Group of order $1225$ is abelian
A group of order $p^2$, where $p$ is a prime, is isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_{p} \times \mathbb{Z}_{p}$. You seem to miss the latter case.
Group of order $1225$ is abelian
A group of order $p^2$, where $p$ is a prime, is isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_{p} \times \mathbb{Z}_{p}$. You seem to miss the latter case.