Hello?
I have a simple question about combinatorial group theory.


For a group $G$, I saw $[G\_k, G\_m] \subset G\_{k+m}$ and these two subgroups need not be equal.
Then is there any known condition that they can be equal?
How about the case that $G$ is (f.g.) free?


Actually, my first question was that:
for a f.g. free group $F$, $F\_m \subset [F\_k,F\_k]$ for some big $m$?



When $[G_k,G_m] = G_{k+m}$?
Presumably $G=G\_1 \ge G\_2 \ge G\_3 \ge \cdots$ is the lower central series of $G$?


So, for a free group $F$ of rank $r$, $F\_2$ = $[F,F]$, and $F/[F\_2,F\_2]$ is the free $r$-generator metabelian group.


$F\_m < [F\_2,F\_2]$ would imply that $F/[F\_2,F\_2]$ is nilpotent of class at most $m$, but since there are 2-generator metabelian groups that are not nilpotent, this is not possible for $r \ge 2$.


