A family of solvable linear groups that do not satisfy the maximal condition on normal subgroups
For integers \(n\geq 3\), \(m\geq 2\) let \(G_{n,m}\) be the group of all upper triangular matrices of degree n having the form \[ \begin{pmatrix} 1&*&...&*&* \\ &*&...&*&* \\ && \ddots & \vdots& \vdots \\ &&&*&* \\ &&&&1 \end{pmatrix} \] over the ring \({\mathbb{Z}}[\frac{1}{m}]\), with diagonal elements of the form \(m^ k\), \(k\in {\mathbb{Z}}\). \textit{H. Abels} has shown [Lond. Math. Soc. Lect. Note Ser. 36, 205-211 (1979; Zbl 0422.20026)] that \(G_{4,m}\) is finitely presented but has infinitely generated centre. In the present paper it is shown that the same holds for all groups \(G_{n,m}\) \(n\geq 4\).
scientific article; zbMATH DE number 3991767