How do I prove $(b)$? I struggle with the proof of injectivity. Any tips?
$Problem$: Let $G$ be a finite Abelian group.
$(a)$ Prove that the group homomorphisms $\chi : G → \mathbb{C}^*$ are exactly the characters of
irreducible representations of $G$.
Pointwise multiplication endows the set of irreducible characters of $G$ with the structure
of a finite Abelian group. This group is denoted by $\hat{G}$. (Remark: $\hat{G}$ is also called the
Pontryagin dual).
$(b)$ Show that the map
$$\mathcal{H} :G \rightarrow \hat{\hat{G}}$$
$$a \mapsto (\chi \mapsto \chi(a))$$
is an isomorphism of groups.
Best Answer
You want to prove that the map $$ x \mapsto (\chi \mapsto \chi(x)) $$ is injective. I assume that you have already proved that this is a homomorphism. So all you need to show is that the kernel of this map is trivial. That is, you want to show that $$ \ker (x \mapsto (\chi \mapsto \chi(x))) = \{e\} $$ where $e$ is the identity in $G$. Now the kernel is exactly all $x$ such that $$ \chi \mapsto \chi(x) $$ is the trivial map from $\hat{G}$ to $\mathbb{C}^\times$. If $x$ is in the kernel, then $\chi(x) = 1$ for all $\chi \in \hat{G}$. That is, $x$ is in the intersection of the kernels of all the characters $\chi: G\to \mathbb{C}^\times$. Hence $x$ is the identity.