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A group $G$ is a free group if and only if it is isomorphic to the free group on some set.
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.
- 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
- Results about free groups can be found here.