# Auto-invariance property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Definition

### Definition with symbols

A subgroup property is termed an **auto-invariance property** if there is a group-closed automorphism property such that for any subgroup , has property in , iff any automorphism of satisfying property , sends to within itself.

### In terms of the function restriction formalism

A subgroup property is termed an **auto-invariance property** if there is a group-closed automorphism property , such that we can write the following function restriction expression for :

Function

In other words, a subgroup has subgroup property if every automorphism with property , for the whole group, restricts to a function from the subgroup to itself.

This is equivalent to the following function restriction expressions:

Automorphism

and:

Endomorphism

In other words, the restriction is automatically guaranteed to be an automorphism of the subgroup.

### Equivalence of definitions

The equivalence of definitions follows from the elementary observation: restriction of automorphism to subgroup invariant under it and its inverse is automorphism.

## Examples

### Normal subgroups

`Further information: normal subgroup`

The property of normality is an auto-invariance property, where the group-closed automorphism property in question is the property of being an inner automorphism.

### Characteristic subgroups

`Further information: characteristic subgroup`

The property of being characteristic is an auto-invariance property, where the group-closed automorphism property in question is the property of being *any* automorphism.

## Relation with other metaproperties

### Weaker metaproperties

- Endo-invariance property
- Invariance property
- Strongly join-closed subgroup property
- Strongly intersection-closed subgroup property
- Automorphism-based relation-implication-expressible subgroup property
- Normalizer-closed subgroup property
- Centralizer-closed subgroup property
- Commutator-closed subgroup property