On Abelian subgroups of finitely generated metabelian groups. (Q2873308)

In 1959 P. Hall introduced the class \(\mathcal H\) consisting of groups \(G\) satisfying the following properties:{\parindent=6mm\begin{itemize}\item[(1)] \(G\) is a (finite or) countable Abelian group;\item[(2)] \(G=T\oplus K\), where \(T\) is a bounded torsion group and \(K\) is torsion-free;\item[(3)] \(K\) contains a free Abelian subgroup \(F\) such that \(K/F\) is a torsion group with trivial \(p\)-subgroups for all primes except for those of a finite set \(\pi=\pi(K)\).NEWLINENEWLINE\end{itemize}} The members of the class \(\mathcal H\) are called \textit{Hall groups}.NEWLINENEWLINE In the article under review the authors investigate the Abelian subgroups of finitely generated metabelian groups. The main results obtained are the following:NEWLINENEWLINE Theorem 1.1. Let \(H\) be an Abelian group. Then \(H\) is a subgroup of a finitely generated Abelian-by-polycyclic group if and only if \(H\) is a Hall group. Moreover, every Hall group embeds in the commutator subgroup of a 2-generated metabelian group.NEWLINENEWLINE Theorem 1.3. There exists a 2-generated metabelian group containing continuously many pairwise non-isomorphic Abelian subgroups.NEWLINENEWLINE The above result gives a negative answer to Baumslag's conjecture in 1988 (see [\textit{G. Baumslag}, in: Combinatorial group theory, Proc. AMS Spec. Sess., College Park, MD, USA 1988, Contemp. Math. 109, 1-9 (1990; Zbl 0715.20020)]).NEWLINENEWLINE Proposition 6.1. Let \(H\) be a finitely generated Abelian group. Then \(H\) is isomorphic to the center of a 2-generated metabelian, nilpotent group \(G\) which is finite if so is \(H\).NEWLINENEWLINE Proposition 6.2. Let \(H\) be a finitely generated Abelian group. Then \(H\) is a normal subgroup of a 2-generated metabelian group \(G\) such that \(G/H\) is finite cyclic.