There is a theorem in finite group theory, that if $a$, $b$, and $c$ are integers all greater than $1$, there exists a finite group $G$ with elements $x$ and $y$ such that: $x$ has order $a$, $y$ has order $b$, and $xy$ has order $c$. I think the first person to prove this was G.A. Miller, whose proof looked at lots of separate cases, and had tons of long, tedious calculations in symmetric groups (I will try and find the paper and post the reference later). I don't know who discovered the more modern proofs, but Derek Holt posted a proof on the group-pub that is one of the most elegant things I've ever seen. Unfortunately, it doesn't seem to be available on the archive of the list, so I will just post it here verbatim: