# Complemented fully invariant subgroup

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

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

- It is both a permutably complemented subgroup and a fully invariant subgroup.
- It is both a lattice-complemented subgroup and a fully invariant subgroup.
- It is both a complemented normal subgroup and a fully invariant subgroup.

## Relation with other properties

### Stronger properties

- Normal Sylow subgroup
- Normal Hall subgroup
- Fully invariant direct factor
- Complemented homomorph-containing subgroup