Difference between revisions of "Hypoabelian group"

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(Definition)
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===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]]
 
* The [[perfect core]] is [[trivial group|trivial]]
* The [[hypoAbelianization]] is the whole group
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* The [[hypoabelianization]] is the whole group
 
* The [[derived series]] terminates at the identity
 
* The [[derived series]] terminates at the identity
 
* There is no nontrivial [[perfect group|perfect]] subgroup.
 
* 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 [[normal series]] where all the successive quotients are abelian
  
 
==Relation with other properties==
 
==Relation with other properties==

Revision as of 14:24, 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 solvability|Find other variations of solvability |

This is an opposite of perfectness

Definition

Symbol-free definition

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

Relation with other properties

Stronger properties

Weaker properties