Introduction

The algorithmic theory of some solvable groups appears now to be rather complete. In the near future two rather large papers will appear which will give complete details of the algorithms for polycyclic-by-finite groups devised by me together with Gilbert Baumslag, Derek J S Robinson and Dan Segal [BCRS]. In addition, Dan Segal will give additional algorithms for automorphisms of these groups which will conclude with a positive solution for the isomorphism problem [DS]. Another development that bears watching is the actual implementation of some of these algorithms for high speed computers. Indeed, Charles Sims has almost got the so-called “polycyclic quotient algorithm” devised by Gil Baumslag, myself and Chuck Miller [BCM IV] running; something that very few of us dreamed could happed (see [CS]). So for the first time a practical theory of computing in group theory with some real depth seems likely. No one can tell where this could lead.

The property of polycyclic-by-finite groups that facilitates this development is that they are coherent. That is, they and all their subgroups are finitely presented. Another important feature is that these groups have an easily recognized (finite) presentation. Indeed, if one writes down a so-called polycyclic presentation (see below), which inevitably presents a solvable group, then the polycyclic quotient algorithm referred to above enables you to effectively decide if the group presented is polycyclic [BCM IV].