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 $\order P = \order Q$, 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.

Also see

 * Sylow Theorems