View source for Abelian group

{{pivotal group property}}

{{basicdef}}

==History==

===Origin of the term===

The term '''Abelian group''' comes from Niels Henrick Abel, a mathematician who worked with groups even before the formal theory was laid down, in order to prove unsolvability of the quintic.

==Definition==

===Symbol-free definition===

An '''Abelian group''' is a group where any two elements commute.

===Definition with symbols===

A group <math>G</math> is termed '''Abelian''' if for any elements <math>x</math> and <math>y</math> in <math>G</math>, <math>xy = yx</math>.

===Equivalent formulations===

* A group is Abelian if its [[center]] is the whole group.
* A group is Abelian if its [[commutator subgroup]] is trivial.

==Examples==

[[Cyclic group]]s are good examples of Abelian groups. Further, any direct product of cyclic groups is also an Abelian group. Further, every [[finitely generated group|finitely generated]] Abelian group is obtained this way. This is the famous [[structure theorem for finitely generated Abelian groups]].

The structure theorem can be used to generate a complete listing of finite Abelian groups, as described here: [[classification of finite Abelian groups]].
==Facts==

===Occurrence as subgroups===

Every [[cyclic group]] is Abelian. Since each group is generated by its cyclic subgroups, every group is generated by a family of Abelian subgroups. A trickier question is: do there exist Abelian [[normal subgroup]]s? A good candidate for an Abelian normal subgroup is the [[center]], which is the collection of elements of the group that commute with ''every'' element of the group.

===Occurrence as quotients===

The maximal Abelian quotient of any group is termed its [[Abelianization]], and this is the quotient by the [[commutator subgroup]]. A subgroup is normal with Abelian quotient group if and only if the subgroup contains the commutator subgroup.

==Metaproperties==

{{varietal}}

Abelian groups form a [[variety of algebras]]. The defining equations for this variety are the equations for a [[group]] along with the commutativity equation.

{{S-closed}}

Any [[subgroup]] of an Abelian group is Abelian -- viz the property of being Abelian is [[subgroup-closed group property|subgroup-closed]]. This follows as a direct consequence of Abelianness being varietal.

{{Q-closed}}

Any [[quotient]] of an Abelian group is Abelian -- viz the property of being Abelian is [[quotient-closed group property|quotient-closed]]. This again follows as a direct consequence of Abelianness being varietal.

{{DP-closed}}

A [[direct product]] of Abelian groups is Abelian -- viz the property of being Abelian is [[direct product-closed group property|direct product-closed]]. This again follows as a direct consequence of Abelianness being varietal.

==Testing==

===The testing problem===

{{further|[[Abelianness testing problem]]}}

{{GAP command for gp|IsAbelian}}

To test whether a group is Abelian, the GAP syntax is:

<pre>IsAbelian (group)</pre>

where <pre>group</pre> either defines the group or gives the name to a group previously defined.

==Study of this notion==

{{msc class|20K}}

==External links==

===Definition links===

* {{wp|Abelian group}}
* {{planetmath|AbelianGroup2}}
* {{mathworld|AbelianGroup}}
* {{sor|A/a010230.htm}}

===Perspective links===
* {{chapman|Abelian_groups}}