# Group in which every retract is a direct factor

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **group in which every retract is a direct factor** is a group satisfying the following equivalent conditions:

- Every retract of the group (i.e., every subgroup that has a normal complement) is a direct factor.
- Every retract of the group is a normal subgroup.

## Formalisms

### In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (retract) satisfies the second property (direct factor), and vice versa.

View other group properties obtained in this way

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Simple group | nontrivial; no proper nontrivial normal subgroup | |FULL LIST, MORE INFO | ||

Splitting-simple group | nontrivial; no proper nontrivial retract | |FULL LIST, MORE INFO | ||

Abelian group | any two elements commute | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Group in which every retract is regular | |FULL LIST, MORE INFO |