This post is part of a three-post set (1 2 3) about groups of order $pqr$, where $p<q<r$ are prime. While designing final exam questions for my group theory class, I was playing around with groups of order 42, for no particular reason, and then a friend noticed something interesting that I wanted to try to prove more generally. Here's a complete writeup of this proof. Is it the most beautiful and elegant proof? Certainly not; you can directly quote some results about groups of square-free ord...
