[Math] How does Cauchy’s theorem follow from Sylow’s theorem

Question

Very quickly, Sylow's first theorem says a sylow p-subgroup of order $p^rm$ exists and Cauchy's theorem says if $p \vert |G|$ then there is an element of order $p$.

It's often said that Cauchys follows easily from Sylows, but I just don't see it! I don't see why a sylow p-subgroup must have an element of order $p$; why couldn't they all be of order $p^n,\ 2<n<r$?

Answer 1

If you have an element $g$ of order $p^n$, consider $$g^{p^{n - 1}}$$ Can you show that this is an element of order $p$?