A faithful matrix representation of the holomorph of the Abels groups (Q1123267)

Let \(G_{nm}\) be the group of all upper triangular matrices of order n with entries in the ring \({\mathbb{Z}}[1/m]\), diagonal elements of the form \(m^ k\), \(k\in {\mathbb{Z}}\), and the (1,1) and (n,n)-entries equal to one. For \(n\geq 4\) and \(m\geq 2\) the group \(G_{nm}\) is finitely presented but does not satisfy the maximality condition for normal - even central - subgroups, as was first observed by the reviewer for \(n=4\) and m a prime, thus answering a problem of P. Hall's. The author proves by his method of decomposable coordinates that the holomorph of the group \(G_{nm}\), \(n\geq 3\), \(m\geq 2\), has a faithful representation by matrices with entries in \({\mathbb{Z}}[1/m]\).