The algorithmic theory of polycyclic-by-finite groups

Abstract

A group is said to be polycyclic-by-finite, or a PF-group for short, if it has a polycyclic normal subgroup of finite index. Equivalently, PF-groups are exactly the groups which have a series of finite length whose infinite factors are cyclic. By a well-known theorem of P. Hall every PF-group is finitely presented-and in fact PF-groups form the largest known sectionclosed class of finitely presented groups. It is this fact that makes PF-groups natural objects of study from the algorithmic standpoint. The general aim of the algorithmic theory of PF-groups can be described as the collection of information about a PF-group which can, in principle