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.

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

• Results about free groups can be found here.

Linguistic Note

A free group is so called because it is free of non-trivial relations.