Proposition 8

Let $G$ be a nilpotent group of class $\leq n$ and let $H$ be a subgroup of $G$. There exists a sequence of subgroups

such that $H^{k+1}$ is normal in $H^k$ and $H^k/H^{k+1}$ is commutative for all $k \leq n$.

Choose a sequence $(G^k)$ of subgroups of $G$ satisfying the conditions of Proposition 7 (b) for all $k$; $G^k$ is normal in $G$. Write:

It is necessary to verify that $H^{k+1}$ is normalized by $H^k = H.G^k$; as it is normalized by $H$, it suffices to verify that it is by $G^k$. Now, if $s \in G^k$ and $h \in H$,

and $(G, G^k).H$ is contained in $G^{k+1}.H = H^{k+1}$; hence $s.H^{k+1}.s^{-1} = H^{k+1}$, which shows that $H^{k+1}$ is normal in $H^k$.

Finally, the canonical homomorphism $G^k/G^{k+1} \to H^k/H^{k+1}$ is obviously surjective; as the first group is commutative, so is the second.