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.

Explicitly, a subgroup $H$ of a group $G$ is normal-extensible automorphism-invariant if, whenever $\sigma$ is a normal-extensible automorphism of $G$ (i.e., an automorphism of $G$ such that, for any group $K$ containing $G$ as a normal subgroup, $\sigma$ extends to an automorphism of $K$), then $\sigma (H)=H$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
strongly intersection-closed subgroup property	Yes	Follows from invariance implies strongly intersection-closed	Suppose $H_{i},i\in I$ are all normal-extensible automorphism-invariant subgroups of a group $G$. Then, the intersection of subgroups $\bigcap _{i\in I}H_{i}$ is also a normal-extensible automorphism-invariant subgroup of $G$.
strongly join-closed subgroup property	Yes	Follows from endo-invariance implies strongly join-closed	Suppose $H_{i},i\in I$ are all normal-extensible automorphism-invariant subgroups of a group $G$. Then, the join of subgroups $\left\langle H_{i}\right\rangle _{i\in I}$ is also a normal-extensible automorphism-invariant subgroup of $G$.
trim subgroup property	Yes	invariance implies identity-true, endo-invariance implies trivially true	In any group $G$, both the whole group $G$ and the trivial subgroup $\{e\}$ are normal-extensible automorphism-invariant.
transitive subgroup property	No	normal-extensible automorphism-invariance is not transitive	It is possible to have groups $H\leq K\leq G$ such that $H$ is normal-extensible automorphism-invariant in $K$ and $K$ is normal-extensible automorphism-invariant in $G$, but $H$ is not normal-extensible automorphism-invariant in $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under all automorphisms	(obvious)		|FULL LIST, MORE INFO
normal-potentially relatively characteristic subgroup	there exists a group containing the whole group as a normal subgroup, in which the subgroup is invariant under the automorphisms preserving the whole group.			|FULL LIST, MORE INFO
normal-potentially characteristic subgroup	there exists a group containing the whole group as a normal subgroup, in which the subgroup becomes a characteristic subgroup	(via normal-potentially relatively characteristic subgroup)		|FULL LIST, MORE INFO
characteristic-extensible automorphism-invariant subgroup	invariant under all characteristic-extensible automorphisms: automorphisms that can always be extended to groups in which the whole group is characteristic			|FULL LIST, MORE INFO
characteristic-potentially characteristic subgroup	there exists a group containing the whole group in which both the group and the subgroup are characteristic	(via normal-potentially characteristic, alternatively via characteristic-extensible automorphism-invariant)		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup	invariant under inner automorphisms	normal-extensible automorphism-invariant implies normal	normal not implies normal-extensible automorphism-invariant	|FULL LIST, MORE INFO

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