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→Weaker properties

{{basicdef}}

{{pivotal group property}}

==History==

===Origin of the term===

The term '''~~Abelian ~~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. The word ''abelian'' is usually begun with a small ''a''. {{quotation|'''wikinote''': Some older content on the wiki uses capital A for Abelian. We're trying to update this content.}}

<section begin=beginner/>

==Definition==

An '''abelian group''' is a [[group]] where any two elements commute. In 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> (here <math>xy</math> denotes the product of <math>x</math> and <math>y</math> in <math>G</math>). Note that <math>x,y</math> are allowed to be equal, though equal elements commute anyway, so we can restrict attention if we wish to unequal elements. <center>{{#widget:YouTube|id=~~=Symbol-free definition===~~uMVm9oSoa6A}}</center>

===~~Definition with symbols~~Full definition===

===Equivalent formulations===

==Facts==

===Occurrence as subgroups===

Every [[cyclic group]] is ~~Abelian~~abelian. Since each group is generated by its cyclic subgroups, every group is generated by a family of ~~Abelian ~~abelian subgroups. A trickier question is: do there exist ~~Abelian ~~abelian [[normal subgroup]]s? A good candidate for an ~~Abelian ~~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 ~~abelian quotient of any group is termed its [[~~Abelianization~~abelianization]], and this is the quotient by the [[~~commutator ~~derived subgroup]]. A subgroup is an [[abelian-quotient subgroup]] (i.e., normal with ~~Abelian ~~abelian quotient group ) if and only if the subgroup contains the ~~commutator ~~derived subgroup.

==Metaproperties==

{| class="sortable" border="1"! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols|-| [[satisfies metaproperty::varietal group property]] || Yes || || The collection of abelian groups forms a subvariety of the variety of groups. In particular, it is closed under taking subgroups, quotients, and arbitrary direct products|-| [[satisfies metaproperty::subgroup-closed group property]] || Yes || [[abelianness is subgroup-closed]] || If <math>G</math> is an abelian group and <math>H</math> is a subgroup of <math>G</math>, then <math>H</math> is abelian.|-| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[abelianness is quotient-closed]] || If <math>G</math> is an abelian group and <math>H</math> is a normal subgroup of <math>G</math>, the [[quotient group]] <math>G/H</math> is abelian.|-| [[satisfies metaproperty::direct product-closed group property]] || Yes || [[abelianness is direct product-closed]] || Suppose <math>G_i, i \in I</math>, are abelian groups. Then, the external direct product <math>\prod_{~~varietal~~i \in I}G_i</math> is also abelian.|}

===Weaker properties==={| class="sortable" border="1"! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions |-| [[stronger than::nilpotent group]] || [[lower central series]] reaches identity, [[upper central series]] reaches whole group || [[abelian implies nilpotent]] || [[nilpotent not implies abelian]] {{strictness examples|nilpotent group|abelian group}} || {{~~Q~~intermediate notions short|nilpotent group|abelian group}}|-~~closed~~| [[stronger than::solvable group]] || [[derived series]] reaches identity, has [[normal series]] with abelian factor groups || [[abelian implies solvable]] || [[solvable not implies abelian]] {{strictness examples|solvable group|abelian group}} || {{intermediate notions short|solvable group|abelian group}}|-| [[stronger than::metabelian group]] || has [[abelian normal subgroup]] with abelian quotient group || || {{strictness examples|metabelian group|abelian group}} || {{intermediate notions short|metabelian group|abelian group}}|-| [[stronger than::virtually abelian group]] || has abelian subgroup of finite index || || {{strictness examples|virtually abelian group|abelian group}} || {{intermediate notions short|virtually abelian group|abelian group}}|-| [[stronger than::FZ-group]] || center has finite index || || {{strictness examples|FZ-group|abelian group}} || {{intermediate notions short|FZ-group|abelian group}}|-| [[stronger than::FC-group]] || every conjugacy class is finite || || {{strictness examples|FC-group|abelian group}} || {{intermediate notions short|FC-group|abelian group}}|}

==Formalisms== {{obtainedbyapplyingthe|diagonal-in-square operator|normal subgroup}} A group <math>G</math> is an abelian group if and only if, in the [[external direct product]] ~~of Abelian groups is Abelian -- viz ~~<math>G \times G</math>, the ~~property of being Abelian ~~diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[~~direct product-closed group property|direct product-closed~~normal subgroup]]~~. This again follows as a direct consequence of Abelianness being varietal~~.

==Testing==

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

The abelianness testing problem is the problem of testing whether a group (described using some [[group description rule]], such as an [[encoding of a group]] or a [[multi-encoding of a group]]) is abelian. Algorithms for the abelianness testing problem range from the [[brute-force black-box group algorithm for abelianness testing]] (that involves testing for ''every'' pair of elements whether they commute, and is quadratic in the order of the group) to the [[generating set-based black-box group algorithm for abelianness testing]] (that involves testing only on a generating set, and is quadratic in the size of the generating set). {{GAP command for gp|test = IsAbelian|class = AbelianGroups}} To test whether a group is abelian, the GAP syntax is: <tt>IsAbelian (group)</tt> where <tt>group</tt> either defines the group or gives the name to a group previously defined. ==Study of this notion==

==External links==

===Definition links===

* {{wp-defined|~~Abelian_group~~Abelian group}}* {{planetmath-defined|AbelianGroup2}}

* {{mathworld|AbelianGroup}}

* {{sor-defined|A/a010230.htm}}

===Perspective links===

* {{chapman|Abelian_groups}}

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