Classification of finitely generated abelian groups

Classification of finitely generated abelian groups: If $G$ is a finitely generated abelian group, then there exist integers $r\ge 0$ and $n_1,\dots,n_t\ge 2$ with $n_1\mid n_2\mid\cdots\mid n_t$ such that

G \cong \mathbb Z^{\,r}\;\oplus\;\bigoplus_{i=1}^t \mathbb Z/n_i\mathbb Z.

The integers $r$ and the invariant factors $n_i$ are uniquely determined by $G$ .

This is obtained by applying the structure theorem over a PID to the PID $\mathbb Z$ : a finitely generated abelian group is a finitely generated module and decomposes as a direct sum of cyclic pieces.