Groups in derived series are normal in the whole group

Question

I am considering a finitely generated, solvable group $G$ which has subexponential growth. Let $G=G_0\vartriangleright G_1\vartriangleright....\vartriangleright G_d\vartriangleright G_{d+1}=1$ denote the derived series of G.

It should be the case that $G_d$ is normal in $G$. Does anyone know why this is true? I don't know whether all of the assumptions above are necessary for the result.

Answer 1

In fact all the subgroups $G_i$ are normal in $G$ for let $N$ be a normal subgroup of $G$ then $N'$ is normal in $G$. To see this let $[n_1,n_2]$ be a commutator in $N'$ then $[n_1,n_2]^g = [n_1^g, n_2^g] = [n'_1,n'_2] \in N'$ $ \forall g \in G$.