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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
A subgroup of a group is termed isomorph-conjugate if any subgroup isomorphic to that subgroup is conjugate to it in the whole group.
Definition with symbols
A subgroup of a group is termed isomorph-conjugate if whenever there is a subgroup isomorphic to , and are conjugate subgroups in .
In terms of the relation implication operator
The property of being isomorph-conjugate can be viewed in terms of the relation implication operator with the relation on the left being that of a subgroup pair being isomorphic and the relation on the right being that of a subgroup pair being conjugate in the whole group.
Relation with other properties
- Isomorph-free subgroup
- Order-dominating subgroup
- Order-unique subgroup
- Order-conjugate subgroup
- Sylow subgroup: For full proof, refer: Sylow implies isomorph-conjugate
- Intermediately isomorph-conjugate subgroup
Conjunction with other properties
- Any normal isomorph-conjugate subgroup is isomorph-free. Thus, this subgroup property is normal-to-characteristic
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
An isomorph-conjugate subgroup of an isomorph-conjugate subgroup is not necessarily isomorph-conjugate. For full proof, refer: Isomorph-conjugacy is not transitive
This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed
An intersection of two isomorph-conjugate subgroups of a group need not be isomorph-conjugate. For full proof, refer: Isomorph-conjugacy is not intersection-closed
One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.GAP-codable subgroup property
View the GAP code for testing this subgroup property at: IsIsomorphConjugateSubgroup
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
This subgroup property can be tested using GAP code, though there is no direct built-in GAP function for it. The GAP code is available at GAP:IsIsomorphConjugateSubgroup, and is invoked as follows: