Orthogonality Relations for Characters

Theorem
Let $G$ be a finite abelian group with identity $e$.

Let $G^*$ be the dual group of characters $\chi : G \to \C^\times$, and $\chi_0$ the trivial character on $G$.

Let $\psi: G \to \C^\times$ be any character.

Let $y \in G$ be arbitrary. Then


 * $\displaystyle \sum_{x \in G} \psi(x) =

\left\{ \begin{array}{rl} 0,&\psi \neq \chi_0 \end{array} \right. $
 * G|,&\psi = \chi_0\\

and


 * $\displaystyle \sum_{\chi \in G^*} \chi(y) =

\left\{ \begin{array}{rl} 0,&y\neq e \end{array} \right. $
 * G^*|,&y=e\\