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## Meditation on the Sylow theorems II

In Part I we discussed some conceptual proofs of the Sylow theorems. Two of those proofs involve reducing the existence of Sylow subgroups to the existence of Sylow subgroups of $S_n$ and $GL_n(\mathbb{F}_p)$ respectively. The goal of this post is to understand the Sylow $p$-subgroups of $GL_n(\mathbb{F}_p)$ in more detail and see what we can learn from them about Sylow subgroups in general.

## Meditation on the Sylow theorems I

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 $p$-group fixed point theorem (PGFPT): If $P$ is a finite $p$-group and $X$ is a finite set on which $P$ acts, then the subset $X^P$ of fixed points satisfies $|X^P| \equiv |X| \bmod p$. In particular, if $|X| \not \equiv 0 \bmod p$ 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.

## The cohomology of the n-torus

The goal of this post is to compute the cohomology of the $n$-torus $X = (S^1)^n \cong \mathbb{R}^n/\mathbb{Z}^n$ in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is $\mathbb{Z}$ by default. At the end we will interpret this computation in terms of cohomology operations.

Theorem: Let $G$ be a finite $p$-group acting on a finite set $X$. Let $X^G$ denote the subset of $X$ consisting of those elements fixed by $G$. Then $|X^G| \equiv |X| \bmod p$; in particular, if $p \nmid |X|$ then $G$ has a fixed point.