Finite abelian group is isomorphic to its Pontryagin dual

Statement

Suppose $G$ is a finite abelian group and ${\hat {G}}$ is its Pontryagin dual. Then, ${\hat {G}}$ is also a finite abelian group and $G\cong {\hat {G}}$. However, there is no canonical isomorphism between the groups, and groups acting on $G$ may have corresponding actions on ${\hat {G}}$ with very different orbit structures.