# Direct projection-invariant subgroup

From Groupprops

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]

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is said to be **direct projection-invariant** if it is invariant under any projection map from the group to a direct factor.

### Definition with symbols

A subgroup of a group is said to be direct projection-invariant if given any direct factor of and the projection map , we have .

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### 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

This subgroup property can be expressed in the function restriction formalism as the invariance property corresponding to projection on a direct factor, or in other words, as:

projection on a direct factor Function

## Relation with other properties

### Stronger properties

- Fully characteristic subgroup
- Retraction-invariant subgroup
- Direct projection-invariant direct factor
- Direct projection-invariant central factor

### Weaker properties

## Metaproperties

### 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

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

The whole group is direct projection-invariant as a subgroup of itself. The trivial subgroup isalso direct projection-invariant.