Structure Theorem for Finitely Generated Abelian Groups

Every finitely generated abelian group is isomorphic to

$\mathbb{Z}^r \times \mathbb{Z}_{n_1} \times \cdots \times \mathbb{Z}_{n_k}$

with $n_1 \mid n_2 \mid \cdots \mid n_k$.

Why it matters

It classifies finitely generated abelian groups up to isomorphism.