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