Characters of Finite Group Map to Roots of Unity

Theorem

Let \(G\) be a finite abelian group. Then for any group character \(f : G \to \mathbb{C}^\ast\) and any \(g \in G\), \(f(g)\) is a \(|G|^{\text{th}}\) root of unity.

Proof

From this theorem, we know that \(g^{|G|} = \mathrm{id}\), and then from the properties of group homomorphisms we have that

\[ 1 = f(\mathrm{id}) = f(g^{|G|}) = f(g)^{|G|}\]

and hence \(f(g)\) is a \(|G|^{\text{th}}\) root of unity.