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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.

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.