Let $\mathbb{Q}$ denote the group of rational numbers (with addition as the binary operation) and let $\mathbb{Z}$ denote the subgroup of integers. The Pontryagin dual of a group $G$ is the group $G^* = \operatorname{Hom}(G,\mathbb{Q/Z})$. (It is most useful when G is abelian).
- Show that $G^*$ is finite if $G$ is a finite group.
- Suppose $G = \mathbb{Z}/n$ for some non-negative integer $n$. Show that $G^*$ is isomorphic to $\mathbb{Z}/n$. Compute $G^*$ (up to isomorphism) for the symmetric group on 3 letters.
for 1 I know if $G$ is finite all its elements are of finite order, and $\mathbb{Q/Z}$ has infinite elements, all of finite order. But I don't know where to go from there or if that's the right idea to begin with. First part of 2 sounds like it follows from similar work that would be done in 1, right now though all I can say is that $G^*$ would be finite. For the second part of 2 I know how the $S_3$ group works but I don't know how to go about finding all the homomorphisms of it to $\mathbb{Q/Z}$, except again, 1 tells me it would be a finite amount. Any help would be appreciated.