The skew linear groups considered in the previous two chapters have the property that their finitely generated subgroups are isomorphic to linear groups, usually of unbounded degree. Here we consider a class of skew linear groups that do not, in general, have this property.

Specifically we study skew linear groups over division rings D of the form D = F(G), where F is a central subfield of D, G is a polycyclicby-finite subgroup of D*, and D is generated as a division ring by and G. Skew linear groups of this kind were first considered by Lichtman. It follows immediately from Goldie's theorem (1.4.2) that D is the (classical) ring of quotients of the Noetherian (Passman, 10.2.7) subalgebra F[G] of D = F(G). Thus each element of D can be written in the forms ac–1 = d–1b for elements a, b, c, d of F[G], with c and d non-zero. This fact plays a crucial role throughout the chapter.

In Section 4.1 below we show that the above class of skew linear groups is a little wider than might be at first apparent. We also indicate examples of such skew linear groups that are finitely generated and soluble but are not (group-theoretically) isomorphic to any linear group, thus confirming the remark above.