Third Sylow Theorem

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

Some sources call this the fourth Sylow theorem, and merges it with what we call the Fifth Sylow Theorem.

Others call this the second Sylow theorem.

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 $$P$$.