# Direct factor of fully invariant subgroup

From Groupprops

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: direct factor and fully characteristic subgroup

View other such compositions|View all subgroup properties

## Contents

## Definition

### Symbol-free definition

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

- It can be expressed as a direct factor of a fully invariant subgroup.
- It is a direct factor in its fully invariant closure.

### In terms of the composition operator

The subgroup property of being a direct factor of fully invariant is obtained by applying the composition operator to the subgroup properties of being a direct factor and of being fully invariant.