# 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]

## 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 $H \le K \le G$ 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) Characteristic-potentially characteristic subgroup, Normal-potentially characteristic subgroup, Normal-potentially relatively characteristic subgroup|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) 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) Normal-potentially characteristic subgroup, Normal-potentially relatively characteristic subgroup|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