Definition:Free Group/Definition 2

Definition

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

• Results about free groups can be found here.

Linguistic Note

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

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): free group
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generator: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): free group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generator: 2.