11 - Examples

Summary

It is easy to construct polycyclic groups, and in this final chapter I shall describe various ways of doing so which illustrate points discussed so far. In a way, this chapter may be seen as a continuation of Chapter 4, section B: to construct groups with specified properties, we must in each case elucidate the arithmetical meaning of those properties, and then have to solve the resulting problem of number theory.

Section A constructs non-isomorphic polycyclic groups with isomorphic finite quotients; the simplest examples of these are directly related to algebraic number fields with class number greater than 1. In section B we consider groups which do not split over their Fitting subgroup, hence cannot be isomorphic to arithmetic groups.

The rest of the chapter concentrates on torsion-free nilpotent groups of class 2. These groups have a particularly simple description in ‘arithmetical’ terms, which is explained in section C. Those of Hirsch number at most 6 can be completely classified, in terms of equivalence classes of binary integral quadratic forms. This classification is given in section D; it illustrates a theme which I have tried to bring out throughout the book, by isomorphically embedding a small segment of the theory of polycyclic groups into an interesting and classical branch of number theory.

Finally, section E discusses unitriangular groups of matrices over rings of algebraic integers.