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Let $A$ be a finite, Abelian, additive group. Let $A^{*} = Hom(A, \mathbb{Q}/\mathbb{Z})$ denote the group of homomorphisms $f$ from $A$ to $\mathbb{Q}/\mathbb{Z}$. Take for granted that $A^{*}$ is an Abelian group (I have already proved this). Prove that $A$ is isomorphic to $A^{*}$ if $A$ is cyclic.

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    That is indeed the case!2012-10-30

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Hint: If $A$ is cyclic, and $a$ generates it, then every homomorphism $f:A\to G$ is completely determined by $f(a)$.