Range of Characters

Theorem
Let $G$ be a finite abelian group of order $m$.

Let $\chi : G \to \C^\times$ be a character on $G$.

Then for any $g \in G$, $\chi(g)$ is an $m^\text{th}$ root of unity.

If $e$ is the identity of $G$ then $\chi(g) = 1$.

Proof
The claim that $\chi(e) = 1$ is shown by Group Homomorphism Preserves Identity.

Now let $g \in G$ be arbitrary and let $k$ be the order of $g$.

By Order of Element Divides Order of Finite Group, we have $m = k\ell$ for some integer $\ell$.

Therefore, $g^m = (g^k)^\ell = e^\ell = e$.

By the homomorphism property we have


 * $1 = \chi(e) = \chi(g^m) = \chi(g)^m$

This proves the theorem.