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Abelian group

578 bytes added, 15:35, 11 April 2017
Weaker properties
* 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>.
<section begin=beginner/>
===Occurrence as quotients===
The maximal abelian quotient of any group is termed its [[abelianization]], and this is the quotient by the [[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.
==Metaproperties==
| [[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.
|}
 
==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]] <math>G \times G</math>, the diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[normal subgroup]].
==Relation with other properties==
| [[weaker than::homocyclic group]] || direct product of isomorphic cyclic groups || || {{strictness examples|abelian group|homocyclic group}} || {{intermediate notions short|abelian group|homocyclic group}}||
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| [[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}}||
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| [[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}}||
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| [[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}}||
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| [[weaker than::finite abelian group]] || abelian and a [[finite group]] || || {{strictness examples|abelian group|finite group}} || {{intermediate notions short|abelian group|finite abelian group}}||
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| [[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}}
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| [[stronger than::FZ-group]] || center has finite index || || {{strictness examples|FZ-group|abelian group}} || {{intermediate notions short|FZ-group|abelian group}}
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| [[stronger than::FC-group]] || every conjugacy class is finite || || {{strictness examples|FC-group|abelian group}} || {{intermediate notions short|FC-group|abelian group}}
|}
* [[Supersolvable group]] is a group that has a [[normal series]] where all the successive quotient groups 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 [[subnormal series]] where all the successive quotent groups are [[cyclic group]]s. An abelian group is polycyclic if and only if it is finitely generated.
 
==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]] <math>G \times G</math>, the diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[normal subgroup]].
==Testing==
! Book !! Page number !! Chapter and section !! Contextual information !! View
|-
| {{booklink-defined-tabular|DummitFoote|17|Formal definition (definition as point (2) in general definition of group)|}}||
|-
| {{booklink-defined-tabular|AlperinBell|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|42||definition introduced in paragraph (immediately after definition of group)|}}|||-* | {{booklink-defined-tabular|Herstein|28||Formal definition|}}|||-* | {{booklink-defined-tabular|RobinsonGT|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|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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