# Group in which every retract is regular

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 regular** is a group in which every retract is a regular retract.

## Definitions used

Term | Definition |
---|---|

Retract | A subgroup of a group is a retract if there exists a normal subgroup of such that and is trivial. |

Regular retract | A subgroup of a group is a regular retract if there exists a subgroup of such that , intersects the normal closure of trivially and intersects the normal closure of trivially. |

## 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 (regular retract), 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 |
---|---|---|---|---|

Group in which every retract is a direct factor | |FULL LIST, MORE INFO | |||

Group in which every retract is a free factor | every retract is a free factor | |FULL LIST, MORE INFO |