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 and respectively. The goal of this post is to understand the Sylow -subgroups of in more detail and see what we can learn from them about Sylow subgroups in general.

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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 -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.

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The cohomology of the n-torus

The goal of this post is to compute the cohomology of the -torus in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is by default. At the end we will interpret this computation in terms of cohomology operations.

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The p-group fixed point theorem

The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let be a finite -group acting on a finite set . Let denote the subset of consisting of those elements fixed by . Then ; in particular, if then has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

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