If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?
Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$