Permutation Group Algorithms

274 Pages·2003·1.229 MB·English

by  Ákos Seress

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Permutation group algorithms comprise one of the workhorses of symbolic
algebrasystemscomputingwithgroupsandplayanindispensableroleinthe
proof of many deep results, including the construction and study of sporadic
finitesimplegroups.Thisbookdescribesthetheorybehindpermutationgroup
algorithms,uptothemostrecentdevelopmentsbasedontheclassificationof
finitesimplegroups.Rigorouscomplexityestimates,implementationhints,and
advancedexercisesareincludedthroughout.
Thecentralthemeisthedescriptionofnearlylinear-timealgorithms,which
are extremely fast in terms of both asymptotic analysis and practical running
time.Asignificantpartofthepermutationgrouplibraryofthecomputational
groupalgebrasystemGAPisbasedonnearlylinear-timealgorithms.
Thebookfillsasignificantgapinthesymboliccomputationliterature.Itis
recommendedforeveryoneinterestedinusingcomputersingrouptheoryand
issuitableforadvancedgraduatecourses.
A´kosSeressisaProfessorofMathematicsatTheOhioStateUniversity.
CAMBRIDGETRACTSINMATHEMATICS
GeneralEditors
B.BOLLOBA´S,W.FULTON,A.KATOK,F.KIRWAN,P.SARNAK
152 Permutation Group Algorithms
A´kosSeress
TheOhioStateUniversity
Permutation Group
Algorithms
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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Published in the United States by Cambridge University Press, New York
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Information on this title: www.cambridge.org/9780521661034
© Ákos Seress 2002
This book is in copyright. 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 Press.
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 appropriate.
Contents
1 Introduction page1
1.1 AListofAlgorithms 4
1.2 NotationandTerminology 6
1.2.1 Groups 7
1.2.2 PermutationGroups 9
1.2.3 AlgorithmicConcepts 10
1.2.4 Graphs 11
1.3 ClassificationofRandomizedAlgorithms 12
2 Black-BoxGroups 16
2.1 ClosureAlgorithms 18
2.1.1 OrbitComputations 18
2.1.2 ClosureofAlgebraicStructures 23
2.2 RandomElementsofBlack-BoxGroups 24
2.3 RandomSubproducts 30
2.3.1 DefinitionandBasicProperties 30
2.3.2 ReducingtheNumberofGenerators 33
2.3.3 ClosureAlgorithmswithoutMembershipTesting 37
2.3.4 DerivedandLowerCentralSeries 38
2.4 RandomPrefixes 40
2.4.1 DefinitionandBasicProperties 40
2.4.2 Applications 44
3 PermutationGroups:AComplexityOverview 48
3.1 Polynomial-TimeAlgorithms 48
3.2 NearlyLinear-TimeAlgorithms 51
3.3 Non-Polynomial-TimeMethods 52
v
vi Contents
4 BasesandStrongGeneratingSets 55
4.1 BasicDefinitions 55
4.2 TheSchreier–SimsAlgorithm 57
4.3 ThePowerofRandomization 62
4.4 ShallowSchreierTrees 64
4.5 StrongGeneratorsinNearlyLinearTime 70
4.5.1 Implementation 75
5 FurtherLow-LevelAlgorithms 79
5.1 ConsequencesoftheSchreier–SimsMethod 79
5.1.1 PointwiseStabilizers 79
5.1.2 Homomorphisms 80
5.1.3 TransitiveConstituentandBlock
Homomorphisms 81
5.1.4 ClosuresandNormalClosures 83
5.2 WorkingwithBaseImages 84
5.3 PermutationGroupsasBlack-BoxGroups 93
5.4 BaseChange 97
5.5 BlocksofImprimitivity 100
5.5.1 BlocksinNearlyLinearTime 101
5.5.2 TheSmallestBlockContainingaGivenSubset 107
5.5.3 StructureForests 111
6 ALibraryofNearlyLinear-TimeAlgorithms 114
6.1 ASpecialCaseofGroupIntersectionandApplications 115
6.1.1 IntersectionwithaNormalClosure 115
6.1.2 CentralizerintheSymmetricGroup 117
6.1.3 TheCenter 120
6.1.4 CentralizerofaNormalSubgroup 120
6.1.5 CoreofaSubnormalSubgroup 124
6.2 CompositionSeries 125
6.2.1 ReductiontothePrimitiveCase 126
6.2.2 TheO’Nan–ScottTheorem 129
6.2.3 NormalSubgroupswithNontrivialCentralizer 133
6.2.4 GroupswithaUniqueNonabelianMinimal
NormalSubgroup 139
6.2.5 Implementation 146
6.2.6 AnElementaryVersion 149
6.2.7 ChiefSeries 155
6.3 QuotientswithSmallPermutationDegree 156
6.3.1 SolvableRadicaland p-Core 157
Contents vii
7 SolvablePermutationGroups 162
7.1 StrongGeneratorsinSolvableGroups 162
7.2 Power-ConjugatePresentations 165
7.3 WorkingwithElementaryAbelianLayers 166
7.3.1 SylowSubgroups 167
7.3.2 ConjugacyClassesinSolvableGroups 172
7.4 TwoAlgorithmsforNilpotentGroups 175
7.4.1 AFastNilpotencyTest 176
7.4.2 TheUpperCentralSeriesinNilpotentGroups 179
8 StrongGeneratingTests 183
8.1 TheSchreier–Todd–Coxeter–SimsProcedure 184
8.1.1 CosetEnumeration 184
8.1.2 Leon’sAlgorithm 186
8.2 Sims’sVerifyRoutine 188
8.3 TowardStrongGeneratorsbyaLasVegasAlgorithm 191
8.4 AShortPresentation 197
9 BacktrackMethods 201
9.1 TraditionalBacktrack 202
9.1.1 PruningtheSearchTree:Problem-Independent
Methods 203
9.1.2 PruningtheSearchTree:Problem-Dependent
Methods 205
9.2 ThePartitionMethod 207
9.3 Normalizers 211
9.4 ConjugacyClasses 214
10 Large-BaseGroups 218
10.1 LabeledBranchings 218
10.1.1 Construction 222
10.2 AlternatingandSymmetricGroups 225
10.2.1 NumberTheoreticandProbabilisticEstimates 228
10.2.2 ConstructiveRecognition:Findingthe
NewGenerators 235
10.2.3 ConstructiveRecognition:TheHomomorphismλ 239
10.2.4 ConstructiveRecognition:TheCaseofGiants 244
10.3 ARandomizedStrongGeneratorConstruction 246
Bibliography 254
Index 262