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 ?