# Subgroup of free group

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

## 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