dual isomorphism

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.

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 $$M=\mathbb{Z}/n\mathbb{Z}$$?
• @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 $C$. – egreg Feb 5 at 10:09