# Retract

From Groupprops

## Definition

### Equivalent definitions in tabular format

No. | Shorthand | A subgroup of a group is termed a retract if ... | A subgroup of a group is termed a retract of if ... |
---|---|---|---|

1 | image of idempotent endomorphism | there is an idempotent endomorphism of the group whose image is precisely that subgroup. This idempotent endomorphism is termed the retraction. | there is an endomorphism of such that and the image of is precisely . |

2 | normal complement | it has a normal complement: a normal subgroup that intersects it trivially, and that together with it generates the whole group. | there is a normal subgroup of such that and is trivial. |

3 | homomorphism extension | any homomorphism from the subgroup to any group extends to a homomorphism from the whole group to that group | for any homomorphism of groups to any group , there exists a homomorphism such that . |

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

## History

### Introduction of the concept

The concept of retract is fairly old, and came about in the beginning of the study of group theory. Retracts were first encountered as the right part in short exact sequences that split.

### Introduction of the term

The term *retract* is not very standard, and the concept is often referred to without the use of this formal term. The term *retract* actually comes from the set-theoretic/topological equivalent notion.

`Further information: Retractions and functors`

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### Monadic second-order description

This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups

View other monadic second-order subgroup properties

## Relation with other properties

### Stronger properties

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

direct factor | direct factor implies retract | retract not implies direct factor | |FULL LIST, MORE INFO | |

free factor | free factor implies retract | retract not implies free factor | |FULL LIST, MORE INFO | |

regular retract | |FULL LIST, MORE INFO |

### Weaker properties

### Related properties

Property | Meaning | Proof of one non-implication | Proof of other non-implication | Notions stronger than both | Notions weaker than both |
---|---|---|---|---|---|

normal subgroup | invariant under all inner automorphisms | retract not implies normal | normal not implies direct factor | direct factor is the conjunction | SCDIN-subgroup, Subgroup having a left transversal that is also a right transversal, Subgroup having a symmetric transversal|FULL LIST, MORE INFO |

## Metaproperties

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

transitive subgroup property | Yes | retract is transitive | If are groups such that is a retract of and is a retract of , then is a retract of . |

trim subgroup property | Yes | In any group, the whole group and the trivial subgroup are retracts. | |

finite-intersection-closed subgroup property | No | It is possible to have a group and subgroups and of such that both and are retracts but is not a retract of . |

## Testing

### GAP code

GAP-codable subgroup propertyOne can write code to test this subgroup property inGAP (Groups, Algorithms and Programming), though there is no direct command for it.

View the GAP code for testing this subgroup property at: IsRetract

View other GAP-codable subgroup properties | View subgroup properties with in-built commands