Let $p$ be a prime number and $G$ be a group with $|G|=p^n$. Show that G contains an element of order $p$.
I would immediately say: "use Cauchy's theorem!", but this question is from a course that hasn't introduced this yet. Is there another way to prove this?
It's clear that every element has an order $p^r$ for some $r\leq n$ because of Lagrange's theorem. How can we proceed?