In this chapter we are going to take a closer look at the way in which a polyclic group can sit inside GLn(ℤ). As we saw in Chapter 7, a good understanding of isomorphisms between polycyclic groups would seem to depend on knowing when soluble subgroups of GLn(ℤ) are conjugate, and an algorithm for deciding this question is sketched in section D, below. Sections B and C investigate the normalizer in GLn(ℤ) of an arbitrary soluble-by-finite subgroup. Using the fact (proved in Chapter 5) that the holomorph of a polycyclic-by-finite group G can be embedded in some GLn(ℤ), we shall derive some rather striking consequences: (a) Aut G has a normal subgroup K, isomorphic to an arithmetic group, with (Aut G)/K finitely generated and abelian-by-finite; (b) Aut G is a finitely presented group; (c) the finite subgroups of Aut G lie in finitely many conjugacy classes; and (d) for each natural number m, there exist only finitely many non-isomorphic extensions of G by a group of order m.

An important tool in these investigations is the Zariski topology in a linear group. This has been lurking in the background to some of the previous chapters; but the time has come for a fuller discussion of the topic, and this is given in section A.

The Zariski topology

Let k be a field and m a natural number.