Abelian group of prime-square order ($G\cong C_{p^2}\text{ or }{C}_p\times C_p$)

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 $a\ne 1$ and $b\notin ⟨a⟩$. The group homomorphism $(i,j)\mapsto a^i{b}^j$ from $C_p\times C_p$ is injective: $b^j\in ⟨a⟩$ with $j\ne 0$ would imply $b\in ⟨a⟩$, using the inverse of $j$ modulo $p$. Equal finite orders then make this an isomorphism.