I found the following exercise on the 12th page of Presentations of Groups:
- Let $F$ be free of rank $r$. Show that $F/F'$ is free abelian of rank $r$.
First of all, $F'$ is not defined. I'm assuming it stands for a general normal subgroup of $F$.
So, what seems very strange here is that $F$ is not supposed to be abelian, but if I prove that $F/F'$ has the same rank that $F$ has, then, $F/F'$ and $F$ are isomorphic, right? But how can a non-necessarily abelian group be isomorphic to an abelian one?
It seems that someone has already posted a question here related to this: Let free group, F and F/F' have same rank.
But the answers don't address the issue I'm pointing out here.