# Abelian direct factor

From Groupprops

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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:

- It is abelian as a group and is a direct factor of the whole group.
- It is both a central subgroup and a direct factor of the whole group.
- It is both a central subgroup and a permutably complemented subgroup of the whole group.
- It is both a central subgroup and a complemented normal subgroup of the whole group.
- It is both a central subgroup and a complemented central factor of the whole group.
- 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 |