# fundamental theorem of finitely generated abelian groups

###### Theorem 1 (Fundamental Theorem of Finitely Generated Abelian Groups).

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

 $G\cong\mathbb{Z}^{r}\oplus\mathbb{Z}/n_{1}\mathbb{Z}\oplus\mathbb{Z}/n_{2}% \mathbb{Z}\oplus\ldots\oplus\mathbb{Z}/n_{s}\mathbb{Z}$

for some integers $r,n_{i}$ satisfying:

 $r\geq 0;\quad\forall i,n_{i}\geq 2;\quad n_{i+1}\mid n_{i}\ \text{for }1\leq i% \leq s-1.$
Title fundamental theorem of finitely generated abelian groups FundamentalTheoremOfFinitelyGeneratedAbelianGroups 2013-03-22 13:54:12 2013-03-22 13:54:12 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 20E34 classification of finitely generated abelian groups AbelianGroupsOfOrder120 FinitelyGenerated AbelianGroup2 fundamental theorem of finitely generated abelian groups