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Abelian group

3,097 bytes added, 15:35, 11 April 2017
Weaker properties
===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::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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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.
{{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-inclosed group property]] || Yes || [[abelianness is subgroup-square operatorclosed]] ||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 Gin 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}}
Abelian * [[Supersolvable group]] is a group that has a [[normal series]] where all the successive quotient groups form are [[cyclic group]]s. An abelian group is supersolvable if and only if it is [[finitely generated abelian group|finitely generated]].* [[Polycyclic group]] is a group that has a [[variety of algebrassubnormal series]]. The defining equations for this variety where all the successive quotent groups are the equations for a [[cyclic group]] along with the commutativity equations. An abelian group is polycyclic if and only if it is finitely generated.
Any [[subgroup]] of an abelian group is abelian -- viz., the property of being abelian is [[subgroup{{obtainedbyapplyingthe|diagonal-closed group property|subgroupin-closed]]. This follows as a direct consequence of abelianness being varietal. {{proofatsquare operator|[[Abelianness is normal subgroup-closed]]}}
{{Q-closed}} Any [[quotient]] of A group <math>G</math> is an abelian group is abelian -- viz if and only if, in the property of being abelian is [[quotient-closed group property|quotient-closed]]. This again follows as a external direct consequence of abelianness being varietal. {{proofat|[[Abelianness is quotient-closedproduct]]}} {<math>G \times G</math>, the diagonal subgroup <math>\{DP-closed}(g,g) \mid g \in G \A [[direct product]] of abelian groups is abelian -- viz the property of being abelian </math> is a [[direct product-closed group property|direct product-closednormal subgroup]]. This again follows as a direct consequence of abelianness being varietal. {{proofat|[[Abelianness is direct product-closed]]}}
{{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==
===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)}} || [ 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)}} || [ Google Books]|-* | {{booklink-defined-tabular|FGTAsch}}, Page |1|1.1 (Elementary group theory)|definition introduced in paragraph)}} || [ Google Books]|}
==External links==
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