Proofs of Sylow theorems 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.

Let $p$ be a prime dividing the order of the finite group $G$. The existence of a Sylow $p$-subgroup can be proved in a standard manner by induction on $|G|$. Let $x \in G$ be a non-central element (i.e. $x \not \in Z(G)$). If the index of the centralizer $|G:C_G(x)|$ is not divisible by $p$ then we may apply induction and find a Sylow $p$-subgroup of $G$ inside $C_G(x)$. So we may assume that $p$ divides the indeces of the centralizers of the non-central elements. By the orbit-stabilizer equation $|G| = |Z(G)| + \sum_i |G:C_G(x_i)|$ where the $x_i$ are representatives of non-central conjugacy classes. In particular $p$ divides $|Z(G)|$. By Cauchy theorem there exists $x \in Z(G)$ of order $p$. By induction there exists a Sylow $p$-subgroup $H/\langle x \rangle$ of $G/\langle x \rangle$, implying that $H$ is a Sylow $p$-subgroup of $G$.
Now let $P$ be a Sylow $p$-subgroup of $G$, and let $\Omega$ be the set of conjugates of $P$ in $G$, $\Omega = \{P^g\ |\ g \in G\}$. $P$ acts on $\Omega$ by conjugation, with $P$ as unique fixed point. Indeed, if $P$ fixes $R \in \Omega$ then $P$ normalizes $R$, so that $PR$ is a $p$-subgroup of $G$ containing $P$ ($|PR| = |P| \cdot |R|/|P \cap R|$), hence by maximality $P=R$. Note that each orbit of an action of a $p$-group has size a power of $p$. In particular partitioning $\Omega$ into $P$-orbits gives $|\Omega| \equiv 1 \mod(p)$. We are left to prove that every Sylow $p$-subgroup of $G$ belongs to $\Omega$. Suppose by contradiction that there exists a Sylow $p$-subgroup $Q$ of $G$ such that $Q \not \in \Omega$. $Q$ acts on $\Omega$, and by the previous argument there are no fixed points under this action, thus $p$ divides $|\Omega|$, contradiction.
