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 $P$.

Also known as
Some sources call this the fourth Sylow theorem, and merge it with what we call the Fifth Sylow Theorem.

Others call this the second Sylow theorem.