By Ben Blum-Smith, Contributing Editor| blogs.ams.org
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...| Spencer's blog
This post is part of a three-post set (1 2 3) about groups of order $pqr$, where $p<q<r$ are prime. While thinking about such $pqr$-groups, I've collected a bunch of theorems and lemmas into a little mental toolbox. An interesting commonality between a lot of these tools is that they tell us about the relationship between various different animals in the subgroup lattice; an interesting difference is that some of them go “upwards,” some of them go “downwards,”1 and some of them even g...| Spencer's blog
This post is part of a three-post set (1 2 3) about groups of order $pqr$, where $p| Spencer's blog