Feeds:
Posts
Comments

Archive for November, 2020

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.

(more…)

Read Full Post »

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.

(more…)

Read Full Post »