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 be a finite group and a prime dividing . By Sylow’s Theorem, has a Sylow -subgroup . Since is a -group, is nontrivial. Now is an abelian -group, so that (by Cauchy’s Theorem for abelian groups) there exists an element of order .