# 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 of a group is termed an **AEP subgroup** of if given any automorphism of , there is an automorphism of such that the restriction of to is precisely .

## 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

- Direct factor
- Base of a wreath product
- Fully normalized subgroup
- Intermediately AEP-subgroup
- Sectionally AEP-subgroup
- Normal AEP-subgroup
- Characteristic AEP-subgroup
- Automorphism-faithful AEP-subgroup
- Characteristic AEP-subgroup

### Weaker properties

- Subgroup in which every subgroup characteristic in the whole group is characteristic:
`For full proof, refer: Characteristic upper-hook AEP implies characteristic`

### Incomparable properties

- Retract:
`For full proof, refer: Retract not implies AEP, AEP not implies retract` - EEP-subgroup

## 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 transitiveABOUT 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 doesnotsatisfy 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 conditionABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition

If and is an AEP-subgroup of , need not be an AEP-subgroup of . `For full proof, refer: AEP does not satisfy intermediate subgroup condition`