Group Types of Order Prime Squared

Theorem
Let $p$ be a prime number.

Let $G$ be a group of order $p^2$.

Then $G$ is isomorphic either to $\Z_{p^2}$ or to $\Z_p \times \Z_p$, where $\Z_p$ denotes the additive group of integers modulo $p$.

Proof
From Group of Order Prime Squared is Abelian, $G$ is an abelian group.

From Abelian Group of Prime-power Order is Product of Cyclic Groups, $G$ is either:
 * the cyclic group of order $p^2$
 * the direct product of the cyclic group of order $p$ with itself.

The result follows from Finite Cyclic Group is Isomorphic to Integers under Modulo Addition.