Algorithms for Polycyclic Groups Habilitationsschrift Eingereicht beim Fachbereich f¨ur Mathematik und Informatik Universit¨at Gesamthochschule Kassel von Bettina Eick aus Bremerv¨orde Kassel, im November 2000 3 Preface A group is called polycyclic if there exists a polycyclic series through the group; that is, a subnormal series of finite length with cyclic factors. Each polycyclic group has a normal series of finite length whose factors are finitely generated abelian groups. This can be considered as a finiteness condition on polycyclic groups which makes them accessible for algorithmic purposes and it is the aim of this book to develop a variety of algorithmic methods for polycyclic groups. The structural investigation of polycyclic groups has been initiated 1938 by Hirsch in [29, 30, 31, 32, 33] and their central position in infinite group theory has long been recognized since. Polycyclic groups form a large class of groups; in particular, each finitely generated nilpotent group and each supersolvable group is polycyclic. Vice versa, each polycyclic group is finitely generated and solvable. In fact, polycyclic groups can be characterized as those solvable groups in which each subgroup is finitely generated. The algorithmic decidability of group theoretic questions in polycyclic groups has been considered by Baumslag, Cannonito, Robinson & Segal in [3] and by Segal in [69]. They proved that a large variety of interesting problems is decidable. The algorithms introduced for this purpose are algorithms in the classical sense and have not been invented for computer implementations. But clearly they can be considered as the fundamental initial step towards an algorithmic theory for polycyclic groups. Algorithms for polycyclic groups which are suitable for implementations and practical for interesting computations are the main issue here. The central goal for each of the considered problems is to reduce it either to applications of well-known methods from linear algebra or number theory or otherwise to solve it by methods which have proved to be practical by implementations. A number of the algorithms described here have been implemented in the computer algebra system Gap [74] and a part of these programs is available in the ‘Polycyclic’ share package [23]. These implementations have been used to exploit the practicality of their underlying methods. This book is intended to give an overview on algorithms for polycyclic groups with a particular interest in infinite polycyclic groups. It includes methods which have been known for some while and it also presents a variety of new developments in this area. In the following we give a brief overview on the 11 chapters of this book. 1-3 We introduce constructive polycyclic sequences which form the fundamental basis of our algorithms for polycyclic groups. We show that they are closely related to polycyclic presentations and we exploit their applications in handling subgroups, factor groups and homomorphisms of polycyclic groups. Most of the topics in these chapters are well-known in principle and another approach to them can be found in [71]. 4-5 We describe algorithms to compute constructive polycyclic sequences for polycyclic permutation groups and polycyclic matrix groups over finite fields or the rational numbers. Permutation groups and matrix groups over finite fields are finite and thus comparatively easy to handle. Our methods for these finite groups are based on an algorithm of Sims [70]. Rational matrix groups can also be infinite and we need a more detailed investigation in this case. Our approaches to rational matrix groups are based on algorithms by Dixon [16] and Ostheimer [59]. 6 Cohomology groups can be used to construct complements and extensions. We outline methods to compute cohomology groups for polycyclic groups which generalize the methods for finite polycyclic groups by Celler, Neub¨user & Wright [9] and by Plesken and Br¨uckner in [62, 7]. Then we exploit the cohomology groups for infinite polycyclic groups. In particular, we present an algorithm for determining an almost complement in an infinite polycyclic group. 4 7 The determination of orbits and stabilizers is one of the most fundamental problems in algorithmic group theory. Methods to determine finite orbits are well-known, but the development of algorithms to solve orbit stabilizer problems for infinite polycyclic groups acting as groups of automorphisms on finitely generated abelian groups can be considered as one of the major steps towards an algorithmic theory of infinite polycyclic groups. We present methods for this purpose in this chapter. A part of these methods has been developed in joint work with Ostheimer [25]. 8-9 We present a variety of algorithms designed to investigate the structure of polycyclic groups given by constructive polycyclic sequences. Chapter 8 contains algorithms for more general questions on subgroups of polycyclic groups such as the determination of centralizers, normalizers and intersec- tions. In Chapter 9 we consider more specific problems. In particular, we introduce a variety of algorithms to exhibit a number of group theoretic structure theorems for infinite polycyclic groups. 10 The algorithms in this book are often described as ‘practical’ or ‘effective’; that is, they are suitable for implementations and they are expected to have interesting applications. In this chapter we describe some examples of such applications and we use these examples to include a report on the performance of some of our methods. 11 This book provides a basis for the investigation of polycyclic groups by computational methods. However, for a number of interesting questions on polycyclic groups there are no practical algorithms known yet and some questions are not even known to be decidable in infinite polycyclic groups. We close this book with a collection of open problems of this type. The algorithms presented here all apply to polycyclic groups in general, but the main focus in this book is towards infinite polycyclic groups. In fact, most of the new developments described in this book are algorithms for infinite polycyclic groups. The computational theory of finite polycyclic groups has been developed over the last 30 years and a large number of practical algorithms for such groups are known. Implementations of them are available in the computer algebra systems Gap [74] and Magma [5]. In the course of the book we include references to finite group methods corresponding to our algorithms. The introduced algorithms make use of a variety of group theoretic properties of polycyclic groups. A number of them are recalled briefly when they are used. For more background on the structure of polycyclic groups we recommend the introduction in [67], pages 147ff, and the book by Segal [68]. Acknowledgments Many thanks are due to Gretchen Ostheimer who introduced me to the interesting field of polycyclic integral matrix groups and her work on algorithms for these groups. Further, I thank Werner Nickel who took part in implementing a basic machinery for polycyclic groups in the ‘Polycyclic’ share package of Gap. The Gap system has provided a useful framework for the implementation of algorithms for polycyclic groups and I acknowledge the support of the Gap team in this project. Finally, I thank Hans Ulrich Besche, Alexander Hulpke, Gunter Malle and Eamonn O’Brien for reading and commenting this work. Contents 1 Introduction to polycyclic sequences 9 1.1 Polycyclicsequences . .. .......... 9 1.2 Normalformsandexponentvectors . ........... 10 1.3 On the determination of constructive polycyclic sequences.................. 11 2 Polycyclic presentations 13 2.1 Definition and relation to polycyclic sequences . ................. 13 2.2 Collectedwordsandcollection . ............ 14 2.3 Consistency of polycyclic presentations . ................. 16 2.4 On the determination of polycyclic presentations . .................. 17 3 Subgroups, factor groups and homomorphisms 19 3.1 Subgroups and induced polycyclic sequences . ................ 19 3.2 Canonicalpolycyclicsequences . ............. 22 3.3 Normalclosuresofsubgroups . ........... 23 3.4 Induced polycyclic sequences for factor groups . .................. 24 3.5 Homomorphisms ................................... ...... 25 4 Polycyclic groups with finite action 29 4.1 Normalsubgroupsandblocks . .......... 29 4.2 Determiningfiniteorbits. ........... 30 4.3 Determininganabeliannormalseries. .............. 31 4.4 Extending constructive polycyclic sequences . .................. 32 5 Polycyclic matrix groups 35 5.1 Structure theory for rational polycyclic matrix groups .................... 35 5.2 Radicalsofrationalmodules. ............ 36 5.3 Semisimplerationalmatrixgroups . ............. 39 5.4 Moduleseriesandinducedactions . ............ 41 5.5 Constructive polycyclic sequences in rational matrix groups ................. 42 6 Cohomology groups 49 6.1 Definitionofcohomologygroups . ........... 49 6.2 Extensionsandcomplements . .......... 50 6.3 Determining the first and second cohomology groups . ................ 52 6.4 Finiteness conditions for cohomology groups . ................. 56 6.5 Almostcomplements ............................... ........ 57 5 6 CONTENTS 7 Orbits and stabilizers for polycyclic groups 61 7.1 Affineactionsandkernelsofderivations . .............. 62 7.2 Actionsonfreeabeliangroups . ........... 64 7.3 Stabilizers of elements in free abelian groups. .................. 65 7.4 Stabilizers of subgroups of free abelian groups . .................. 67 7.5 Actions on finitely generated