# Complete direct factor

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: direct factor with a group property (itself viewed as a subgroup property): complete group

View a complete list of such conjunctions

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): complete group

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed a **complete direct factor** or **complete normal subgroup** if it satisfies the following equivalent conditions:

- It is both a complete group (as a group by itself) and a normal subgroup of the whole group.
- It is both a complete group (as a group by itself) and a direct factor of the whole group.
- It is both a complete group (as a group by itself) and a complemented normal subgroup of the whole group.