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.
Archive for November, 2020
Meditation on the Sylow theorems II
Posted in math, math.GR, tagged finite fields, fixed point theorems, group actions on November 2, 2020| 1 Comment »
Meditation on the Sylow theorems I
Posted in math, math.GT, tagged fixed point theorems, group actions on November 1, 2020| 8 Comments »
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.