4 - Arithmetical methods

Summary

This chapter illustrates how certain kinds of group-theoretical problem can be translated into questions of algebraic number theory. We have already seen an example of this in Chapter 2, where the Dirichlet Units Theorem played a key role; using deeper arithmetical results we shall be able to derive some more subtle properties of polycyclic groups. The basic idea is that in certain aspects of its internal structure, a polycyclic group resembles a subgroup of the semi-direct product o⁺]o*, where o is the ring of integers in some algebraic number field: the precise sense in which this holds was stated as Proposition 1 of Chapter 2.

The first two sections, A and B, are concerned with a theorem of Baer, which says that supersolubility in a polycyclic group is determined by the finite quotients of the group; their main purpose is to give a simple introduction to the method. In section C we prove that polycyclic groups are conjugacy separable: i.e., conjugacy of elements in such a group is controlled by the finite quotients of the group. This important theorem is one of the highlights of the subject, and was only established within the last decade (by V.N. Remeslennikov and E. Formanek). As well as being aesthetically attractive because of its simplicity and depth, this result has powerful consequences for the behaviour of soluble subgroups in GL_n(ℤ); these will play an important role in the later chapters.