Hyperabelian skew-linear groups (Q798443)

For any group G let hph(G) be the least ordinal \(\gamma\) such that G is the union of \(\gamma\) normal subgroups containing each other and with locally nilpotent subsequent factors. Also hah(G) is defined similarly for abelian factors. Of course, hph(G)\(\leq hah(G).\) Theorem 1. Let \(p=0\) or a prime and let 1\(\leq\beta \leq\gamma \) be ordinals. Then there exists a division ring D of characteristic p that is locally finite-dimensional over its centre and a torsion-free hyperabelian subgroup G of \(D^* (=D-\{0\})\) such that \(hph(G)=\beta\) and \(hah(G)=\gamma\). - Let D be as theorem 1 and \(G\subset GL(n,D)\) with \(hph(G)=1\). Then it is known that \(hah(G)<\infty\). Let \(d(n,p)=\max\{hap(G)| G\subset GL(n,D),hph(G)=1\}.\) Theorem 2. \(d(n,p)=2+[\log_ 2(n)]\) if \(n=1\) or \(p=0\) and \(=1+[-\log_ 2(n)]\) if \(n>1\) and \(p>0\).