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# 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:

$r\geq 0;\quad\forall i,n_{i}\geq 2;\quad n_{{i+1}}\mid n_{i}\ \text{for }1\leq i% \leq s-1.$ |

Defines:

fundamental theorem of finitely generated abelian groups

Keywords:

finitely generated, abelian group

Related:

AbelianGroupsOfOrder120, FinitelyGenerated, AbelianGroup2

Synonym:

classification of finitely generated abelian groups

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

20E34*no label found*

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