Semilinear and Skew Linear Groups

Abstract

In Chap. 4 we studied a polycyclic group by considering it as a subgroup of some GL(n,Z). Is there some sort of analogous way of obtaining properties of finitely generated abelian-by-polycyclic groups? Firstly a consideration of linear groups will not suffice, since soluble linear groups must be nilpotent-by-abelian-by-finite. Indeed not even all finitely generated, abelian-by-polycyclic, nilpotent-by-abelian-by-finite groups (that is, in the notation of Chap. 1, G∩AP∩NAF groups) need be isomorphic to even quasi-linear groups (a quasi-linear group is a subgroup of a direct product of a finite number of linear groups; equivalently a quasi-linear group is any group of automorphisms of a Noetherian module over a commutative ring, see Wehrfritz 1979c, p. 55).