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Let $C$ be the category of finitely generated abelian groups. If $M$ is a finitely generated abelian group, then define its dual as $M^* = \operatorname{Hom}(M,\mathbb{Q}/\mathbb{Z})$.

Now I want to show that $M$ and $M^{**}$ are canonically isomorphic.

I know that we need to use the structure theorem of abelian groups, but don't get how to use it to show isomorphisam.

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The structure theorem tells you that $M$ is a direct sum of cyclic groups. Thus you are reduced to prove the statement for a cyclic group.

For instance, if $M=\mathbb{Z}$, the statement is essentially obvious. Can you prove it for $M=\mathbb{Z}/n\mathbb{Z}$?

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  • $\begingroup$ @mathsstudent Much more is true: if $C$ is a finite cyclic group, then there is a unique subgroup of $\mathbb{Q}/\mathbb{Z}$ isomorphic to $C$. $\endgroup$ – egreg Feb 5 at 10:09

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