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.