## Use Sylow’s Theorem to prove Cauchy’s Theorem

Use Sylow’s Theorem to prove Cauchy’s Theorem. (Note that we only used Cauchy’s Theorem for abelian groups in the proof of Sylow’s Theorem, so this line of reasoning is not circular.)

Let $G$ be a finite group and $p$ a prime dividing $|G|$. By Sylow’s Theorem, $G$ has a Sylow $p$-subgroup $P$. Since $P$ is a $p$-group, $Z(P)$ is nontrivial. Now $Z(P)$ is an abelian $p$-group, so that (by Cauchy’s Theorem for abelian groups) there exists an element $x \in Z(P) \leq P \leq G$ of order $p$.