The celebrated Nielsen-Schreier theorem in group theory says subgroup of a free group is free. This was proved for finitely generated subgroups of free groups by Jakob Nielsen in 1921, which involved the machinery of Nielsen transformations. It was proven in full generality by Otto Schreier in 1926.
But the standard proof of the theorem is done using topology in modern days. If $F$ is a free group, one can realize $F$ as fundamental group of wedge of circles $X$ with circles labelled by the generators of $F$. If $H$ is a subgroup of $F$, then using covering space theory pick a covering space $p : E \to X$ with $p_*\pi_1(E) = H$. Since covering space of a graph is a graph, and fundamental group of a graph is free (quotient the maximal tree out), $H$ is free and we're done.
Whom was this proof originally done by? I always thought it was done by Serre in his famous Trees. But apparently there are controversies about this, as someone recently told me that this might have been done by Dehn earlier than Serre. I'd appreciate a definitive answer to this one.