On soluble skew linear groups over finite-dimensional division algebras

Let D be a division ring of finite degree d and let n be a positive integer. If G is any soluble subgroup of GL(n, D), we prove that G has derived length at most 9 + log(2)d + (11/3)log(2)n and that G has a unipotent-by-abelian (abelian if G is completely reducible) normal subgroup of finite index dividing b(n).d(2n), where b(n) is an integer-valued function of n only. Actually, we derive bounds rather better than those quoted above, but rather more involved to state. (c) 2006 Elsevier B.V. All rights reserved.