Definition

The free group of a set is

Where denotes the free product of the cyclic groups generated by all .

Universal property

In short, group homomorphims “play nice” with a basis set of the free group.

Let be any group and be the free group of a set . Then for any function

there exists a unique group homomorphism

such that is an extension of , i.e.

This gives the following diagram:

Not every group is a free group. For example is a free product, but not a free group.

@hatcher2002 - Chapter 1.2