PlanetMath: fundamental theorem of finitely generated abelian groups

(more info)

Math for the people, by the people.

Main Menu

fundamental theorem of finitely generated abelian groups

(Theorem)

Theorem 1 (Fundamental Theorem of Finitely Generated Abelian Groups)

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

$\displaystyle G\cong \mathbb{Z}^{r}\oplus\mathbb{Z}/n_1\mathbb{Z}\oplus\mathbb{Z}/n_2\mathbb{Z}\oplus\ldots\oplus\mathbb{Z}/n_s\mathbb{Z}$

for some integers satisfying:

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

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

(view preamble | get metadata)

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

Log in to rate this entry.
(view current ratings)

Cross-references: integers, expression, abelian groups, finitely generated
There are 5 references to this entry.

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 8590 times total.

Classification:

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

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact