# Application of Schur orthogonality

Lemma. Let be a finite group of order , and write for the set of irreducible characters of . Then

1.

2.

Proof. First consider the case . This is just an example; it could also be something much better. Then the second statement is obvious, and the first is left as an exercise to the reader. The general case is similar.

Here is a trivial consequence:

Corollary. Let be a positive integer, and let . Then

Proof 1. Without loss of generality, has exact order . Set , let , and note that

Part 1 of the lemma gives the result.

Proof 2. Set as before, let be the homomorphism , and the homomorphism . Then part 1 of the lemma does not give the result, but part 2 does.

In fact, the corollary also implies the lemma, because both are true ().

## 3 thoughts on “Application of Schur orthogonality”

1. Thanks for this super clear explanation! Using parentheses as variable names really makes conjugation look simpler.

• OK, I see two typos where you have two periods ending the statement of parts 1 and 2 of the lemma. Otherwise I also want to thank you for the very clear explanation of this otherwise difficult material.