{Z},A) \cong A_n$, where $A_n=\lbrace a \in A | na=0 \rbrace$.

I manage to prove the iff part. My question is what map should I defined so that I can prove the two modules are isomorphic?

Initially I defined the map $f:$Hom$_{\mathbb{Z}}(\mathbb{Z}/n \mathbb{Z},A)$ $\rightarrow A_n$ where $f(\phi_a)=a$. But this brings me nowhere.

Can anyone give some hints?

Idonknow
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    $f$ is indeed the isomorphism. You should note that $φ: \mathbb{Z}/n\mathbb{Z} \to A$ is uniquelly determined by $φ(1)$. – user87690 Aug 25 '13 at 16:02
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    In addition to checking that $f$ is a homomorphism you should prove that $a\mapsto \phi_a$ is its inverse. If a function has an inverse, it is a bijection. – Jyrki Lahtonen Aug 25 '13 at 16:11

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