PlanetMath: fundamental theorem of finitely generated abelian groups

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fundamental theorem of finitely generated abelian groups

(Theorem)

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: $$G\cong \Ints^{r}\oplus\Ints/n_1\Ints\oplus\Ints/n_2\Ints\oplus\ldots\oplus\Ints/n_s\Ints$$ for some integers $r,n_i$ satisfying: $$r\geq 0;\quad \forall i, n_i\geq 2;\quad n_{i+1}\mid n_i\ \text{for } 1\leq i\leq s-1.$$

"fundamental theorem of finitely generated abelian groups" is owned by alozano. [ full author list (2) | owner history (1) ]

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See Also: abelian groups of order $120$

, finitely generated group, abelian group

Other names:

classification of finitely generated abelian groups

Also defines:

fundamental theorem of finitely generated abelian groups

Keywords:

finitely generated, abelian group

Attachments:: abelian groups of order (Example) by alozano
proof of fundamental theorem of finitely generated abelian groups (Proof) by puuhikki

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Cross-references: integers, expression, abelian groups, finitely generated, theorem
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This is version 3 of fundamental theorem of finitely generated abelian groups, born on 2003-08-25, modified 2008-05-19.
Object id is 4652, canonical name is FundamentalTheoremOfFinitelyGeneratedAbelianGroups.
Accessed 10402 times total.

Classification:

AMS MSC:

20E34 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: General structure theorems)

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