Difference between revisions of "Hypoabelian group"

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{{group property}}
 
{{group property}}
  
{{variationof|solvability}}
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{{variation of|solvable group}}
 
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{{opposite of|perfect group}}
{{oppositeof|perfectness}}
 
 
 
 
==Definition==
 
==Definition==
  
 
===Symbol-free definition===
 
===Symbol-free definition===
  
A [[group]] is termed '''hypoAbelian''' if the following equivalent conditions are satisfied:
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A [[group]] is termed '''hypoabelian''' if the following equivalent conditions are satisfied:
  
* The [[perfect core]] is [[trivial group|trivial]]
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# The [[defining ingredient::perfect core]] is [[trivial group|trivial]]
* The [[hypoAbelianization]] is the whole group
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# The [[defining ingredient::hypoabelianization]] is the quotient by the trivial subgroup, and hence, isomorphic to the whole group.
* The [[derived series]] terminates at the identity
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# The transfinite [[derived series]] terminates at the identity. (Note that this is the ''transfinite'' derived series, where the successor of a given subgroup is its [[commutator subgroup]] and subgroups at limit ordinals are given by intersecting all previous subgroups.)
* There is no nontrivial [[perfect group|perfect]] subgroup.
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# There is no nontrivial [[perfect group|perfect]] subgroup.
* There is a descending [[strongly normal series]] where all the successive quotients are Abelian
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# There is a descending transfinite [[normal series]] where all the successive quotients are abelian
  
 
==Relation with other properties==
 
==Relation with other properties==
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===Stronger properties===
 
===Stronger properties===
  
* [[Solvable group]]
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{| class="wikitable" border="1"
* [[Hypocentral group]]
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! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
 
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|-
===Weaker properties===
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| [[Weaker than::Solvable group]] || [[derived series]] terminates at identity in finitely many steps || [[solvable implies hypoabelian]] || [[hypoabelian not implies solvable]] || {{intermediate notions short|hypoabelian group|solvable group}}
 
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|-
* [[Locally solvable group]]
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| [[Weaker than::Hypocentral group]] || transfinite [[lower central series]] terminates at identity || [[hypocentral implies hypoabelian]] || [[hypoabelian not implies hypocentral]] || {{intermediate notions short|hypoabelian group|hypocentral group}}
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|-
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| [[Weaker than::Residually solvable group]] || intersection of all finite members of [[derived series]] is identity || [[residually solvable implies hypoabelian]] || [[hypoabelian not implies residually solvable]] || {{intermediate notions short|hypoabelian group|residually solvable group}}
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|-
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| [[Weaker than::Free group]] || Free on a given generating set || (via residually solvable) || (via residually solvable) || {{intermediate notions short|hypoabelian group|free group}}
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|}

Latest revision as of 14:35, 24 October 2009

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of solvable group|Find other variations of solvable group |

This is an opposite of perfect group

Definition

Symbol-free definition

A group is termed hypoabelian if the following equivalent conditions are satisfied:

  1. The perfect core is trivial
  2. The hypoabelianization is the quotient by the trivial subgroup, and hence, isomorphic to the whole group.
  3. The transfinite derived series terminates at the identity. (Note that this is the transfinite derived series, where the successor of a given subgroup is its commutator subgroup and subgroups at limit ordinals are given by intersecting all previous subgroups.)
  4. There is no nontrivial perfect subgroup.
  5. There is a descending transfinite normal series where all the successive quotients are abelian

Relation with other properties

Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Solvable group derived series terminates at identity in finitely many steps solvable implies hypoabelian hypoabelian not implies solvable Residually solvable group|FULL LIST, MORE INFO
Hypocentral group transfinite lower central series terminates at identity hypocentral implies hypoabelian hypoabelian not implies hypocentral |FULL LIST, MORE INFO
Residually solvable group intersection of all finite members of derived series is identity residually solvable implies hypoabelian hypoabelian not implies residually solvable |FULL LIST, MORE INFO
Free group Free on a given generating set (via residually solvable) (via residually solvable) Residually nilpotent group, Residually solvable group|FULL LIST, MORE INFO