If a group $G$ has order $p^n$, where $p$ is prime and $n \geq 1$, does there exist some element $a\in G$ s.t. the order of $a$ is $p$?
I happen to know that this is true by Cauchy's theorem, but that theorem has not been presented yet in the book. I only have this so far:
Let $a \in G$, then the order of the cyclic subgroup generated by $a$ divides the order of the group $G$ (by Lagrange's theorem).
But I get stuck because I don't know how to show that the $a$ has order $p$.