# Definitions for **abelian group**

### This page provides all possible meanings and translations of the word **abelian group**

### Princeton's WordNet

Abelian group, commutative group(noun)

a group that satisfies the commutative law

### Wiktionary

abelian group(Noun)

a group in which the group operation is commutative

### Freebase

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order. Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel. The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research.

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"abelian group." *Definitions.net.* STANDS4 LLC, 2014. Web. 25 Oct. 2014. <http://www.definitions.net/definition/abelian group>.