nilpotent subgroup of finite index in finitely generated abelien by nilpotent

Question

let $N$ be nilpotent subgroup of finite index in finitely generated abelien by nilpotent(i.e there exist normal subgroup $M$ abelien such that $G/M$ is nilpotent ), proceed by induction on the order of $G/N$ and consider first of all the special case where $G/N$ cyclic i wont to prove that there exist normal subgroup $H$ of index some prime $q$ in $G$ where $H$ contain $N$.

Answer 1

There's a bit of redundancy here, it seems.

Assume $G$ is abelian-by-nilpotent, so that there is an abelian normal subgroup $M$ such that $G/M$ is nilpotent. Let $N$ be a proper, normal subgroup of finite index.

Then $G/N \ne 1$ is also abelian-by-nilpotent. This is because

• $$M N / N \cong M /(M \cap N)$$ is a normal subgroup of $G/N$ which is abelian, as a quotient of the abelian group $M$, and
• $$ (G / N) / ( M N / N) \cong G /M N \cong (G/M) / (MN /M) $$ is nilpotent, as a quotient of the nilpotent group $G / M$.

Now all you have to prove is that a finite, abelian-by-nilpotent group $H \ne 1$ has a normal subgroup of index a prime. To do this, just note that $H' < H$.