§ Sylow Theorem 1

I've always wanted a proof I can remember, and I think I've found one.

§ Lemma: (papb)p(ab)\binom{pa}{pb} \equiv_p \binom{a}{b}:

§ Continuing: Size of Ω\Omega modulo pp:

§ Lemma: size of stabilizer of subset when action is free:

§ Continuing: Showing that Stab(S)=pn|Stab(S) = p^n.

Combining the above, we find that Stab(S)S|Stab(S)| \leq |S|. So the stabilizer of size S=pk|S| = p^k it is in some sense "maximal": it has the largest size a stabilizer could have!