# AEP-subgroup

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

### 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.

## 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