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→Weaker properties

===Equivalent formulations===

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===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-~~in~~closed group property]] || Yes || [[abelianness is subgroup-~~square operator~~closed]] ||~~normal ~~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.|-~~A ~~| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[abelianness is quotient-closed]] || If <math>G</math> is an abelian group ~~if ~~and ~~only if~~<math>H</math> is a normal subgroup of <math>G</math>, ~~in ~~the [[~~external ~~quotient group]] <math>G/H</math> is abelian.|-| [[satisfies metaproperty::direct product-closed group property]] || Yes || [[abelianness is direct product-closed]] || Suppose <math>~~G ~~G_i, i \~~times G~~in I</math>, are abelian groups. Then, the ~~diagonal subgroup ~~external direct product <math>\prod_{ ~~(g,g) \mid g ~~i \in ~~G \~~I}G_i</math> is ~~a [[normal subgroup]]~~also abelian.|}

==Relation with other properties==

{| class="sortable" border="1"

! Property !! Meaning !! Proof of implication !! ~~pProof ~~Proof of strictness (reverse implication failure) !! Intermediate notions !! Comparison

|-

| [[weaker than::cyclic group]] || generated by one element || [[cyclic implies abelian]] || [[abelian not implies cyclic]] {{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}}||

|-

| [[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}}

|}

==~~Metaproperties~~=Incomparable properties=== ~~{{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|

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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