Third Sylow Theorem

Theorem
All the Sylow $p$-subgroups of a finite group are conjugate.

Proof
Suppose $$P$$ and $$Q$$ are Sylow $p$-subgroups of $$G$$.

By the Second Sylow Theorem, $$Q \subseteq P$$.

But since $$\left|{P}\right| = \left|{Q}\right|$$, it follows that $$Q$$ must equal a conjugate of $$P$$.