On soluble skew linear groups over finite-dimensional division algebras

Let DD be a division ring of finite degree dd and let nn be a positive integer. If GG is any soluble subgroup of GL(n,D), we prove that GG has derived length at most 9+log2d+(11/3)log2n9+log2d+(11/3)log2n and that GG has a unipotent-by-abelian (abelian if GG is completely reducible) normal subgroup of finite index dividing b(n).d2nb(n).d2n, where b(n)b(n) is an integer-valued function of nn only. Actually, we derive bounds rather better than those quoted above, but rather more involved to state.