# Quasiautomorphism-invariant subgroup

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

### Symbol-free definition

A subgroup of a group is termed **quasiautomorphism-invariant** if any quasiautomorphism of the whole group sends the subgroup to within itself.

A quasiautomorphism of a group is a quasihomomorphism of groups from the group to itself, with a two-sided inverse that is also a quasihomomorphism.

## 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 following function restriction expression can be used for a quasiautomorphism-invariant subgroup:

Quasiautomorphism Function

In other words, every quasiautomorphism of the whole group restricts to a function from the subgroup to itself.

An alternative expression is as the balanced subgroup property (function restriction formalism) corresponding to quasiautomorphisms:

Quasiautomorphism Quasiautomorphism

In other words, every quasiautomorphism of the whole group restricts to a quasiautomorphism from the subgroup to itself.

## Relation with other properties

### Stronger properties

- Quasiendomorphism-invariant subgroup
- 1-automorphism-invariant subgroup
- Quasihomomorph-containing subgroup

### Weaker properties

## 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