Difference between revisions of "Homomorph-dominating subgroup"

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(New page: {{wikilocal}} {{subgroup property}} ==Definition== A subgroup <math>H</math> of a group <math>G</math> is termed '''homomorph-dominating''' in <math>G</math> if, for any homomorp...)
 
(Weaker properties)
 
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===Weaker properties===
 
===Weaker properties===
  
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* [[Stronger than::Endomorph-dominating subgroup]]
 
* [[Isomorph-conjugate subgroup]] if the whole group is a [[co-Hopfian group]] -- it is not isomorphic to any proper subgroup of itself.
 
* [[Isomorph-conjugate subgroup]] if the whole group is a [[co-Hopfian group]] -- it is not isomorphic to any proper subgroup of itself.
  

Latest revision as of 18:23, 19 September 2008

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

A subgroup H of a group G is termed homomorph-dominating in G if, for any homomorphism \varphi \in \operatorname{Hom}(H,G), there exists g \in G such that \varphi(H) \le gHg^{-1}.

Relation with other properties

Stronger properties

Weaker properties

Conjunction with other properties

A homomorph-containing subgroup is precisely the same as a subgroup that is both normal and homomorph-dominating. For full proof, refer: Homomorph-dominating and normal equals homomorph-containing