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→Weaker properties

==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 =~~==Symbol-free definition===~~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.

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===Equivalent formulations===

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==Notation==

===Non-examples===

Not every group is abelian. The smallest non-abelian group is [[symmetric group:S3|the symmetric group on three letters]]: the group of all permutations on three letters, under composition. Its being non-abelian hinges on the fact that the order in which permutations are performed~~, ~~matters.

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===Occurrence as quotients===

The maximal abelian quotient of any group is termed its [[abelianization]], and this is the quotient by the [[~~commutator ~~derived subgroup]]. A subgroup is an [[abelian-quotient subgroup]] (i.e., normal with abelian quotient group) if and only if the subgroup contains the ~~commutator ~~derived subgroup.

==~~Formalisms~~Metaproperties==

{~~{obtainedbyapplyingthe~~|~~diagonal~~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-~~in~~closed group property]] || Yes || [[abelianness is quotient-~~square operator~~closed]] ||If <math>G</math> is an abelian group and <math>H</math> is a normal subgroupof <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_{i \in I}G_i</math> is also abelian.|}

==~~Metaproperties~~=Stronger properties===

{| class="sortable" border="1"! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Comparison|-| [[weaker than::cyclic group]] || generated by one element || [[cyclic implies abelian]] || [[abelian not implies cyclic]] {~~varietal~~{strictness examples|abelian group|cyclic group}} || {{intermediate notions short|abelian group|cyclic group}} |||-| [[weaker than::homocyclic group]] || direct product of isomorphic cyclic groups || || {{strictness examples|abelian group|homocyclic group}} || {{intermediate notions short|abelian group|homocyclic group}}|||-| [[Weaker than::residually cyclic group]] || every non-identity element is outside a normal subgroup with a cyclic quotient group || || {{strictness examples|abelian group|residually cyclic group}} || {{intermediate notions short|abelian group|residually cyclic group}} || |-| [[Weaker than::locally cyclic group]] || every finitely generated subgroup is cyclic || || {{strictness examples|abelian group|locally cyclic group}} || {{intermediate notions short|abelian group|locally cyclic group}} || |-| [[Weaker than::epabelian group]] || abelian group whose exterior square is the trivial group || || {{strictness examples|abelian group|epabelian group}} || {{intermediate notions short|abelian group|epabelian group}} || |-| [[weaker than::finite abelian group]] || abelian and a [[finite group]] || || {{strictness examples|abelian group|finite group}} || {{intermediate notions short|abelian group|finite abelian group}}|||-| [[weaker than::finitely generated abelian group]] || abelian and a [[finitely generated group]] || || {{strictness examples|abelian group|finitely generated group}} || {{intermediate notions short|abelian group|finitely generated abelian group}}|||}

==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|

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

<~~pre~~tt>IsAbelian (group)</~~pre~~tt>

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

==Study of this notion==

==References==

===Textbook references===

==External links==

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