# 1-automorphism-invariant subgroup

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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 of a group is termed a -**automorphism-invariant** subgroup if any 1-automorphism of sends to itself. In other words, for every 1-automorphism of , .

## 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 Function

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

Alternative function restriction expressions are:

- 1-automorphism 1-endomorphism
- 1-automorphism 1-automorphism. This shows that the property of being a 1-automorphism-invariant subgroup is a balanced subgroup property (function restriction formalism) with respect to 1-automorphisms.

## 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 form a subgroup for any particular , then that subgroup is 1-automorphism-invariant. This is because 1-automorphisms preserve the orders of elements.

## Relation with other properties

### Stronger properties

### Weaker properties

- Quasiautomorphism-invariant subgroup
- Strong quasiautomorphism-invariant subgroup
- Characteristic subgroup
- Normal subgroup

## 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 transitiveABOUT 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-closedABOUT 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