First Sylow Theorem/Proof 2

Proof
Let $\mathbb S = \set {S \subseteq G: \order S = p^n}$, that is, the set of all of subsets of $G$ which have exactly $p^n$ elements.

Now let $G$ act on $\mathbb S$ by the rule:
 * $\forall S \in \mathbb S: g * S = g S = \set {x \in G: x = g s: s \in S}$

From Set of Orbits forms Partition, the orbits partition $\mathbb S$.

Let these orbits be represented by $\set {S_1, S_2, \ldots, S_n}$, so that:


 * $\size {\mathbb S} = \size {\Orb {S_1} } + \card {\Orb {S_2} } + \ldots + \size {\Orb {S_r} }$

Thus each $\Orb {S_i}$ is the orbit under $*$ of some $S_i$ whose order is $p^n$.

By the Orbit-Stabilizer Theorem:
 * $\order {\Orb {S_i} } = \dfrac {\order G} {\order {\Stab {S_i} } }$ for all $i \in \set {1, 2, \ldots, n}$

where $\Stab {S_i}$ is the stabilizer of $S_i$ under $*$.