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.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 62 \delta$