$D(G)$ is finite if $G$ has finitely many commutators I read somewhere that if a group $G$ has only finitely many commutators then the commutator subgroup $D(G)$ is itself finite.
Do you know a proof of this result?
 A: I am sorry but I do not have a complete answer at the moment.
I will show that if $G$ is finitely generated and there are only finitely many commutators, then $D(G)$ is finite.
We will prove that the center of $G$, $Z(G)$ has finite index in $G$, then a theorem by Schur will allow us to conclude.
If $G$ is finitely generated, take $g_1,\ldots,g_n\in G$ to be a generating set. We have
$$Z(G)=\bigcap_{i=1}^n Z(g_i),$$
where $Z(g)$ is the centralizer of $g$ in $G$.
It suffices to prove that every $Z(g_i)$ has finite index in $G$; this is the same as asking to verify that every element $g_i$ has finitely many conjugates.
Take $h\in G$. We have $hgh^{-1}=g[g^{-1},h]$. Since there are only finitely many commutators, every $g\in G$ has finitely many conjugates.
Hence $Z(g_i)$ is of finite index in $G$, for every $i=1\ldots n$.
Now, we have the following theorem
Theorem (Schur) If $[G:Z(G)]<\infty$, then $D(G)$ is finite.
I learned about this theorem in the italian book Gruppi by Antonio Machì. There is now a translated edition where it is Theorem 2.38 on page 84.
