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Permutation Group Algorithms This page intentionally left blank Permutation group algorithms comprise one of the workhorses of symbolic algebra systems computing with groups and play an indispensable role in the proof of many deep results, including the construction and study of sporadic finite simple groups. This book describes the theory behind permutation group algorithms, up to the most recent developments based on the classification of finite simple groups. Rigorous complexity estimates, implementation hints, and advanced exercises are included throughout. The central theme is the description of nearly linear-time algorithms, which are extremely fast in terms of both asymptotic analysis and practical running time. A significant part of the permutation group library of the computational group algebra system GAP is based on nearly linear-time algorithms. The book fills a significant gap in the symbolic computation literature. It is recommended for everyone interested in using computers in group theory and is suitable for advanced graduate courses. Akos´ Seress is a Professor of Mathematics at The Ohio State University. CAMBRIDGE TRACTS IN MATHEMATICS General Editors B. BOLLOBAS,´ W. LTON, A. KATOK, F. KIRWAN, P. SARNAK 152 Permutation Group Algorithms Akos´ Seress The Ohio State University Permutation Group Algorithms Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States by Cambridge University Press, New York wwcambridgorg Information on this title: wwcambrid gor g/9780521661034 © Ákos Seress 2002 This book is in copyrigh Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Pres First published in print format 2003 ISBN-13 978-0-511-06647-4 eBook (NetLibrary) ISBN-10 0-511-06647-3 eBook (NetLibrary) ISBN-13 978-0-521-66103-4 hardback ISBN-10 0-521-66103-X hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriat Contents 1Introductionpage1 1.1AListofAlgorithms4 1.2NotationandTerminology6 1.2.1Groups7 1.2.2PermutationGroups9 1.2.3AlgorithmicConcepts10 1.2.4Graphs11 1.3ClassificationofRandomizedAlgorithms12 2Black-BoxGroups16 2.1ClosureAlgorithms18 2.1.1OrbitComputations18 2.1.2ClosureofAlgebraicStructures23 2.2RandomElementsofBlack-BoxGroups24 2.3RandomSubproducts30 2.3.1DefinitionandBasicProperties30 2.3.2ReducingtheNumberofGenerators33 2.3.3ClosureAlgorithmswithoutMembershipTesting37 2.3.4DerivedandLowerCentralSeries38 2.4RandomPrefixes40 2.4.1DefinitionandBasicProperties40 2.4.2Applications44 3PermutationGroups:AComplexityOverview48 3.1Polynomial-TimeAlgorithms48 3.2NearlyLinear-TimeAlgorithms51 3.3Non-Polynomial-TimeMethods52 v vi Contents 4BasesandStrongGeneratingSets55 4.1BasicDefinitions55 4.2TheSchreier–SimsAlgorithm57 4.3ThePowerofRandomization62 4.4ShallowSchreierTrees64 4.5StrongGeneratorsinNearlyLinearTime70 4.5.1Implementation75 5FurtherLow-LevelAlgorithms79 5.1ConsequencesoftheSchreier–SimsMethod79 5.1.1PointwiseStabilizers79 5.1.2Homomorphisms80 5.1.3 Transitive Constituent and Block Homomorphisms81 5.1.4ClosuresandNormalClosures83 5.2WorkingwithBaseImages84 5.3PermutationGroupsasBlack-BoxGroups93 5.4BaseChange97 5.5BlocksofImprimitivity100 5.5.1BlocksinNearlyLinearTime101 5.5.2TheSmallestBlockContainingaGivenSubset107 5.5.3StructureForests111 6ALibraryofNearlyLinear-TimeAlgorithms114 6.1ASpecialCaseofGroupIntersectionandApplications115 6.1.1IntersectionwithaNorm a lClosure115 6.1.2CentralizerintheSymmetricGroup117 6.1.3TheCenter120 6.1.4CentralizerofaNormalSubgroup120 6.1.5CoreofaSubnorm a lSubgroup124 6.2CompositionSeries125 6.2.1ReductiontothePrimitiveCase126 6.2.2TheO’Nan–ScottTheorem129 6.2.3NormalSubgroupswithNontrivialCentralizer133 6.2.4 Groups with a Unique Nonabelian Minimal NormalSubgroup139 6.2.5Implementation146 6.2.6AnElementaryVersion149 6.2.7ChiefSeries155 6.3QuotientswithSmallPermutationDegree156 6.3.1SolvableRadicalandp-Core157 Contents vii 7SolvablePermutationGroups162 7.1StrongGeneratorsinSolvableGroups162 7.2Power-ConjugatePresentations165 7.3WorkingwithElementaryAbelianLayers166 7.3.1SylowSubgroups167 7.3.2ConjugacyClassesinSolvableGroups172 7.4TwoAlgorithmsforNilpotentGroups175 7.4.1AFastNilpotencyTest176 7.4.2TheUpperCentralSeriesinNilpotentGroups179 8StrongGeneratingTests183 8.1TheSchreier–Todd–Coxeter–SimsProcedure184 8.1.1CosetEnumeration184 8.1.2Leon’sAlgorithm186 8.2Sims’sVerifyRoutine188 8.3TowardStrongGeneratorsbyaLasVegasAlgorithm191 8.4AShortPresentation197 9BacktrackMethods201 9.1TraditionalBacktrack202 9.1.1 Pruning the Search Tree: Problem-Independent Methods203 9.1.2 Pruning the Search Tree: Problemependent Methods205 9.2ThePartitionMethod207 9.3Normalizers211 9.4ConjugacyClasses214 10Large-BaseGroups218 10.1LabeledBranchings218 10.1.1Construction222 10.2AlternatingandSymmetricGroups225 10.2.1NumberTheoreticandProbabilisticEstimates228 10.2.2 Constructive Recognition: Finding the NewGenerators235 10.2.3ConstructiveRecognition:TheHomomorphismλ239 10.2.4ConstructiveRecognition:TheCaseofGiants244 10.3ARandomizedStrongGeneratorConstruction246 Bibliography254 Index262 1 Introduction Computational group theory (CGT) is a subfield of symbolic algebra; it deals with the design, analysis, and implementation of algorithms for manipulating groups. It is an interdisciplinary area between mathematics and computer sci- ence. The major areas of CGT are the algorithms for finitely presented groups, polycyclic and finite solvable groups, permutation groups, matrix groups, and representation theory. The topic of this book is the third of these areas. Permutation groups are the oldest type of representations of groups; in fact, the work of Galois on permutation groups, which is generally considered as the start of group theory as a separate branch of mathematics, preceded the abstract definition of groups by about a half a century. Algorithmic questions permeated permutation group theory from its inception. Galois group computations, and the related problem of determining all transitive permutation groups of a given degree, are still active areas of research (see [Hulpke, 1996]). Mathieu’s constructions of his simple groups also involved serious computations. Nowadays, permutation group algorithms are among the best developed parts of CGT, and we can handle groups of degree in the hundreds of thousands. The basic ideas for handling permutation groups appeared in [Sims, 1970, 1971a]; even today, Sims’s methods are at the heart of most of the algorithms. At first glance, the efficiency of permutation group algorithms may be surpris- ing. The input consists of a list of generators. On one hand, this representation is very efficient, since afew permutationsin Sn can describe an object of size up to n!. On the other hand, the succinctness of such a representation G =S ne- cessitates nontrivial algorithms to answer even such basic questions as finding the order of G or testing membership of agiven permutationin G. Initially, it is not even clear how to prove in polynomial time in the input length that a certain permutation g is in G, because writing g as a product of the given generators S for G may require an exponentially long word. Sims’s seminal 1 2 Introduction idea was to introduce the notions of base and strong generating set. This data structure enables us to decide membership in G constructively, by writing any given element of G as a short product of the strong generators. The technique for constructing a strong generating set can also be applied to other tasks such as computing normal closures of subgroups and handling homomorphisms of groups. Therefore, a significant part of this book is devoted to the description of variants and applications of Sims’s method. A second generation of algorithms uses divide-and-conquer techniques by utilizing the orbit structure and imprimitivity block structure of the input group, thereby reducing the problems to primitive groups. Although every abstract group has a faithful transitive permutation representation, the structure of prim- itive groups is quite restricted. This extra information, partly obtained as a consequence of the classification of finite simple groups, can be exploited in the design of algorithms. We shall also describe some of the latest algorithms, which use an even finer divide-and-conquer technique. A tower of normal subgroups is constructed such that the factor groups between two consecutive normal subgroups are the products of isomorphic simple groups. Abelian factors are handled by linear al- gebra, whereas the simple groups occurring in nonabelian factors are identified with standard copies of these groups, and the problems are solved in the stan- dard copies. This identification process works in the more general black-box group setting, when we do not use the fact that the input group is represented by permutations: The algorithms only exploit the facts that we can multiply and invert group elements and decide whether two group elements are equal. This generality enables us to use the same algorithms for matrix group inputs. Computations with matrix groups is currently the most active area of CGT. Dealing with permutation groups is the area of CGT where the complexity analysis of algorithms is the
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