# Verbal subgroup of free group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: verbal 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

This article describes a property that arises as the conjunction of a subgroup property: fully invariant 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 **verbal subgroup of free group** is a subgroup of a group satsifying the following equivalent conditions:

- It is a verbal subgroup in the whole group and the whole group is a free group.
- It is a fully invariant subgroup in the whole group and the whole group is a free group.

### Equivalence of definitions

`For full proof, refer: fully invariant implies verbal in reduced free group`

## Relation with other properties

### Weaker properties

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

Characteristic subgroup of free group | |FULL LIST, MORE INFO | |||

Normal subgroup of free group | |FULL LIST, MORE INFO | |||

Subgroup of free group | |FULL LIST, MORE INFO |