Editing Free group (section)

== Examples ==
The group ('''Z''',+) of [[integer]]s is free of rank 1; a generating set is ''S'' = {1}. The integers are also a [[free abelian group]], although all free groups of rank <math>\geq 2</math> are non-abelian. A free group on a two-element set ''S'' occurs in the proof of the [[Banach–Tarski paradox]] and is described there.

On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order.

In [[algebraic topology]], the [[fundamental group]] of a [[bouquet of circles|bouquet of ''k'' circles]] (a set of ''k'' loops having only one point in common) is the free group on a set of ''k'' elements.