There are two proofs of Nielsen-Schreier that I know of. The theorem states that every subgroup of a free group is free. The first proof uses topology and covering space theory and is rather elegant. The second uses combinatorial techniques on a free group of words with no relations.
Is there a more algebraic proof which somehow just uses the universal property of free groups and maybe other properties of groups that are proved more "algebraically"?
I'm interested because groups are defined purely algebraically by equations, and some proofs that a subgroup of a free abelian group is free abelian have a far more algebraic flavour. So perhaps there is some proof of Nielsen-Schreier that also has a more algebraic flavour?
Ideally I would like a proof that does not involve combinatorial properties of a group of words on generators; in other words preferably no facts from combinatorial group theory.