# Group in which every retract is a free 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 free factor** is a group with the property that every retract of the group is a free factor of the group, i.e., a factor in an internal free product.

## 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 |

### 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 |