loops/doc/chap0.txt

  
  
                               The LOOPS Package
  
  
                  Computing with quasigroups and loops in GAP
  
  
                                 Version 3.4.0
  
  
                                 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
  © 2017 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 QuasigroupIsomorph
      6.11-7 LoopIsomorph
      6.11-8 IsomorphicCopyByPerm
      6.11-9 IsomorphicCopyByNormalSubloop
      6.11-10 Discriminator
      6.11-11 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 Left Bruck Loops and Right Bruck Loops
      9.3-1 LeftBruckLoop
      9.3-2 RightBruckLoop
    9.4 Moufang Loops
      9.4-1 MoufangLoop
    9.5 Code Loops
      9.5-1 CodeLoop
    9.6 Steiner Loops
      9.6-1 SteinerLoop
    9.7 Conjugacy Closed Loops
      9.7-1 RCCLoop and RightConjugacyClosedLoop
      9.7-2 LCCLoop and LeftConjugacyClosedLoop
      9.7-3 CCLoop and ConjugacyClosedLoop
    9.8 Small Loops
      9.8-1 SmallLoop
    9.9 Paige Loops
      9.9-1 PaigeLoop
    9.10 Nilpotent Loops
      9.10-1 NilpotentLoop
    9.11 Automorphic Loops
      9.11-1 AutomorphicLoop
    9.12 Interesting Loops
      9.12-1 InterestingLoop
    9.13 Libraries of Loops Up To Isotopism
      9.13-1 ItpSmallLoop
  A Files
  B Filters