On soluble-by-finite subgroups of division algebras

It is shown that a soluble-by-finite subgroup G of a finite-dimensional division algebra D contains an abelian normal subgroup of index dividing 60deg(D)2. Moreover, the group of outer automorphisms of G induced by the elements of D is also soluble-by-finite. The question of whether a division algebra generated by a soluble-by-finite group is necessarily is a crossed product is also addressed.