# Characteristic direct factor of abelian group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: characteristic direct factor with a group property imposed on theambient group: abelian group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

This article describes a property that arises as the conjunction of a subgroup property: fully invariant direct factor with a group property imposed on theambient group: abelian group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

A subgroup of a group is termed a **characteristic direct factor** of if the following equivalent conditions are satisfied:

- is an abelian group and is a characteristic direct factor of (i.e., is both a characteristic subgroup of and a direct factor in ).
- is an abelian group and is a fully invariant direct factor of (i.e., is a fully invariant subgroup as well as a direct factor of ). See also equivalence of definitions of fully invariant direct factor for other equivalent formulations of this.

### Equivalence of definitions

`Further information: equivalence of definitions of characteristic direct factor of abelian group`