Isomorph-containing subgroup

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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]

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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

• Isomorph-free subgroup: For a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent.
• Homomorph-containing subgroup: Also related:
  • Subhomomorph-containing subgroup
  • Subisomorph-containing subgroup
  • Variety-containing subgroup
• Fully invariant direct factor: For full proof, refer: Equivalence of definitions of fully invariant direct factor

Weaker properties

• Characteristic subgroup: For proof of the implication, refer Isomorph-containing implies characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic not implies isomorph-containing. Also related:
  • Intermediately injective endomorphism-invariant subgroup
  • Injective endomorphism-invariant subgroup
  • Intermediately characteristic subgroup
  • Normal subgroup
• Normal-isomorph-containing subgroup

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