On Monday, March 25, 2013 1:07:13 PM UTC, G. A. Edgar wrote:
> > > It would be very interesting if we find a result in group theory where > > the simplest proof involves topological groups.
Not quite what you asked but I think the simplest proofs of the fact that subgroups of free groups are free involve elementary algebraic topology and the fundamental group of a graph. (I would expect that all others on this thread know that, though). Despite this being different from what you asked, I think it's related, because presumably the reason what you're asking for would be "very interesting" is the idea of settling elementary questions in group theory with techniques outside elementary group theory. Clearly, it is easy to see that it is obvious that, in a trivial sense, Nielsen-Schreier is a good candidate for providing such a surprise.