4 - DIVISION RINGS ASSOCIATED WITH POLYCYCLIC GROUPS

Summary

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.