# Definition:Free Group

## Definition

### Definition 1

A group $G$ is a free group if and only if it is isomorphic to the free group on some set.

### Definition 2

A group $G$ is a free group if and only if it has a presentation of the form $\gen S$, where $S$ is a set.

That is, it has a presentation without relators.

In this context, free means free of non-trivial relations.

## Also see

• Results about free groups can be found here.