Let $G$ be a group of order $p^2$, where $p$ is prime. Show that $G$ must have a subgroup order of order $p$.
What I have so far:
$G^{p^2} =e .$
If $G$ has an element $g$ of order $p^2$, then $g^p$ is of order $p$. $\langle g^p\rangle$ is a subgroup of order $p$.
$G$ must have an element $a$ of order $p$ by Lagrange's Theorem. $\langle a\rangle$ is a subgroup of order $p$.
Is this sufficient? Or am I missing some details?