Isomorph-containing subgroup: Difference between revisions

From Groupprops
No edit summary
Line 3: Line 3:


==Definition==
==Definition==
{{quick phrase|[[quick phrase::contains all isomorphic subgroups]], [[quick phrase::weakly closed in any ambient group]]}}


A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''isomorph-containing subgroup''' if it satisfies the following equivalent conditions:
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''isomorph-containing subgroup''' if it satisfies the following equivalent conditions:


# Whenever <math>K \le G</math> is a subgroup of <math>G</math> isomorphic to <math>H</math>, <math>K \le H</math>.
# Whenever <math>K \le G</math> is a subgroup of <math>G</math> isomorphic to <math>H</math>, <math>K \le H</math>.
# If <math>G</math> is a subgroup of <math>L</math>, <mah>H</math> is [[defining ingredient::weakly closed subgroup|weakly closed]] in <math>G</math> with respect to <math>L</math>.
# If <math>G</math> is a subgroup of <math>L</math>, <math>H</math> is [[defining ingredient::weakly closed subgroup|weakly closed]] in <math>G</math> with respect to <math>L</math>.


===Equivalence of definitions===
===Equivalence of definitions===

Revision as of 21:18, 12 November 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

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≤G is a subgroup of G isomorphic to H, K≤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

Weaker properties

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

Retrieved from "https://groupprops.subwiki.org/w/index.php?title=Isomorph-containing_subgroup&oldid=22569"