In the last post, I linked to a video of me explaining and proving Lagrange’s theorem. This theorem does a lot to empower us to get a deeper understanding of the structure of groups (for instance, allowing us to prove that there is only one single group of size p, where p is any prime number).
This time, we’ll be going a level deeper and proving a much much more powerful theorem: Cauchy’s theorem! While Lagrange’s theorem gives us a necessary condition for the existence of subgroups (that the subgroup’s size divides the group’s size), Cauchy’s theorem gives us a sufficient condition (that the subgroup’s size is a prime factor of the group’s size)!
This is a really powerful tool to add to our toolkit for analyzing groups. It also really nicely complements Lagrange’s theorem, in that Lagrange’s theorem is most powerful when applied to group sizes that are prime or have few prime factors, while Cauchy’s theorem is most powerful when applied to group sizes that are highly composite.
Anyway, here are the links to the two videos in which I go in depth into all of this!
Even though these two videos together are about 35 minutes, the proof itself is actually quite short! Most of the work done in the videos is just building up some powerful concepts (actions, orbits, stabilizers). These concepts allow us to prove the theorem in just a few lines! (In fact, if you look closely at the proof, you might notice a a way to get an even stronger result with just a few more lines!)