If $p$ is a prime then all the non trivial subgroups of $G$ with $\lvert G\rvert=p^2$ are cyclic

Question

If $p$ is a prime then all the non trivial subgroups of $G$ with $\lvert G\rvert=p^2$ are cyclic.

I tried looking online where does this result come from, but could not find any direct result. I found that all groups with order $p^2$ are isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z_p} \times \mathbb{Z_p}$; is this a consequence of this result?

Answer 1

You do not need the classification of groups of order $p^2$. Just use Lagrange theorem to conclude that a nontrivial subgroup (that is, different of the trivial one and $G$) has order $p$.

Now, groups of prime order $p$ are known to be cyclic (the order of a nontrivial element cannot be $1$, so it is $p$).

Answer 2

If $H\le G$ is nontrivial, then $|H|=p$ (Lagrange). Then $1\ne x\in H\Longrightarrow \langle x\rangle=H$ (Lagrange, again), so $H$ is cyclic.