Difference between revisions of "Homomorph-containing subgroup"

From Groupprops
Jump to: navigation, search
(Stronger properties)
Line 19: Line 19:
 
===Weaker properties===
 
===Weaker properties===
  
* [[Stronger than::Fully characteristic subgroup]]: Also related:
+
* [[Stronger than::Fully invariant subgroup]]: Also related:
** [[Stronger than::Intermediately fully characteristic subgroup]]
+
** [[Stronger than::Intermediately fully invariant subgroup]]
 
** [[Stronger than::Strictly characteristic subgroup]]
 
** [[Stronger than::Strictly characteristic subgroup]]
 
** [[Stronger than::Intermediately characteristic subgroup]]
 
** [[Stronger than::Intermediately characteristic subgroup]]
Line 35: Line 35:
  
 
{{trim}}
 
{{trim}}
 +
 +
For any group <math>G</math>, the trivial subgroup and the whole group are both homomorph-containing.
 +
 +
{{intransitive}}
 +
 +
We can have subgroups <math>H \le K \le G</math> such that <math>H</math> is a homomorph-containing subgroup of <math>K</math> and <math>K</math> is a homomorph-containing subgroup of <math>G</math> but <math>H</math> is not a homomorph-containing subgroup of <math>G</math>. {{proofat|[[Homomorph-containment is not transitive]]}}
  
 
{{intsubcondn}}
 
{{intsubcondn}}
 +
 +
If <math>H \le K \le G</math> and <math>H</math> is a homomorph-containing subgroup of <math>G</math>, <math>H</math> is also a homomorph-containing subgroup of <math>K</math>. {{proofat|[[Homomorph-containment satisfies intermediate subgroup condition]]}}
  
 
{{join-closed}}
 
{{join-closed}}
 +
 +
If <math>H_i, i \in I</math>, are all homomorph-containing subgroups of <math>G</math>, then so is the [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math>. {{proofat|[[Homomorph-containment is strongly join-closed]]}}
  
 
{{quot-transitive}}
 
{{quot-transitive}}
 +
 +
If <math>H \le K \le G</math> are groups such that <math>H</math> is a homomorph-containing subgroup of <math>G</math> and <math>K/H</math> is a homomorph-containing subgroup of <math>G/H</math>, then <math>K</math> is a homomorph-containing subgroup of <math>G</math>. {{proofat|[[Homomorph-containment is quotient-transitive]]}}

Revision as of 18:51, 15 June 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

For any group G, the trivial subgroup and the whole group are both homomorph-containing.

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

We can have subgroups H \le K \le G such that H is a homomorph-containing subgroup of K and K is a homomorph-containing subgroup of G but H is not a homomorph-containing subgroup of G. For full proof, refer: Homomorph-containment is not transitive

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

If H \le K \le G and H is a homomorph-containing subgroup of G, H is also a homomorph-containing subgroup of K. For full proof, refer: Homomorph-containment satisfies 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

If H_i, i \in I, are all homomorph-containing subgroups of G, then so is the join of subgroups \langle H_i \rangle_{i \in I}. For full proof, refer: Homomorph-containment is strongly join-closed

Quotient-transitivity

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

If H \le K \le G are groups such that H is a homomorph-containing subgroup of G and K/H is a homomorph-containing subgroup of G/H, then K is a homomorph-containing subgroup of G. For full proof, refer: Homomorph-containment is quotient-transitive