classify groups of order $p^2$ simple or not

Question

I wonder $p$-groups of order $p^2$ simple or not. I know all of them are abelian. So center equal whole group.Hence, it is not useful to test simplicity.

• How we can classify them according to whether they are simple or not. For example;

Let $G$ be a group such that $|G|=25=5^2$. Then $G$ has Sylow $5$-subgroup(s) order $25$ such that $n_5\equiv 1$ (mod $5)$.

• I could not imagine how I would continue i.e., is it simple or not ?

Answer 1

The abelian groups of order $p^2$ are $\mathbb{Z}/p^2\mathbb{Z}, \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} $ and both are not simple.

Answer 2

Note that if p is the smallest prime dividing the order of G, then any subgroup of index p is normal, and since group of prime power order have subgroup of index p, hence they can not be simple.