free group

Definition

The free group of a set $S$ is

$${\prod _{s\in S}}^{\ast }\mathbb{Z}\cdot s$$

Where ${\prod }^{\ast }$ denotes the free product of the cyclic groups generated by all $s\in S$.

Universal property

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

Statement

Let $G$ be any group and $F(S)$ be the free group of a set $S$. Then for any function

$\phi :S⟶G$

there exists a unique group homomorphism

$\tilde{\phi }:F(S)⟶G$

such that $\tilde{\phi }$ is an extension of $\phi$, i.e.

$\tilde{\phi }{|}_S=\phi .$

This gives the following diagram:

Proof

todo

Notes

• Every group is a quotient of a free group.

Proof:

todo

• Not every group is a free group. For example ${\mathbb{Z}}_2\ast {\mathbb{Z}}_2$ is a free product, but not a free group.

References

@hatcher2002 - Chapter 1.2