# Characteristically complemented subgroup

From Groupprops

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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]

## Definition

### Symbol-free definition

A subgroup of a group is termed a **characteristically complemented subgroup** or **characteristic retract** if it satisfies the following equivalent conditions:

- There is a retraction (viz an idempotent endomorphism) on the group, whose image is that subgroup, and whose kernel is a characteristic subgroup.
- There is a characteristic subgroup that is a permutable complement to it.
- There is a characteristic subgroup that is a lattice complement to it.

The corresponding characteristic subgroup (there may be more than one such) is termed a complemented characteristic subgroup.

## Relation with other properties

### Stronger properties

- Characteristically complemented characteristic subgroup
- Characteristically complemented normal 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

If are groups such that is characteristically complemented in and is characteristically complemented in , then is characteristically complemented in . This follows essentially from the fact that characteristicity is quotient-transitive. `For full proof, refer: Characteristically complemented is transitive`

`Further information: Characteristicity is quotient-transitive, Quotient-transitive and stronger than normality implies complementary property is transitive`

### Quotient-transitivity

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

View a complete list of quotient-transitive subgroup properties