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.

  • Did you look at the Wikipeda page on this theorem? It attributes the topological proof (or at least a topological proof) to Baer and Levi in the 1930s. – KCd Dec 16 '15 at 12:19
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    Oh, of course you saw the Wikipedia page: it is the link in your first sentence! Evidently you did not read it carefully enough. – KCd Dec 16 '15 at 12:32
  • @KCd Right, oh well! I guess that should be an answer. I'd be happy to accept if you answer it. – Balarka Sen Dec 16 '15 at 14:02
up vote 7 down vote accepted

See the end of the Wikipedia link in your first sentence. The source is Rotman's Introduction to the Theory of Groups (1995), which reads on p.383:"There are today several different proofs of this theorem, some "algebraic" and some "geometric". The first geometric proof was given by Baer and Levi in 1936, and this is the proof we present. There is another elegant geometric proof, due to J.-P.Serre, using the notion of groups acting on trees".

  • The question should not have been asked, since the OP could have answered it just by reading the Wikipedia link more carefully. I prefer to leave my answer as is to make a point that everything needed is already on Wikipedia (including the original paper, as the first reference there). – KCd Dec 16 '15 at 21:05
  • @Danu, I have edited the answer to be CW now. You're welcome to edit it further if you wish. – KCd Dec 16 '15 at 23:51

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