The post that made me google ‘latex does not exist’.
Lemma. Let be a finite group of order , and write for the set of irreducible characters of . Then
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 ().