# Subgroup of free group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: subgroup with a group property imposed on theambient group: free group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

A **subgroup of free group** is a subgroup of a free group (i.e., the whole group is free).

## Relation with other properties

### Stronger properties

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

Normal subgroup of free group | a normal subgroup when the whole group is free | |FULL LIST, MORE INFO | ||

Retract of free group | a retract when the whole group is free | |FULL LIST, MORE INFO | ||

Free factor of free group | a free factor when the whole group is free | Retract of free group|FULL LIST, MORE INFO | ||

Characteristic subgroup of free group | a characteristic subgroup when the whole group is free | |FULL LIST, MORE INFO | ||

Verbal subgroup of free group | a verbal subgroup when the whole group is free | |FULL LIST, MORE INFO |

### Weaker properties

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

Free subgroup | freeness is subgroup-closed | |FULL LIST, MORE INFO |