# Abelian direct factor

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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): abelian group
View a complete list of such conjunctions
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: central subgroup and direct factor
View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup of a group is an abelian direct factor or central direct factor or complemented central subgroup if it satisfies the following conditions:

1. It is abelian as a group and is a direct factor of the whole group.
2. It is both a central subgroup and a direct factor of the whole group.
3. It is both a central subgroup and a permutably complemented subgroup of the whole group.
4. It is both a central subgroup and a complemented normal subgroup of the whole group.
5. It is both a central subgroup and a complemented central factor of the whole group.
6. It is both a central subgroup and a lattice-complemented subgroup of the whole group.

## Relation with other properties

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Central subgroup contained in the center |FULL LIST, MORE INFO
Direct factor factor in an internal direct product |FULL LIST, MORE INFO
Complemented central factor central factor having a permutable complement Direct factor|FULL LIST, MORE INFO
Complemented normal subgroup normal subgroup having a permutable complement Direct factor|FULL LIST, MORE INFO
Abelian hereditarily normal subgroup |FULL LIST, MORE INFO
Abelian normal subgroup Central subgroup|FULL LIST, MORE INFO
Permutably complemented subgroup Direct factor|FULL LIST, MORE INFO
Lattice-complemented subgroup Direct factor|FULL LIST, MORE INFO