# Auto-invariance property

## Definition

### Definition with symbols

A subgroup property $p$ is termed an auto-invariance property if there is a group-closed automorphism property $q$ such that for any subgroup $H \le G$, $H$ has property $p$ in $G$, iff any automorphism of $G$ satisfying property $p$, sends $H$ to within itself.

### In terms of the function restriction formalism

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

$p = q \to$ Function

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

This is equivalent to the following function restriction expressions:

$p = q \to$ Automorphism

and:

$p = q \to$ 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.