# Characteristically complemented normal 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]

## Definition

A subgroup of a group is termed a **characteristically complemented normal subgroup** or **characteristically complemented direct factor** if it is a direct factor in an internal direct product where the other direct factor is a characteristic subgroup.

## Relation with other properties

### Stronger properties

- Characteristically complemented characteristic subgroup:
*For proof of the implication, refer Characteristically complemented characteristic implies characteristically complemented normal and for proof of its strictness (i.e. the reverse implication being false) refer Characteristically complemented normal not implies characteristically complemented characteristic*.

### 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

### Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.

View a complete list of quotient-transitive subgroup properties