loops/doc/chap0.txt
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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
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Copyright
© 2016 Gábor P. Nagy and Petr Vojtěchovský.
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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