Orthogonality of Group Characters
The following relations are often called the orthogonality relations of characters.
Proof
If is the principal character then clearly
If is not the principal character, then there is some for which . Fix such an and consider that
Now, because is an element of the group, and is thus invertible, the function is a bijection. Thus
given they count the same terms.
This allows us to deduce that
This expression is in and hence it is equal to zero if either or , however by choice of and therefore
Lemma
For any finite abelian group , and element , there exists a character such that .
Proof
If then and the character evaluates to on the cyclic subgroup generated by . If this group is non-trivial, then it has a non-principal character, which can be extended to a character of , however this would contradict the fact that all characters evaluate to on this cyclic subgroup.
%% TODO change this to use splitting into product of cyclic groups and fix all the characters on each group to be the principal character except 1
Theorem
Let be a finite abelian group with . For each
Proof
Suppose that . Then from the properties of a group homomorphism
noting that .
Now suppose that . Therefore, by the previous lemma, there exists a character such that . Then we have that
Because is an element of the group and has an inverse also varies over and hence
This lets us deduce that
Since by choice of , we can conclude that
Corollary
Let be a finite abelian group with . For each
Proof
This follows from the previous result by replacing with . That is we have
and
where because they are roots of unity.
Then in the condition in the cases clearly , as required.