# Changes

## Abelian group

, 15:35, 11 April 2017
Weaker properties
* Its [[defining ingredient::derived subgroup]] $G' = [G,G]$ is trivial.
* (Choose a generating set $S$ for $G$). For any elements $a,b \in S$, $ab = ba$.
* The diagonal subgroup $\{ (g,g) \mid g \in G \}$ is a [[defining ingredient::normal subgroup]] inside $G \times G$.
<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 $G_i, i \in I$, are abelian groups. Then, the external direct product $\prod_{i \in I} G_i$ is also abelian.
|}

==Formalisms==

{{obtainedbyapplyingthe|diagonal-in-square operator|normal subgroup}}

A group $G$ is an abelian group if and only if, in the [[external direct product]] $G \times G$, the diagonal subgroup $\{ (g,g) \mid g \in G \}$ 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}}
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* [[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 $G$ is an abelian group if and only if, in the [[external direct product]] $G \times G$, the diagonal subgroup $\{ (g,g) \mid g \in G \}$ is a [[normal subgroup]].
==Testing==
! Book !! Page number !! Chapter and section !! Contextual information !! View
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| {{booklink-defined-tabular|DummitFoote|17|Formal definition (definition as point (2) in general definition of group)|}}||
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