The LOOPS Package Computing with quasigroups and loops in GAP Version 3.3.0 Gábor P. Nagy Petr Vojtěchovský Gábor P. Nagy Email: mailto:nagyg@math.u-szeged.hu Address: Department of Mathematics, University of Szeged Petr Vojtěchovský Email: mailto:petr@math.du.edu Address: Department of Mathematics, University of Denver ------------------------------------------------------- Copyright © 2016 Gábor P. Nagy and Petr Vojtěchovský. ------------------------------------------------------- Contents (Loops) 1 Introduction 1.1 License 1.2 Installation 1.3 Documentation 1.4 Test Files 1.5 Memory Management 1.6 Feedback 1.7 Acknowledgment 2 Mathematical Background 2.1 Quasigroups and Loops 2.2 Translations 2.3 Subquasigroups and Subloops 2.4 Nilpotence and Solvability 2.5 Associators and Commutators 2.6 Homomorphism and Homotopisms 3 How the Package Works 3.1 Representing Quasigroups 3.2 Conversions between magmas, quasigroups, loops and groups 3.3 Calculating with Quasigroups 3.4 Naming, Viewing and Printing Quasigroups and their Elements 3.4-1 SetQuasigroupElmName and SetLoopElmName 4 Creating Quasigroups and Loops 4.1 About Cayley Tables 4.2 Testing Cayley Tables 4.2-1 IsQuasigroupTable and IsQuasigroupCayleyTable 4.2-2 IsLoopTable and IsLoopCayleyTable 4.3 Canonical and Normalized Cayley Tables 4.3-1 CanonicalCayleyTable 4.3-2 CanonicalCopy 4.3-3 NormalizedQuasigroupTable 4.4 Creating Quasigroups and Loops From Cayley Tables 4.4-1 QuasigroupByCayleyTable and LoopByCayleyTable 4.5 Creating Quasigroups and Loops from a File 4.5-1 QuasigroupFromFile and LoopFromFile 4.6 Creating Quasigroups and Loops From Sections 4.6-1 CayleyTableByPerms 4.6-2 QuasigroupByLeftSection and LoopByLeftSection 4.6-3 QuasigroupByRightSection and LoopByRightSection 4.7 Creating Quasigroups and Loops From Folders 4.7-1 QuasigroupByRightFolder and LoopByRightFolder 4.8 Creating Quasigroups and Loops By Nuclear Extensions 4.8-1 NuclearExtension 4.8-2 LoopByExtension 4.9 Random Quasigroups and Loops 4.9-1 RandomQuasigroup and RandomLoop 4.9-2 RandomNilpotentLoop 4.10 Conversions 4.10-1 IntoQuasigroup 4.10-2 PrincipalLoopIsotope 4.10-3 IntoLoop 4.10-4 IntoGroup 4.11 Products of Quasigroups and Loops 4.11-1 DirectProduct 4.12 Opposite Quasigroups and Loops 4.12-1 Opposite, OppositeQuasigroup and OppositeLoop 5 Basic Methods And Attributes 5.1 Basic Attributes 5.1-1 Elements 5.1-2 CayleyTable 5.1-3 One 5.1-4 Size 5.1-5 Exponent 5.2 Basic Arithmetic Operations 5.2-1 LeftDivision and RightDivision 5.2-2 LeftDivisionCayleyTable and RightDivisionCayleyTable 5.3 Powers and Inverses 5.3-1 LeftInverse, RightInverse and Inverse 5.4 Associators and Commutators 5.4-1 Associator 5.4-2 Commutator 5.5 Generators 5.5-1 GeneratorsOfQuasigroup and GeneratorsOfLoop 5.5-2 GeneratorsSmallest 5.5-3 SmallGeneratingSet 6 Methods Based on Permutation Groups 6.1 Parent of a Quasigroup 6.1-1 Parent 6.1-2 Position 6.1-3 PosInParent 6.2 Subquasigroups and Subloops 6.2-1 Subquasigroup 6.2-2 Subloop 6.2-3 IsSubquasigroup and IsSubloop 6.2-4 AllSubquasigroups 6.2-5 AllSubloops 6.2-6 RightCosets 6.2-7 RightTransversal 6.3 Translations and Sections 6.3-1 LeftTranslation and RightTranslation 6.3-2 LeftSection and RightSection 6.4 Multiplication Groups 6.4-1 LeftMutliplicationGroup, RightMultiplicationGroup and MultiplicationGroup 6.4-2 RelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup and RelativeMultiplicationGroup 6.5 Inner Mapping Groups 6.5-1 LeftInnerMapping, RightInnerMapping, MiddleInnerMapping 6.5-2 LeftInnerMappingGroup, RightInnerMappingGroup, MiddleInnerMappingGroup 6.5-3 InnerMappingGroup 6.6 Nuclei, Commutant, Center, and Associator Subloop 6.6-1 LeftNucles, MiddleNucleus, and RightNucleus 6.6-2 Nuc, NucleusOfQuasigroup and NucleusOfLoop 6.6-3 Commutant 6.6-4 Center 6.6-5 AssociatorSubloop 6.7 Normal Subloops and Simple Loops 6.7-1 IsNormal 6.7-2 NormalClosure 6.7-3 IsSimple 6.8 Factor Loops 6.8-1 FactorLoop 6.8-2 NaturalHomomorphismByNormalSubloop 6.9 Nilpotency and Central Series 6.9-1 IsNilpotent 6.9-2 NilpotencyClassOfLoop 6.9-3 IsStronglyNilpotent 6.9-4 UpperCentralSeries 6.9-5 LowerCentralSeries 6.10 Solvability, Derived Series and Frattini Subloop 6.10-1 IsSolvable 6.10-2 DerivedSubloop 6.10-3 DerivedLength 6.10-4 FrattiniSubloop and FrattinifactorSize 6.10-5 FrattinifactorSize 6.11 Isomorphisms and Automorphisms 6.11-1 IsomorphismQuasigroups 6.11-2 IsomorphismLoops 6.11-3 QuasigroupsUpToIsomorphism 6.11-4 LoopsUpToIsomorphism 6.11-5 AutomorphismGroup 6.11-6 IsomorphicCopyByPerm 6.11-7 IsomorphicCopyByNormalSubloop 6.11-8 Discriminator 6.11-9 AreEqualDiscriminators 6.12 Isotopisms 6.12-1 IsotopismLoops 6.12-2 LoopsUpToIsotopism 7 Testing Properties of Quasigroups and Loops 7.1 Associativity, Commutativity and Generalizations 7.1-1 IsAssociative 7.1-2 IsCommutative 7.1-3 IsPowerAssociative 7.1-4 IsDiassociative 7.2 Inverse Propeties 7.2-1 HasLeftInverseProperty, HasRightInverseProperty and HasInverseProperty 7.2-2 HasTwosidedInverses 7.2-3 HasWeakInverseProperty 7.2-4 HasAutomorphicInverseProperty 7.2-5 HasAntiautomorphicInverseProperty 7.3 Some Properties of Quasigroups 7.3-1 IsSemisymmetric 7.3-2 IsTotallySymmetric 7.3-3 IsIdempotent 7.3-4 IsSteinerQuasigroup 7.3-5 IsUnipotent 7.3-6 IsLeftDistributive, IsRightDistributive, IsDistributive 7.3-7 IsEntropic and IsMedial 7.4 Loops of Bol Moufang Type 7.4-1 IsExtraLoop 7.4-2 IsMoufangLoop 7.4-3 IsCLoop 7.4-4 IsLeftBolLoop 7.4-5 IsRightBolLoop 7.4-6 IsLCLoop 7.4-7 IsRCLoop 7.4-8 IsLeftNuclearSquareLoop 7.4-9 IsMiddleNuclearSquareLoop 7.4-10 IsRightNuclearSquareLoop 7.4-11 IsNuclearSquareLoop 7.4-12 IsFlexible 7.4-13 IsLeftAlternative 7.4-14 IsRightAlternative 7.4-15 IsAlternative 7.5 Power Alternative Loops 7.5-1 IsLeftPowerAlternative, IsRightPowerAlternative and IsPowerAlternative 7.6 Conjugacy Closed Loops and Related Properties 7.6-1 IsLCCLoop 7.6-2 IsRCCLoop 7.6-3 IsCCLoop 7.6-4 IsOsbornLoop 7.7 Automorphic Loops 7.7-1 IsLeftAutomorphicLoop 7.7-2 IsMiddleAutomorphicLoop 7.7-3 IsRightAutomorphicLoop 7.7-4 IsAutomorphicLoop 7.8 Additonal Varieties of Loops 7.8-1 IsCodeLoop 7.8-2 IsSteinerLoop 7.8-3 IsLeftBruckLoop and IsLeftKLoop 7.8-4 IsRightBruckLoop and IsRightKLoop 8 Specific Methods 8.1 Core Methods for Bol Loops 8.1-1 AssociatedLeftBruckLoop and AssociatedRightBruckLoop 8.1-2 IsExactGroupFactorization 8.1-3 RightBolLoopByExactGroupFactorization 8.2 Moufang Modifications 8.2-1 LoopByCyclicModification 8.2-2 LoopByDihedralModification 8.2-3 LoopMG2 8.3 Triality for Moufang Loops 8.3-1 TrialityPermGroup 8.3-2 TrialityPcGroup 8.4 Realizing Groups as Multiplication Groups of Loops 8.4-1 AllLoopTablesInGroup 8.4-2 AllProperLoopTablesInGroup 8.4-3 OneLoopTableInGroup 8.4-4 OneProperLoopTableInGroup 8.4-5 AllLoopsWithMltGroup 8.4-6 OneLoopWithMltGroup 9 Libraries of Loops 9.1 A Typical Library 9.1-1 LibraryLoop 9.1-2 MyLibraryLoop 9.1-3 DisplayLibraryInfo 9.2 Left Bol Loops and Right Bol Loops 9.2-1 LeftBolLoop 9.2-2 RightBolLoop 9.3 Moufang Loops 9.3-1 MoufangLoop 9.4 Code Loops 9.4-1 CodeLoop 9.5 Steiner Loops 9.5-1 SteinerLoop 9.6 Conjugacy Closed Loops 9.6-1 RCCLoop and RightConjugacyClosedLoop 9.6-2 LCCLoop and LeftConjugacyClosedLoop 9.6-3 CCLoop and ConjugacyClosedLoop 9.7 Small Loops 9.7-1 SmallLoop 9.8 Paige Loops 9.8-1 PaigeLoop 9.9 Nilpotent Loops 9.9-1 NilpotentLoop 9.10 Automorphic Loops 9.10-1 AutomorphicLoop 9.11 Interesting Loops 9.11-1 InterestingLoop 9.12 Libraries of Loops Up To Isotopism 9.12-1 ItpSmallLoop A Files B Filters