Filed under: Algebra, Math | Tags: algebra, geometry, graph theory, group theory, Math, serre
Okay, first post for a while. As I promised quite a while back, let’s prove together that subgroups of free groups are free. It’s surprising that this is nontrivial to prove: just try to come up with some subgroups of and you’ll see what I mean. In fact, using only basic algebraic topology and a bit of graph theory, we can come up with a really simple argument that replaces this one. Perhaps that’s an argument in favor of algebraic topology. But I think this angle is sort of interesting, and it should be a fresh experience for me, at least.
The proof is due to Jean-Pierre “Duh Bear” Serre in his book Trees. A heads up if you track this down — Serre has a really weird way of defining graphs. Fortunately, for this proof at least, a little bit of work translates things into the same language of graphs and digraphs that we saw when talking about Cayley graphs. I review that below the fold. It takes a while to set up the machinery, though the proof itself isn’t too long. To recompense, I’ve left out a couple minor details, which you’re probably able to fill in. If some step doesn’t make sense, work it out — or try to disprove it!