Normal-extensible automorphism-invariant 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

A subgroup of a group is termed a normal-extensible automorphism-invariant subgroup' if it is invariant under every normal-extensible automorphism of the whole group.

Formalisms

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

The property of being a normal-extensible automorphism-invariant subgroup can be expressed as the invariance property:

Normal-extensible automorphism $\to$ Function

It can also be expressed as the endo-invariance property:

Normal-extensible automorphism $\to$ Endomorphism

Finally, since the inverse of a normal-extensible automorphism is also normal-extensible, it can be expressed as an auto-invariance property:

Normal-extensible automorphism $\to$ Automorphism

Relation with other properties

Stronger properties

• Characteristic-extensible automorphism-invariant subgroup
  • Strongly potentially characteristic subgroup
  • Strongly potentially relatively characteristic subgroup
• Semi-strongly potentially characteristic subgroup
• Semi-strongly potentially relatively characteristic subgroup

Weaker properties

• Normal subgroup: For proof of the implication, refer Normal-extensible automorphism-invariant implies normal and for proof of its strictness (i.e. the reverse implication being false) refer Normal not implies normal-extensible automorphism-invariant.

Metaproperties

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).
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Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness