# Complemented characteristic subgroup

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Symbol-free definition

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

- It is a characteristic subgroup as well as a permutably complemented subgroup.
- It is a characteristic subgroup as well as a lattice-complemented subgroup.
- It is a characteristic subgroup as well as a complemented normal subgroup, i.e., it occurs as the normal subgroup part in an internal semidirect product.

## Relation with other properties

### Stronger properties

- Left-transitively complemented normal subgroup
- Characteristically complemented characteristic subgroup
- Characteristic direct factor