# 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 of a group is normal-extensible automorphism-invariant if, whenever is a normal-extensible automorphism of (i.e., an automorphism of such that, for any group containing as a normal subgroup, extends to an automorphism of ), then .

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

strongly intersection-closed subgroup property | Yes | Follows from invariance implies strongly intersection-closed | Suppose are all normal-extensible automorphism-invariant subgroups of a group . Then, the intersection of subgroups is also a normal-extensible automorphism-invariant subgroup of . |

strongly join-closed subgroup property | Yes | Follows from endo-invariance implies strongly join-closed | Suppose are all normal-extensible automorphism-invariant subgroups of a group . Then, the join of subgroups is also a normal-extensible automorphism-invariant subgroup of . |

trim subgroup property | Yes | invariance implies identity-true, endo-invariance implies trivially true | In any group , both the whole group and the trivial subgroup are normal-extensible automorphism-invariant. |

transitive subgroup property | No | normal-extensible automorphism-invariance is not transitive | It is possible to have groups such that is normal-extensible automorphism-invariant in and is normal-extensible automorphism-invariant in , but is not normal-extensible automorphism-invariant in . |

## 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 Function

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

Normal-extensible automorphism 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 Automorphism