AEP-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]

Definition

Symbol-free definition

A subgroup of a group is termed an AEP-subgroup or Automorphism Extension Property subgroup if any automorphism of the subgroup extends to an automorphism of the whole group.

Definition with symbols

A subgroup H of a group G is termed an AEP subgroup of G if given any automorphism \sigma of G, there is an automorphism \sigma' of G such that the restriction of \sigma' to H is precisely \sigma.

Formalisms

In terms of the function extension formalism

The property of being an AEP-subgroup is the balanced subgroup property with respect to the function extension formalism for the property of being an automorphism.

Relation with other properties

Stronger properties

Weaker properties

Incomparable properties

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this 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 transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Any AEP subgroup of an AEP subgroup is an AEP subgroup. This follows from the Automorphism Extension Property being a balanced subgroup property.

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

Clearly, both the trivial subgroup and the whole group satisfy AEP, hence AEP is a trim subgroup property.

Intermediate subgroup condition

NO: This subgroup property does not satisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some 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 SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition

If H \le K \le G and H is an AEP-subgroup of G, H need not be an AEP-subgroup of K. For full proof, refer: AEP does not satisfy intermediate subgroup condition