GAP (polycyclic) - Chapter 1: Preface

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1 Preface

### 1 Preface

A group G is called polycyclic if there exists a subnormal series in G with cyclic factors. Every polycyclic group is soluble and every supersoluble group is polycyclic. The class of polycyclic groups is closed with respect to forming subgroups, factor groups and extensions. Polycyclic groups can also be characterised as those soluble groups in which each subgroup is finitely generated.

K. A. Hirsch has initiated the investigation of polycyclic groups in 1938, see [Hir38a], [Hir38b], [Hir46], [Hir52], [Hir54], and their central position in infinite group theory has been recognised since.

A well-known result of Hirsch asserts that each polycyclic group is finitely presented. In fact, a polycyclic group has a presentation which exhibits its polycyclic structure: a pc-presentation as defined in the Chapter Introduction to polycyclic presentations. Pc-presentations allow efficient computations with the groups they define. In particular, the word problem is efficiently solvable in a group given by a pc-presentation. Further, subgroups and factor groups of groups given by a pc-presentation can be handled effectively.

The GAP 4 package Polycyclic is designed for computations with polycyclic groups which are given by a pc-presentation. The package contains methods to solve the word problem in such groups and to handle subgroups and factor groups of polycyclic groups. Based on these basic algorithms we present a collection of methods to construct polycyclic groups and to investigate their structure.

In [BCRS91] and [Seg90] the theory of problems which are decidable in polycyclic-by-finite groups has been started. As a result of these investigation we know that a large number of group theoretic problems are decidable by algorithms in polycyclic groups. However, practical algorithms which are suitable for computer implementations have not been obtained by this study. We have developed a new set of practical methods for groups given by pc-presentations, see for example [Eic00], and this package is a collection of implementations for these and other methods.

We refer to [Rob82], page 147ff, and [Seg83] for background on polycyclic groups. Further, in [Sim94] a variation of the basic methods for groups with pc-presentation is introduced. Finally, we note that the main GAP library contains many practical algorithms to compute with finite polycyclic groups. This is described in the Section on polycyclic groups in the reference manual.

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