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Abelian group

5,705 bytes added, 15:35, 11 April 2017
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.
An '''abelian group''' is a [[group]] where any two elements commute.<center>{{#widget:YouTube|id=uMVm9oSoa6A}}</center>
===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> (here <math>xy</math> denotes the product of <math>x</math> and <math>y</math> in <math>G</math>).
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===Equivalent formulations===
* A group <math>G</math> is termed abelian if its it satisfies the following equivalent conditions: * Its [[defining ingredient::center]] <math>Z(G)</math> is the whole group.* A group is abelian if its Its [[defining ingredient::commutator derived subgroup]] <math>G' = [G,G]</math> is trivial.* (Choose a generating set <math>S</math> for <math>G</math>). For any elements <math>a,b \in S</math>, <math>ab = ba</math>.* The diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[defining ingredient::normal subgroup]] inside <math>G \times G</math>.
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==Notation==
==Examples==
 
{{group property see examples}}
===Some infinite examples===
===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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==Facts==
===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.
==FormalismsMetaproperties==
{{obtainedbyapplyingthe|diagonalclass="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-inclosed group property]] || Yes || [[abelianness is quotient-square operatorclosed]] ||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.|}
A group <math>G</math> is an abelian group if and only if, in the [[external direct product]] <math>G \times G</math>, the diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[normal subgroup]].==Relation with other properties==
==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}}|||}
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}} || {{intermediate notions short|nilpotent group|abelian group}}|-| [[stronger than::solvable group]] || [[derived series]] reaches identity, has [[normal series]] with abelian factor groups form a || [[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}}|-| [[variety stronger than::virtually abelian group]] || has abelian subgroup of algebrasfinite index || || {{strictness examples|virtually abelian group|abelian group}} || {{intermediate notions short|virtually abelian group|abelian group}}|-| [[stronger than::FZ-group]]. The defining equations for this variety are the equations for a || center has finite index || || {{strictness examples|FZ-group|abelian group}} || {{intermediate notions short|FZ-group|abelian group}}|-| [[stronger than::FC-group]] along with the commutativity equation.|| every conjugacy class is finite || || {{strictness examples|FC-group|abelian group}} || {{intermediate notions short|FC-group|abelian group}}|}
{{S-closed}}===Incomparable properties===
Any * [[subgroupSupersolvable group]] is a group that has a [[normal series]] where all the successive quotient groups are [[cyclic group]] of an s. An abelian group is abelian -- viz., the property of being abelian supersolvable if and only if it is [[subgroup-closed finitely generated abelian group property|subgroup-closedfinitely generated]]. This follows as * [[Polycyclic group]] is a direct consequence of abelianness being varietal. {{proofat|group that has a [[subnormal series]] where all the successive quotent groups are [[Abelianness is subgroup-closedcyclic group]]}}s. An abelian group is polycyclic if and only if it is finitely generated.
{{Q-closed}}==Formalisms==
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. {{proofatobtainedbyapplyingthe|[[Abelianness is quotientdiagonal-closed]]in-square operator|normal subgroup}}
{{DP-closed}} A [[direct product]] of abelian groups group <math>G</math> is an abelian -- viz group if and only if, in the property of being abelian is [[external direct product-closed group property|direct product-closed]]. This again follows as <math>G \times G</math>, the diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a direct consequence of abelianness being varietal. {{proofat|[[Abelianness is direct product-closednormal subgroup]]}}.
==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:
<prett>IsAbelian (group)</prett>
where <prett>group</prett> either defines the group or gives the name to a group previously defined.
==Study of this notion==
==References==
===Textbook references===
* {| class="sortable" border="1"! Book !! Page number !! Chapter and section !! Contextual information !! View|-| {{booklink-defined-tabular|DummitFoote}}, Page |17 |Formal definition (definition as Point point (2) in general definition of a group)|}} ||* |-| {{booklink-defined-tabular|AlperinBell}}, Page |2 |1.1 (Rudiments of Group Theory/Review)|definition introduced in paragraph)}} || [https://books.google.com/books?id=EroGCAAAQBAJ&pg=PA2 Google Books]|-* | {{booklink-defined-tabular|Artin}}, Page |42 ||definition introduced in paragraph (defined immediately after the definition of group, as a group where the composition is commutative)}} ||* |-| {{booklink-defined-tabular|Herstein|28||Formal definition}}, Page 28 (formal definition)|||-* | {{booklink-defined-tabular|RobinsonGT}}, Page |2 |1.1 (Binary Operations, Semigroups, and Groups)|formal definition)}} || [https://books.google.com/books?id=EroGCAAAQBAJ&pg=PA2 Google Books]|-* | {{booklink-defined-tabular|FGTAsch}}, Page |1|1.1 (Elementary group theory)|definition introduced in paragraph)}} || [https://books.google.com/books?id=BprbtnlI6HEC&pg=PA1 Google Books]|}
==External links==
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