# Characteristically complemented characteristic subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: characteristically complemented subgroup and characteristic subgroup

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## Contents

## Definition

A subgroup of a group is termed a **characteristically complemented characteristic subgroup** if it is a characteristic subgroup and it has a permutable complement that is also a characteristic subgroup.

## Relation with other properties

### Stronger properties

### Weaker properties

- Left-transitively complemented normal subgroup
- Characteristic direct factor
- Characteristically complemented subgroup
- Complemented characteristic subgroup
- Complemented normal subgroup
- Retract
- 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: |ABOUT 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

Suppose are groups such that is a characteristically complemented characteristic subgroup of and is a characteristically complemented characteristic subgroup of . Then is a characteristically complemented characteristic subgroup of . `For full proof, refer: Characteristically complemented characteristic is transitive`

### 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).

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