Character

Let be a finite Abelian Group. A character on is a Homomorphism

Theorem

The characters on form an Orthonormal Basis of .

Proof

Let and be characters and . If then clearly . Otherwise pick some such that and note:

and thus .

It remains to prove that they span . As is a Finite Abelian Group, write

Given where and let

Easy to show that these are characters and for we have .

Now note that is a vector space of dimension (e.g. by using ) and the characters form an orthonormal set of size so we are done.

Lemma

Let . Then