I have a question related to this question in stack exchange. By the above question, if $G$ is finite abelian group then its Pontryagin dual $\hat{G}=\operatorname{Hom}(G, \mathbb{Q}/\mathbb{Z})$ is finite.
Now if $M$ is a discrete $p$-primary abelian group or a compact pro-$p$ abelian group, we can define $\hat{M}=\operatorname{Hom}(M,\mathbb{Q}_p/\mathbb{Z}_p)$. Then, in the Iwasawa theory, $\hat{M}$ is also called the Pontryagin dual of $M$. This definition in mostly used in the context of Selmer groups of Elliptic curves by Greenberg.
My question: Assume that $M$ is finite. Does it imply that $\hat{M}$ is finite? (I suppose not)
$\Bbb Q_p / \Bbb Z_p$, that is, Prüfer $p$-group is also isomorphic to $\Bbb Z[1/p]/\Bbb Z$. Suppose $G$ has exponent $p^k$; then $Hom(G, \Bbb Q_p / \Bbb Z_p) = Hom(G, \Bbb Z/p^k)$ (which is obviously finite) because image of $p^k$-torsion is $p^k$-torsion.
Also you can use general compact-discrete duality and note that discrete finite group is compact, and any compact discrete group is finite. (it is just a way to convince yourself that it should be true).