How to show every proper subgroup is cyclic? Let $|G| = p^2$ where $p$ is prime. Show that every proper subgroup of $G$ is cyclic.
I don't know how to approach this problem.
Here is what I have:
From Lagrange Theorem, for any subgroup $H\subset G$, $\frac{|G|}{|H|}$. Now, the order of $G$ is $p^2$. We need to show that the order of H is equal to $|\langle a \rangle |$ where $a \in H$. (Not sure!)

If $H$ is a subgroup, by Lagrange's theorem $|H|$ is either $1$ or $p$ ($p^2$ is ruled out because $H$ is proper). Now, every group of prime order is cyclic (why?).
