O. Baues and F. Gruewald in their paper "AUTOMORPHISM GROUPS OF POLYCYCLIC-BY-FINITE GROUPS AND ARITHMETIC GROUPS" have stated that a polycyclic-by-finite group has a unique maximal finite normal subgroup (see page 216, Section 1.2). I was wondering if somone could give a proof for that? Does this result still hold for finitely presented amenable groups, in particular elementary amenable ones?

**My try**: Let $G$ be a virtually polycyclic group. Since a virtually polycyclic group is finitely presented, it is countbale. So assume that $N_1 ,N_2 ,\ldots $ are finite normal subgroups of $G$. For each inetegr $n\geq 1$, $N_1 N_2 \cdots N_n$ is a finite subgroup. On other hand, a given virtually polycyclic group has only finitely many finite subgroups up to isomorphism. Hence the ranks of finite subgroups are bounded. Thus there are only at most $n_G$, where $n_G$ is an integer depending only on $G$, possible different finite subgroups $N_i$ in $G$.