# Does a finitely presented amenable group contain a unique maximal finite normal subgroup?

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 $$G$$.

In a group $$G$$, the polyfinite radical $$W(G)$$ is the union of all finite normal subgroups. So, $$G$$ has a maximal finite normal subgroup iff $$W(G)$$ is finite.

You're asking whether $$W(G)$$ is finite for every finitely presented elementary amenable group. The answer is negative. There are indeed finitely presented solvable groups in which the center has infinite torsion (the torsion part of the center is clearly contained in the polyfinite radical, which is actually the torsion part of the FC-center).

I'm not sure what are the simplest known examples, but here are ones: for a prime $$p$$, the group of matrices $$\begin{pmatrix}1 & a_{12} & a_{13} & a_{14}\\ 0 & t^{n_1}(t+1)^{n_2} & a_{23} & a_{24} \\ 0 & 0 & t^{n_3}(t+1)^{n_4} & a_{34}\\ 0 & 0 & 0 & 1\end{pmatrix}: a_{ij}\in\mathbf{F}_p[t,(t+t^2)^{-1}],n_k\in\mathbf{Z}.$$

• Thanks very much for your nice answer. I got it completely. You mentioned in a comment that the answer is certainly yes for virtually polycyclic groups. I've added my try. Is my argument correct? Commented Jul 16, 2023 at 7:31
• @M.Ramana In a virtually polycyclic group every ascending sequence of subgroups is eventually constant, hence the polyfinite subgroup is finite. A larger class of groups in which this holds is the class of max-n groups (i.e., in which every ascending sequence of normal subgroups is eventually constant). This includes all abelian-by-(virtually polycyclic) groups.
– YCor
Commented Jul 16, 2023 at 15:37
• Dear @YCor , I appreciate your comment. By your answer, finitely presented elementary amenable groups don't have necessarily a unique maximal finite normal subgroup. Is still true for a unique maximal finite subgroup? I mean do finitely presented amenable groups have a unique maximal finite subgroup? Commented Jul 21, 2023 at 5:10
• @M.Ramana the same give counterexamples, but even the infinite dihedral group has maximal finite subgroups, but infinitely many.
– YCor
Commented Jul 21, 2023 at 8:55
• Right. I understood. I really appreciate your help. Commented Jul 22, 2023 at 2:53