Difference between revisions of "Homomorph-containing subgroup"

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(Relation with other properties)
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* [[Weaker than::Order-containing subgroup]]
 
* [[Weaker than::Order-containing subgroup]]
 +
* [[Weaker than::Subhomomorph-containing subgroup]]
 +
* [[Weaker than::Variety-containing subgroup]]
 
* [[Weaker than::Normal Sylow subgroup]]
 
* [[Weaker than::Normal Sylow subgroup]]
 
* [[Weaker than::Normal Hall subgroup]]
 
* [[Weaker than::Normal Hall subgroup]]
 +
 
===Weaker properties===
 
===Weaker properties===
  
* [[Stronger than::Intermediately fully characteristic subgroup]]
+
* [[Stronger than::Fully characteristic subgroup]]: Also related:
* [[Stronger than::Fully characteristic subgroup]]
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** [[Stronger than::Intermediately fully characteristic subgroup]]
* [[Stronger than::Strictly characteristic subgroup]]
+
** [[Stronger than::Strictly characteristic subgroup]]
* [[Stronger than::Intermediately characteristic subgroup]]
+
** [[Stronger than::Intermediately characteristic subgroup]]
* [[Stronger than::Characteristic subgroup]]
+
** [[Stronger than::Characteristic subgroup]]
* [[Isomorph-free subgroup]] in case the subgroup is [[co-Hopfian group|co-Hopfian as a group]]: it is not isomorphic to any proper subgroup of itself.
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* [[Stronger than::Isomorph-containing subgroup]]
  
 
==Facts==
 
==Facts==

Revision as of 18:03, 9 March 2009

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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-containing if for any \varphi \in \operatorname{Hom}(H,G), the image \varphi(H) is contained in H.

Relation with other properties

Stronger properties

Weaker properties

Facts

Metaproperties

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties