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J.deSilva, Fundamentals of Information Theory and Coding Design Lawrence C.Washington, Elliptic Curves: Number Theory and Cryptography Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Juergen Bierbrauer, Introduction to Coding Theory William Kocay and Donald L.Kreher, Graphs, Algorithms, and Optimization Derek F.Holt with Bettina Eick and Eamonn A.O’Brien, Handbook of Computational Group Theory © 2005 by Chapman & Hall/CRC Press DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H ROSEN HANDBOOK OF COMPUTATIONAL GROUP THEORY DEREK F.HOLT BETTINA EICK EAMONN A.O’BRIEN CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C © 2005 by Chapman & Hall/CRC Press Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher The consent of CRC Press does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must be obtained in writing from CRC Press for such copying Direct all inquiries to CRC Press, 2000 N.W Corporate Blvd., Boca Raton, Florida 33431 Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe Visit the CRC Press Web site at www.crcpress.com © 2005 by Chapman & Hall/CRC Press No claim to original U.S Government works International Standard Book Number 1-58488-372-3 Printed in the United States of America Printed on acid-free paper © 2005 by Chapman & Hall/CRC Press Preface This book is about computational group theory, which we shall frequently abbreviate to CGT The origins of this lively and active branch of mathematics can be traced back to the late nineteenth and early twentieth centuries, but it has been flourishing particularly during the past 30 to 40 years The aim of this book is to provide as complete a treatment as possible of all of the fundamental methods and algorithms in CGT, without straying above a level suitable for a beginning postgraduate student There are currently three specialized books devoted to specific areas of CGT, namely those of Greg Butler [But91] and Ákos Seress [Ser03] on algorithms for permutation groups, and the book by Charles Sims [Sim94] on computation with finitely presented groups These texts cover their respective topics in greater detail than we shall here, although we have relied heavily on some of the pseudocode presented in [Sim94] in our treatment of coset enumeration in Chapter 5, The most basic algorithms in CGT tend to be representation specific; that is, there are separate methods for groups given as permutation or matrix groups, groups defined by means of polycyclic presentations, and groups that are defined using a general finite presentation We have devoted separate chapters to algorithms that apply to groups in these different types of representations, but there are other chapters that cover important methods involving more than one type For example, Chapter is about finding presentations of permutation groups and the connections between coset enumeration and methods for finding the order of a finite permutation group We have also included a chapter (Chapter 11) on the increasing number of precomputed stored libraries and databases of groups, character tables, etc that are now publicly available They have been playing a major rôle in CGT in recent years, both as an invaluable resource for the general mathematical public, and as components for use in some advanced algorithms in CGT The library of all finite groups of order up to 2000 (except for order 1024) has proved to be particularly popular with the wider community It is inevitable that our choice of topics and treatment of the individual topics will reflect the authors’ personal expertise and preferences to some extent On the positive side, the final two chapters of the book cover applications of string-rewriting techniques to CGT (which is, however, treated in much greater detail in [Sim94]), and the application of finite state automata to the computation of automatic structures of finitely presented groups On the other hand, there may be some topics for which our treatment is more superficial than it would ideally be One such area is the complexity analysis of the algorithms of CGT During the 1980s and 1990s some, for the most part friendly and respectful, rivalry developed between those whose research in CGT was principally directed towards producing better performance of their code, and those who were more interested in proving theoretical results concerning the complexity of v © 2005 by Chapman & Hall/CRC Press vi the algorithms This study of complexity began with the work of Eugene Luks, who established a connection in his 1982 article [Luk82] between permutation group algorithms and the problem of testing two finite graphs for isomorphism Our emphasis in this book will be more geared towards algorithms that perform well in practice, rather than those with the best theoretical complexity Fortunately, Seress’ book [Ser03] includes a very thorough treatment of complexity issues, and so we can safely refer the interested reader there In any case, as machines become faster, computer memories larger, and bigger and bigger groups come within the range of practical computation, it is becoming more and more the case that those algorithms with the more favourable complexity will also run faster when implemented The important topic of computational group representation theory and computations with group characters is perhaps not treated as thoroughly as it might be in this book We have covered some of the basic material in Chapter 7, but there is unfortunately no specialized book on this topic For a brief survey of the area, we can refer the reader to the article by Gerhard Hiss [His03] One of the most active areas of research in CGT at the present time, both from the viewpoint of complexity and of practical performance, is the development of effective methods for computing with large finite groups of matrices Much of this material is beyond the scope of this book It is, in any case, developing and changing too rapidly to make it sensible to attempt to cover it properly here Some pointers to the literature will of course be provided, mainly in Section 7.8 Yet another topic that is beyond the scope of this book, but which is of increasing importance in CGT, is computational Lie theory This includes computations with Coxeter groups, reflection groups, and groups of Lie type and their representations It also connects with computations in Lie algebras, which is an area of independent importance The article by Cohen, Murray, and Taylor [CMT04] provides a possible starting point for the interested reader The author firmly believes that the correct way to present a mathematical algorithm is by means of pseudocode, since a textual description will generally lack precision, and will usually involve rather vague instructions like “carry on in a similar manner” So we have included pseudocode for all of the most basic algorithms, and it is only for the more advanced procedures that we have occasionally lapsed into sketchy summaries We are very grateful to Thomas Cormen who has made his LATEXpackage ‘clrscode’ for displaying algorithms publicly available This was used by him and his coauthors in the well-known textbook on algorithms [CLRS02] Although working through all but the most trivial examples with procedures that are intended to be run on a computer can be very tedious, we have attempted to include illustrative examples for as many algorithms as is practical At the end of each chapter, or sometimes section, we have attempted to direct the reader’s attention to some applications of the techniques developed © 2005 by Chapman & Hall/CRC Press vii in that chapter either to other areas of mathematics or to other sciences It is generally difficult to this effectively Although there are many important and interesting applications of CGT around, the most significant of them will typically use methods of CGT as only one of many components, and so it not possible to them full justice without venturing a long way outside of the main topic of the book We shall assume that the reader is familiar with group theory up to an advanced undergraduate level, and has a basic knowledge of other topics in algebra, such as ring and field theory Chapter includes a more or less complete survey of the required background material in group theory, but we shall assume that at least most of the topics reviewed will be already familiar to readers Chapter assumes some basic knowledge of group representation theory, such as the equivalence between matrix representations of a group G over a field K and KG-modules, but it is interesting to note that many of the most fundamental algorithms in the area, such as the ‘Meataxe’, use only rather basic linear algebra A number of people have made helpful and detailed comments on draft versions of the book, and I would particularly like to thank John Cannon, Bettina Eick, Joachim Neubüser, and Colva Roney-Dougal in this regard Most of all, I am grateful to my two coauthors, Bettina Eick and Eamonn O’Brien for helping me to write parts of the book The whole of Chapter 8, on computing in polycyclic groups, was written by Bettina Eick, as were Sections 11.1 and 11.4 on the libraries of primitive groups and of small groups, respectively Section 7.8 on computing in matrix groups and Section 9.4 on computing p-quotients of finitely presented groups were written by Eamonn O’Brien August 2004 © 2005 by Chapman & Hall/CRC Press Derek Holt Contents Notation and displayed procedures A Historical Review of Computational Group Theory Background Material 2.1 Fundamentals 2.1.1 Definitions 2.1.2 Subgroups 2.1.3 Cyclic and dihedral groups 2.1.4 Generators 2.1.5 Examples—permutation groups and matrix groups 2.1.6 Normal subgroups and quotient groups 2.1.7 Homomorphisms and the isomorphism theorems 2.2 Group actions 2.2.1 Definition and examples 2.2.2 Orbits and stabilizers 2.2.3 Conjugacy, normalizers, and centralizers 2.2.4 Sylow’s theorems 2.2.5 Transitivity and primitivity 2.3 Series 2.3.1 Simple and characteristically simple groups 2.3.2 Series 2.3.3 The derived series and solvable groups 2.3.4 Central series and nilpotent groups 2.3.5 The socle of a finite group 2.3.6 The Frattini subgroup of a group 2.4 Presentations of groups 2.4.1 Free groups 2.4.2 Group presentations 2.4.3 Presentations of group extensions 2.4.4 Tietze transformations 2.5 Presentations of subgroups 2.5.1 Subgroup presentations on Schreier generators 2.5.2 Subgroup presentations on a general generating set 2.6 Abelian group presentations xvi 9 11 12 13 13 14 15 17 17 19 20 21 22 26 26 27 27 29 31 32 33 33 36 38 40 41 41 44 46 ix © 2005 by Chapman & Hall/CRC Press x 2.7 2.8 Representation theory, modules, extensions, derivations, and complements 48 2.7.1 The terminology of representation theory 49 2.7.2 Semidirect products, complements, derivations, and first cohomology groups 50 2.7.3 Extensions of modules and the second cohomology group 52 2.7.4 The actions of automorphisms on cohomology groups 54 Field theory 56 2.8.1 Field extensions and splitting fields 56 2.8.2 Finite fields 58 2.8.3 Conway polynomials 59 Representing Groups on a Computer 3.1 Representing groups on computers 3.1.1 The fundamental representation types 3.1.2 Computational situations 3.1.3 Straight-line programs 3.1.4 Black-box groups 3.2 The use of random methods in CGT 3.2.1 Randomized algorithms 3.2.2 Finding random elements of groups 3.3 Some structural calculations 3.3.1 Powers and orders of elements 3.3.2 Normal closure 3.3.3 The commutator subgroup, derived series, and lower central series 3.4 Computing with homomorphisms 3.4.1 Defining and verifying group homomorphisms 3.4.2 Desirable facilities Computation in Finite Permutation Groups 4.1 The calculation of orbits and stabilizers 4.1.1 Schreier vectors 4.2 Testing for Alt(⍀) and Sym(⍀) 4.3 Finding block systems 4.3.1 Introduction 4.3.2 The Atkinson algorithm 4.3.3 Implementation of the class merging process 4.4 Bases and strong generating sets 4.4.1 Definitions 4.4.2 The Schreier-Sims algorithm 4.4.3 Complexity and implementation issues 4.4.4 Modifying the strong generating set—shallow Schreier trees © 2005 by Chapman & Hall/CRC Press 61 61 61 62 64 65 67 67 69 72 72 73 73 74 74 75 77 77 79 81 82 82 83 85 87 87 90 93 95 xi 4.5 4.6 4.7 4.8 4.4.5 The random Schreier-Sims method 4.4.6 The solvable BSGS algorithm 4.4.7 Change of base Homomorphisms from permutation groups 4.5.1 The induced action on a union of orbits 4.5.2 The induced action on a block system 4.5.3 Homomorphisms between permutation groups Backtrack searches 4.6.1 Searching through the elements of a group 4.6.2 Pruning the tree 4.6.3 Searching for subgroups and coset representatives 4.6.4 Automorphism groups of combinatorial structures and partitions 4.6.5 Normalizers and centralizers 4.6.6 Intersections of subgroups 4.6.7 Transversals and actions on cosets 4.6.8 Finding double coset representatives Sylow subgroups, p-cores, and the solvable radical 4.7.1 Reductions involving intransitivity and imprimitivity 4.7.2 Computing Sylow subgroups 4.7.3 A result on quotient groups of permutation groups 4.7.4 Computing the p-core 4.7.5 Computing the solvable radical 4.7.6 Nonabelian regular normal subgroups Applications 4.8.1 Card shuffling 4.8.2 Graphs, block designs, and error-correcting codes 4.8.3 Diameters of Cayley graphs 4.8.4 Processor interconnection networks Coset Enumeration 5.1 The basic procedure 5.1.1 Coset tables and their properties 5.1.2 Defining and scanning 5.1.3 Coincidences 5.2 Strategies for coset enumeration 5.2.1 The relator-based method 5.2.2 The coset table-based method 5.2.3 Compression and standardization 5.2.4 Recent developments and examples 5.2.5 Implementation issues 5.2.6 The use of coset enumeration in practice 5.3 Presentations of subgroups 5.3.1 Computing a presentation on Schreier generators 5.3.2 Computing a presentation on the user generators © 2005 by Chapman & Hall/CRC Press 97 98 102 105 105 106 107 108 110 113 114 118 121 124 126 131 132 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Presentations of groups 2.4.1 Free groups 2.4.2 Group presentations 2.4.3 Presentations of group extensions 2.4.4 Tietze transformations 2.5 Presentations of subgroups 2.5.1 Subgroup presentations... Quotients of Finitely Presented Groups 9.1 Finite quotients and automorphism groups of finite groups 9.1.1 Description of the algorithm 9.1.2 Performance issues 9.1.3 Automorphism groups of finite groups... elementary abelian p -group In general, if A, B are subsets of a group G, then we define 2.1.2 Subgroups DEFINITION 2.3 A subset H of a group G is called a subgroup of G if it forms a group under the