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]

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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

• 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.
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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
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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