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 \mathop \in G} \psi \left({x}\right) = \begin{cases}

\left\lvert{G}\right\rvert & : \psi = \chi_0 \\ 0 & : \psi \ne \chi_0 \end{cases}$

and:


 * $\displaystyle \sum_{\chi \mathop \in G^*} \chi \left({y}\right) = \begin{cases}

\left\lvert{G^*}\right\rvert & : y = e \\ 0 & : y \ne e \end{cases}$

Proof
If $\psi = \chi_0$, then it is straightforward that:

If $\psi \neq \chi_0$, then $\exists y \in G$ such that $\psi \left({y}\right) \neq 1$.

As $x$ runs through $G$ in the summation, $yx$ also runs through $G$.

So:

Since by assumption $\psi \left({y}\right) \ne 1$, it must be true that:
 * $\displaystyle \sum_{x \mathop \in G} \psi \left({x}\right) = 0$