# 1-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]
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

## Definition

### Definition with symbols

A subgroup $H$ of a group $G$ is termed a $1$-automorphism-invariant subgroup if any 1-automorphism of $G$ sends $H$ to itself. In other words, for every 1-automorphism $\varphi$ of $G$, $\varphi(H) \subseteq H$.

## 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 1-automorphism-invariant subgroup can be expressed as an invariance property:

1-automorphism $\to$ Function

In other words, $H$ is a 1-automorphism-invariant subgroup of a group $G$ if and only if every 1-automorphism of $G$ restricts to a function from $H$ to itself.

Alternative function restriction expressions are:

## Examples

• In any group, the whole group and the trivial subgroup are 1-automorphism-invariant.
• In a cyclic group, every subgroup is 1-automorphism-invariant.
• If the set of elements of order $n$ form a subgroup for any particular $n$, then that subgroup is 1-automorphism-invariant. This is because 1-automorphisms preserve the orders of elements.

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

### Join-closedness

This subgroup property is not join-closed, viz., it is not true that a join of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not join-closed