Definition:Free Group

Definition
Let $G$ be a group.

Definition 1
The group $G$ is free it is isomorphic to the free group on some set.

Definition 2
The group $G$ is free it has a presentation of the form $\langle S \rangle$ where $S$ is a set, that is, a presentation without relators.

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

Also see

 * Equivalence of Definitions of Free Group
 * Definition:Universal Property of Free Group on Set
 * Definition:Rank of Free Group
 * Definition:Free Abelian Group
 * Definition:Free Product of Groups