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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]
QUICK PHRASES: contains all isomorphic subgroups, weakly closed in any ambient group
- Whenever is a subgroup of isomorphic to , .
- If is a subgroup of , is weakly closed in with respect to .
Equivalence of definitions
Further information: Isomorph-containing iff weakly closed in any ambient group
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
- 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:
- Fully invariant direct factor: For full proof, refer: Equivalence of definitions of fully invariant direct factor
- 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:
- Normal-isomorph-containing subgroup
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
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