Difference between revisions of "Isomorph-containing subgroup"

From Groupprops
Jump to: navigation, search
(Definition)
(Relation with other properties)
Line 21: Line 21:
 
===Stronger properties===
 
===Stronger properties===
  
* [[Weaker than::Isomorph-free subgroup]]: For a finite subgroup, and more generally, for a [[co-Hopfian group|co-Hopfian]] subgroup, the two properties are equivalent.
+
{| class="sortable" border="1"
* [[Weaker than::Homomorph-containing subgroup]]: Also related:
+
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
** [[Weaker than::Subhomomorph-containing subgroup]]
+
|-
** [[Weaker than::Subisomorph-containing subgroup]]
+
[[Weaker than::isomorph-free subgroup]] || no other isomorphic subgroup (for a finite subgroup, and more generally, for a [[co-Hopfian group|co-Hopfian]] subgroup, the two properties are equivalent) || obvious || any example of a non-co-Hopfian group as a subgroup of itself -- such as the [[group of integers]]
** [[Weaker than::Variety-containing subgroup]]
+
|-
* [[Weaker than::Fully invariant direct factor]]: {{proofat|[[Equivalence of definitions of fully invariant direct factor]]}}
+
| [[Weaker than::homomorph-containing subgroup]] || contains any homomorphic image of itself in the group || obvious; isomorphs are also homomorphic images || [[cyclic maximal subgroup of dihedral group:D8]] is isomorph-containing but not homomorph-containing || {{intermediate notions short|isomorph-containing subgroup|homomorph-containing subgroup}}
 +
|-
 +
| [[Weaker than::Subhomomorph-containing subgroup]] || contains any homomorphic image in the whole group of any subgroup of it || (via homomorph-containing) || (via homomorph-containing) || {{intermediate notions short|isomorph-containing subgroup|subhomomorph-containing subgroup}}
 +
|-
 +
| [[Weaker than::subisomorph-containing subgroup]] || contains any subgroup of the whole group isomorphic to any subgroup of itself || (obvious) || [[cyclic maximal subgroup of dihedral group:D8]] is isomorph-containing but not subisomorph-containing || {{intermediate notions short|isomorph-containing subgroup|subisomorph-containing subgroup}}
 +
|-
 +
| [[Weaker than::variety-containing subgroup]] || contains any subgroup of the whole group in the subvariety of the variety of groups generated by it || (obvious) || (via either homomorph-containing or subisomorph-containing) || {{intermediate notions short|isomorph-containing subgroup|variety-containing subgroup}}
 +
|-
 +
| [[Weaker than::fully invariant direct factor]] || both a [[fully invariant subgroup]] and a [[direct factor]] || [[equivalence of definitions of fully invariant direct factor]] ||  || {{intermediate notions short|isomorph-containing subgroup|fully invariant direct factor}}
 +
|}
  
 
===Weaker properties===
 
===Weaker properties===
  
* [[Stronger than::Characteristic subgroup]]: {{proofofstrictimplicationat|[[Isomorph-containing implies characteristic]]|[[Characteristic not implies isomorph-containing]]}} Also related:
+
{| class="sortable" border="1"
** [[Stronger than::Intermediately injective endomorphism-invariant subgroup]]
+
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
** [[Stronger than::Injective endomorphism-invariant subgroup]]
+
|-
** [[Stronger than::Intermediately characteristic subgroup]]
+
| [[Stronger than::characteristic subgroup]] || invariant under any automorphism of the whole group || [[isomorph-containing implies characteristic]] || [[characteristic not implies isomorph-containing]] || {{intermediate notions short|characteristic subgroup|isomorph-containing subgroup}}
** [[Stronger than::Normal subgroup]]
+
|-
* [[Stronger than::Normal-isomorph-containing subgroup]]
+
| [[Stronger than::injective endomorphism-invariant subgroup]] || invariant under any injective endomorphism of the whole group || || Can use same example as for characteristic not implies isomorph-containing, if finite || {{intermediate notions short|injective endomorphism-invariant subgroup|isomorph-containing subgroup}}
 +
 
 +
| [[Stronger than::intermediately injective endomorphism-invariant subgroup]] || injective endomorphism-invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately injective endomorphism-invariant subgroup|isomorph-containing subgroup}}
 +
|-
 +
| [[Stronger than::intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || || || {{intermediate notions short|intermediately characteristic subgroup|isomorph-containing subgroup}}
 +
|-
 +
| [[Stronger than::normal subgroup]] || invariant under all inner automorphisms || || || {{intermediate notions short|normal subgroup|isomorph-containing subgroup}}
 +
|-
 +
| [[Stronger than::normal-isomorph-containing subgroup]] || contains any normal subgroup of the whole group isomorphic to it || (obvious) || any proper nontrivial subgroup of a finite simple non-abelian group || {{intermediate notions short|normal-isomorph-containing subgroup|isomorph-containing subgroup}}
 +
|}
  
 
==Metaproperties==
 
==Metaproperties==

Revision as of 00:31, 18 March 2019

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

QUICK PHRASES: contains all isomorphic subgroups, weakly closed in any ambient group

A subgroup H of a group G is termed an isomorph-containing subgroup if it satisfies the following equivalent conditions:

  1. Whenever K \le G is a subgroup of G isomorphic to H, K \le H.
  2. If G is a subgroup of L, H is weakly closed in G with respect to L.

Equivalence of definitions

Further information: Isomorph-containing iff weakly closed in any ambient group

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
isomorph-free subgroup no other isomorphic subgroup (for a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent) obvious any example of a non-co-Hopfian group as a subgroup of itself -- such as the group of integers
homomorph-containing subgroup contains any homomorphic image of itself in the group obvious; isomorphs are also homomorphic images cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not homomorph-containing |FULL LIST, MORE INFO
Subhomomorph-containing subgroup contains any homomorphic image in the whole group of any subgroup of it (via homomorph-containing) (via homomorph-containing) Homomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Subisomorph-containing subgroup|FULL LIST, MORE INFO
subisomorph-containing subgroup contains any subgroup of the whole group isomorphic to any subgroup of itself (obvious) cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not subisomorph-containing Right-transitively isomorph-containing subgroup|FULL LIST, MORE INFO
variety-containing subgroup contains any subgroup of the whole group in the subvariety of the variety of groups generated by it (obvious) (via either homomorph-containing or subisomorph-containing) Homomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Subisomorph-containing subgroup|FULL LIST, MORE INFO
fully invariant direct factor both a fully invariant subgroup and a direct factor equivalence of definitions of fully invariant direct factor Complemented homomorph-containing subgroup, Complemented isomorph-containing subgroup, Homomorph-containing subgroup|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup invariant under any automorphism of the whole group isomorph-containing implies characteristic characteristic not implies isomorph-containing Injective endomorphism-invariant subgroup, Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
injective endomorphism-invariant subgroup invariant under any injective endomorphism of the whole group Can use same example as for characteristic not implies isomorph-containing, if finite Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO intermediately injective endomorphism-invariant subgroup injective endomorphism-invariant in every intermediate subgroup |FULL LIST, MORE INFO
intermediately characteristic subgroup characteristic in every intermediate subgroup Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
normal subgroup invariant under all inner automorphisms Characteristic subgroup, Injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
normal-isomorph-containing subgroup contains any normal subgroup of the whole group isomorphic to it (obvious) any proper nontrivial subgroup of a finite simple non-abelian group |FULL LIST, MORE INFO

Metaproperties

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

For full proof, refer: Isomorph-containment is not transitive

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