Karl Jones: Created page with "In abstract algebra, an '''abelian group''', also called a '''commutative group''', is a group in which the result of applying the group operation..."

2016-09-22T18:03:37Z

Created page with "In abstract algebra, an '''abelian group''', also called a '''commutative group''', is a group in which the result of applying the group operation..."

New page

In [[abstract algebra]], an '''abelian group''', also called a '''commutative group''', is a [[Group (mathematics)|group]] in which the result of applying the group operation to two group elements does not depend on the order in which they are written (the axiom of [[Commutative property|commutativity)]].

== Description ==

Abelian groups generalize the [[arithmetic]] of [[addition]] of [[Integer|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, from which many other basic concepts, such as [[Module (mathematics)|modules]] and [[Vector space|vector spaces]] are developed.

The theory of abelian groups is generally simpler than that of their [[Non-abelian group|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.

== See also ==

* [[Abelianization]]
* [[Class field theory]]
* [[Commutator subgroup]]
* [[Dihedral group of order 6]], the smallest non-Abelian group
* [[Elementary abelian group]]
* [[Pontryagin duality]]
* [[Pure injective module]]
* [[Pure projective module]]

== External links ==

* [ Abelian group] @ Wikipedia.org

[[Category:Mathematics]]

Abelian group - Revision history

Karl Jones: Created page with "In abstract algebra, an '''abelian group''', also called a '''commutative group''', is a group in which the result of applying the group operation..."