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:

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

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

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

Title	fundamental theorem of finitely generated abelian groups
Canonical name	FundamentalTheoremOfFinitelyGeneratedAbelianGroups
Date of creation	2013-03-22 13:54:12
Last modified on	2013-03-22 13:54:12
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Theorem
Classification	msc 20E34
Synonym	classification of finitely generated abelian groups
Related topic	AbelianGroupsOfOrder120
Related topic	FinitelyGenerated
Related topic	AbelianGroup2
Defines	fundamental theorem of finitely generated abelian groups