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 ().

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.

!