fundamental theorem of finitely generated abelian groups

Theorem 1 (Fundamental Theorem of Finitely Generated AbelianGroups).

Let $G$ be a finitely generated abelian group. Then there is aunique expression of the form:

$$G\cong {\mathbb{Z}}^r\oplus \mathbb{Z}/n_1\mathbb{Z}\oplus \mathbb{Z}/n_2\mathbb{Z}\oplus \mathrm{\dots }\oplus \mathbb{Z}/n_s\mathbb{Z}$$

for some integers $r\mathrm{,}{n}_i$ satisfying:

$$r\ge 0;\forall i,n_i\ge 2;n_{i+1}\mid n_i\mathit{\text{for }}1\le i\le s-1.$$