TITLE: Proofs of Sylow theorems
QUESTION [15 upvotes]: It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know.

REPLY [24 votes]: I love Wielandt's proof for the existence of Sylow subgroups (Sylow I). Isaacs uses this proof in his books  Finite Group Theory  and  Algebra: A Graduate Course.   As Isaacs mentions, the idea of the proof is not very natural and does not generalize to other situations well but it is simply beautiful. First a lemma:
Lemma: Let $p,a,b$ be natural numbers where $p$ is prime and $a \geq b$. Then $$\binom{pa}{pb} \equiv \binom{a}{b} \pmod{p}$$
Proof. Consider the polynomial $(x + 1)^{pa} = (x^p+1)^a \in \mathbb{F}_p[x]$. Computing the coefficient of the $x^{pb}$ term in two different ways yields the result.
Proof of Sylow I: Let $|G| = p^nm$ such that $p \nmid m$. Let $$\Omega = \{ X \subseteq G: |X| = p^n\} $$
(Note that we are taking every subset of $G$ with $p^n$ elements). $G$ acts on $\Omega$ by left multiplication. Observe that
$$|\Omega| = \binom{p^nm}{p^n} \equiv m \pmod{p}$$
by repeated usage of the lemma. Hence $p \nmid |\Omega|$, therefore $\Omega$ has an orbit $\mathcal{O}$ such that $p \nmid |\mathcal{O}|$. Now let $X \in \mathcal{O}$ and let $H$ be the stabilizer subgroup of $X$. Since $|G:H| = |\mathcal{O}|$ (orbit-stabilizer theorem), we deduce that $p^n$ divides $|H|$; in particular $p^n \leq |H|$. On the other hand, for $x \in X$ by definition of stabilizing $Hx \subseteq X$ and hence $$|H| = |Hx| \leq |X| = p^n$$
Thus $H$ is a Sylow $p$-subgroup.