Third Sylow Theorem/Proof 1

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, $Q$ is a subset of a conjugate of $P$.

But since $\order P = \order Q$, it follows that $Q$ must equal a conjugate of $P$.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $11$: The Sylow Theorems: Corollary $11.11$