= Abelian group of prime-square order
{title2=$G\cong C_{p^2}\text{ or }C_p\times C_p$}
= Abelian groups of prime-square order
{synonym}
An <abelian group> of order $p^2$ is cyclic if it has an element of order $p^2$. Otherwise <Lagrange's theorem> makes every nonidentity element have order $p$. Choose $a\ne1$ and $b\notin\langle a\rangle$. The <group homomorphism> $(i,j)\mapsto a^ib^j$ from $C_p\times C_p$ is injective: $b^j\in\langle a\rangle$ with $j\ne0$ would imply $b\in\langle a\rangle$, using the inverse of $j$ modulo $p$. Equal finite orders then make this an isomorphism.
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