AEP-subgroup

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 $$\sigma$$.

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.

Stronger properties

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

Weaker properties

 * Stronger than::Subgroup in which every subgroup characteristic in the whole group is characteristic:

Incomparable properties

 * Retract:
 * EEP-subgroup

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

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

If $$H \le K \le G$$ and $$H$$ is an AEP-subgroup of $$G$$, $$H$$ need not be an AEP-subgroup of $$K$$.