As an undergraduate the proofs I saw of the Sylow theorems seemed very complicated and I was totally unable to remember them. The goal of this post is to explain proofs of the Sylow theorems which I am actually able to remember, several of which use our old friend
The -group fixed point theorem (PGFPT): If is a finite -group and is a finite set on which acts, then the subset of fixed points satisfies . In particular, if then this action has at least one fixed point.
There will be some occasional historical notes taken from Waterhouse’s The Early Proofs of Sylow’s Theorem.