ANUPQ   ANU p-Quotient  3.2 29 November 2017 Greg Gamble Werner Nickel Eamonn O'Brien Max Horn Greg Gamble Email: mailto:Greg.Gamble@uwa.edu.au Homepage: http://school.maths.uwa.edu.au/~gregg Address: Greg Gamble Department of Mathematics and Statistics Curtin University GPO Box U 1987 Perth WA 6845 Australia Werner Nickel Homepage: http://www.mathematik.tu-darmstadt.de/~nickel/ Eamonn O'Brien Email: mailto:obrien@math.auckland.ac.nz Homepage: http://www.math.auckland.ac.nz/~obrien Address: Department of Mathematics University of Auckland Private Bag 92019 Auckland New Zealand Max Horn Email: mailto:max.horn@math.uni-giessen.de Homepage: http://www.quendi.de/math Address: AG Algebra Mathematisches Institut Justus-Liebig-Universität Gießen Arndtstraße 2 35392 Gießen Germany ------------------------------------------------------- Contents (ANUPQ) 1 Introduction 1.1 Overview 1.2 How to read this manual 1.3 Authors and Acknowledgements 2 Mathematical Background and Terminology 2.1 Basic notions 2.1-1 pc Presentations and Consistency 2.1-2 Exponent-p Central Series and Weighted pc Presentations 2.1-3 p-Cover, p-Multiplicator 2.1-4 Descendants, Capable, Terminal, Nucleus 2.1-5 Laws 2.2 The p-quotient Algorithm 2.2-1 Finding the p-cover 2.2-2 Imposing the Relations of the fp Group 2.2-3 Imposing Laws 2.3 The p-group generation Algorithm, Standard Presentation, Isomorphism Testing 3 Infrastructure 3.1 Loading the ANUPQ Package 3.2 The ANUPQData Record 3.2-1 ANUPQData 3.2-2 ANUPQDirectoryTemporary 3.3 Setting the Verbosity of ANUPQ via Info and InfoANUPQ 3.3-1 InfoANUPQ 3.4 Utility Functions 3.4-1 PqLeftNormComm 3.4-2 PqGAPRelators 3.4-3 PqParseWord 3.4-4 PqExample 3.4-5 AllPqExamples 3.4-6 GrepPqExamples 3.4-7 ToPQLog 3.5 Attributes and a Property for fp and pc p-groups 3.5-1 NuclearRank 3.6 Hints and Warnings regarding the use of Options 3.6-1 ANUPQWarnOfOtherOptions 4 Non-interactive ANUPQ functions 4.1 Computing p-Quotients 4.1-1 Pq 4.1-2 PqEpimorphism 4.1-3 PqPCover 4.2 Computing Standard Presentations 4.2-1 PqStandardPresentation 4.2-2 EpimorphismPqStandardPresentation 4.3 Testing p-Groups for Isomorphism 4.3-1 IsPqIsomorphicPGroup 4.4 Computing Descendants of a p-Group 4.4-1 PqDescendants 4.4-2 PqSupplementInnerAutomorphisms 4.4-3 PqList 4.4-4 SavePqList 5 Interactive ANUPQ functions 5.1 Starting and Stopping Interactive ANUPQ Processes 5.1-1 PqStart 5.1-2 PqQuit 5.1-3 PqQuitAll 5.2 Interactive ANUPQ Process Utility Functions 5.2-1 PqProcessIndex 5.2-2 PqProcessIndices 5.2-3 IsPqProcessAlive 5.3 Interactive Versions of Non-interactive ANUPQ Functions 5.3-1 Pq 5.3-2 PqEpimorphism 5.3-3 PqPCover 5.3-4 PqStandardPresentation 5.3-5 EpimorphismPqStandardPresentation 5.3-6 PqDescendants 5.3-7 PqSetPQuotientToGroup 5.4 Low-level Interactive ANUPQ functions based on menu items of the pq program 5.5 General commands 5.5-1 PqNrPcGenerators 5.5-2 PqFactoredOrder 5.5-3 PqOrder 5.5-4 PqPClass 5.5-5 PqWeight 5.5-6 PqCurrentGroup 5.5-7 PqDisplayPcPresentation 5.5-8 PqSetOutputLevel 5.5-9 PqEvaluateIdentities 5.6 Commands from the Main p-Quotient menu 5.6-1 PqPcPresentation 5.6-2 PqSavePcPresentation 5.6-3 PqRestorePcPresentation 5.6-4 PqNextClass 5.6-5 PqComputePCover 5.7 Commands from the Advanced p-Quotient menu 5.7-1 PqCollect 5.7-2 PqSolveEquation 5.7-3 PqCommutator 5.7-4 PqSetupTablesForNextClass 5.7-5 PqTails 5.7-6 PqComputeTails 5.7-7 PqAddTails 5.7-8 PqDoConsistencyChecks 5.7-9 PqCollectDefiningRelations 5.7-10 PqCollectWordInDefiningGenerators 5.7-11 PqCommutatorDefiningGenerators 5.7-12 PqDoExponentChecks 5.7-13 PqEliminateRedundantGenerators 5.7-14 PqRevertToPreviousClass 5.7-15 PqSetMaximalOccurrences 5.7-16 PqSetMetabelian 5.7-17 PqDoConsistencyCheck 5.7-18 PqCompact 5.7-19 PqEchelonise 5.7-20 PqSupplyAutomorphisms 5.7-21 PqExtendAutomorphisms 5.7-22 PqApplyAutomorphisms 5.7-23 PqDisplayStructure 5.7-24 PqDisplayAutomorphisms 5.7-25 PqWritePcPresentation 5.8 Commands from the Standard Presentation menu 5.8-1 PqSPComputePcpAndPCover 5.8-2 PqSPStandardPresentation 5.8-3 PqSPSavePresentation 5.8-4 PqSPCompareTwoFilePresentations 5.8-5 PqSPIsomorphism 5.9 Commands from the Main p-Group Generation menu 5.9-1 PqPGSupplyAutomorphisms 5.9-2 PqPGExtendAutomorphisms 5.9-3 PqPGConstructDescendants 5.9-4 PqPGSetDescendantToPcp 5.10 Commands from the Advanced p-Group Generation menu 5.10-1 PqAPGDegree 5.10-2 PqAPGPermutations 5.10-3 PqAPGOrbits 5.10-4 PqAPGOrbitRepresentatives 5.10-5 PqAPGSingleStage 5.11 Primitive Interactive ANUPQ Process Read/Write Functions 5.11-1 PqRead 5.11-2 PqReadAll 5.11-3 PqReadUntil 5.11-4 PqWrite 6 ANUPQ Options 6.1 Overview 6.1-1 AllANUPQoptions 6.1-2 ANUPQoptions 6.2 Detailed descriptions of ANUPQ Options 7 Installing the ANUPQ Package 7.1 Testing your ANUPQ installation 7.2 Running the pq program as a standalone A Examples A.1 The Relators Option A.2 The Identities Option and PqEvaluateIdentities Function A.3 A Large Example A.4 Developing descendants trees A.4-1 PqDescendantsTreeCoclassOne