# Free groups, covering spaces and Artin's theorem

## TL;DR

This paper offers a didactic proof of Artin's theorem demonstrating that the commutator subgroup of a free group on two generators is not finitely generated, using the infinite grid approach.

## Contribution

It provides a new, pedagogically valuable proof of Artin's theorem employing the infinite grid method, differing from previous proofs.

## Key findings

- The commutator subgroup of a free group on two generators is not finitely generated.
- The proof employs an infinite grid approach for clarity and didactic value.
- The approach offers an alternative perspective to existing proofs.

## Abstract

In this expository note we provide a proof of Artin's theorem which states that the commutator subgroup of a free group on two generators is not finitely generated. The proof employs the infinite grid as in two other proofs in the literature mentioned in the note but takes a somewhat different approach which seems to be of didactic value.