Previous: [[A worked example - the abelianization functor]] Here's what I was hiding from you about free groups the whole time, the reason why we only talked about *quotients* of free groups and not *subgroups* - subgroups of free groups are pretty pathological! It is unfortunately hard to actually prove many of the things I will say here, so take them on faith for now. We will come back to these later with full force. Take the easiest nonabelian free group, $F_2$, and look at its subgroups. Are they free? Intuition says they should be, because we have no relations in $F_2$ and thus should have no relations in any subgroup. This is in fact true But when we're working with words, this becomes really hard to prove. We do in fact have some relations: they are the cancellation relations $aa^{-1} = e$ and so on! How can we really know that we didn't do something strange with catastrophic cancellation? The Nielsen-Schreier theorem, stating that subgroups of free groups are free, is in fact not easy to prove. Their original, combinatorial proof is very hard. However, *topology* provides a proof that we will see soon. Till then, let's look at the a very weird thing about $F_2$. Consider the subgroup of $F_2$ generated by $ab, a^2b^2, a^3, b^3, \ldots, a^nb^n, \ldots$. We will see that this has to be free, and that none of these elements have relations between them. What this means is that the free group on countably many generators is a subgroup of $F_2$! This also means that $F_k$ is a subgroup of $F_2$ for any natural number $k$. The smallest free group that we cannot find inside $F_2$ is the free group on uncountably many generators, because that has to have uncountably many elements while $F_2$ has countably many elements. This is our first indication that the problem of looking at subgroups of free groups is not entirely trivial. Next: [[F_2 and the Banach-Tarski paradox]]