  
  
                              [1X The LOOPS Package [101X
  
  
                 [1X Computing with quasigroups and loops in [5XGAP[105X [101X
  
  
                                     3.4.3
  
  
                                14 November 2022
  
  
                                Gábor Péter Nagy
  
                               Petr Vojtěchovský
  
  
  
  Gábor Péter Nagy
      Email:    [7Xmailto:nagyg@math.u-szeged.hu[107X
      Homepage: [7Xhttp://www.math.u-szeged.hu/~nagyg/[107X
      Address:  [33X[0;14YBolyai Institute, University of Szeged[133X
                [33X[0;14Y6725 Szeged, Aradi vertanuk tere 1[133X
                [33X[0;14YHungary[133X
  
  
  Petr Vojtěchovský
      Email:    [7Xmailto:petr@math.du.edu[107X
      Homepage: [7Xhttp://www.math.du.edu/~petr/[107X
      Address:  [33X[0;14YDepartment of Mathematics, University of Denver[133X
                [33X[0;14Y2280 S. Vine Street[133X
                [33X[0;14YDenver, CO 80208[133X
                [33X[0;14YUSA[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  LOOPS  package  provides  researchers  in nonassociative algebra with a
  computational  tool  that  integrates  standard  notions of loop theory with
  libraries of loops and group-theoretical algorithms of GAP. The package also
  expands GAP toward nonassociative structures.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2005-2017 Gábor P. Nagy and Petr Vojtěchovský.[133X
  
  [33X[0;0YThe [5XLOOPS[105X package is free software; you can redistribute it and/or modify it
  under     the     terms     of    the    GNU    General    Public    License
  ([7Xhttps://www.fsf.org/licenses/gpl.html[107X)  as  published  by the Free Software
  Foundation;  either  version 2 of the License, or (at your option) any later
  version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  thank  the following people for sending us remarks and comments, and for
  suggesting  new  functionality of the package: Muniru Asiru, Bjoern Assmann,
  Andreas  Distler,  Aleš  Drápal,  Graham  Ellis,  Steve  Flammia, Kenneth W.
  Johnson,  Michael K. Kinyon, Olexandr Konovalov, Frank Lübeck, Jonathan D.H.
  Smith, David Stanovský and Glen Whitney.[133X
  
  [33X[0;0YThe  library  of Moufang loops of order 243 was generated from data provided
  by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy
  closed  loops  of  order  less  than  28 was generated from data provided by
  Katharina  Artic.  The  library  of  right  Bruck  loops of order 27, 81 was
  obtained jointly with Izabella Stuhl.[133X
  
  [33X[0;0YGábor  P.  Nagy  was  supported by OTKA grants F042959 and T043758, and Petr
  Vojtěchovský  was  supported  by the 2006 and 2016 University of Denver PROF
  grants and the Simons Foundation Collaboration Grant 210176.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Loops)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YInstallation[133X
    1.2 [33X[0;0YDocumentation[133X
    1.3 [33X[0;0YTest Files[133X
    1.4 [33X[0;0YMemory Management[133X
    1.5 [33X[0;0YFeedback[133X
  2 [33X[0;0YMathematical Background[133X
    2.1 [33X[0;0YQuasigroups and Loops[133X
    2.2 [33X[0;0YTranslations[133X
    2.3 [33X[0;0YSubquasigroups and Subloops[133X
    2.4 [33X[0;0YNilpotence and Solvability[133X
    2.5 [33X[0;0YAssociators and Commutators[133X
    2.6 [33X[0;0YHomomorphism and Homotopisms[133X
  3 [33X[0;0YHow the Package Works[133X
    3.1 [33X[0;0YRepresenting Quasigroups[133X
    3.2 [33X[0;0YConversions between magmas, quasigroups, loops and groups[133X
    3.3 [33X[0;0YCalculating with Quasigroups[133X
    3.4 [33X[0;0YNaming, Viewing and Printing Quasigroups and their Elements[133X
      3.4-1 [33X[0;0YSetQuasigroupElmName and SetLoopElmName[133X
  4 [33X[0;0YCreating Quasigroups and Loops[133X
    4.1 [33X[0;0YAbout Cayley Tables[133X
    4.2 [33X[0;0YTesting Cayley Tables[133X
      4.2-1 [33X[0;0YIsQuasigroupTable and IsQuasigroupCayleyTable[133X
      4.2-2 [33X[0;0YIsLoopTable and IsLoopCayleyTable[133X
    4.3 [33X[0;0YCanonical and Normalized Cayley Tables[133X
      4.3-1 CanonicalCayleyTable
      4.3-2 CanonicalCopy
      4.3-3 NormalizedQuasigroupTable
    4.4 [33X[0;0YCreating Quasigroups and Loops From Cayley Tables[133X
      4.4-1 [33X[0;0YQuasigroupByCayleyTable and LoopByCayleyTable[133X
    4.5 [33X[0;0YCreating Quasigroups and Loops from a File[133X
      4.5-1 [33X[0;0YQuasigroupFromFile and LoopFromFile[133X
    4.6 [33X[0;0YCreating Quasigroups and Loops From Sections[133X
      4.6-1 CayleyTableByPerms
      4.6-2 [33X[0;0YQuasigroupByLeftSection and LoopByLeftSection[133X
      4.6-3 [33X[0;0YQuasigroupByRightSection and LoopByRightSection[133X
    4.7 [33X[0;0YCreating Quasigroups and Loops From Folders[133X
      4.7-1 [33X[0;0YQuasigroupByRightFolder and LoopByRightFolder[133X
    4.8 [33X[0;0YCreating Quasigroups and Loops By Nuclear Extensions[133X
      4.8-1 NuclearExtension
      4.8-2 LoopByExtension
    4.9 [33X[0;0YRandom Quasigroups and Loops[133X
      4.9-1 [33X[0;0YRandomQuasigroup and RandomLoop[133X
      4.9-2 RandomNilpotentLoop
    4.10 [33X[0;0YConversions[133X
      4.10-1 IntoQuasigroup
      4.10-2 PrincipalLoopIsotope
      4.10-3 IntoLoop
      4.10-4 IntoGroup
    4.11 [33X[0;0YProducts of Quasigroups and Loops[133X
      4.11-1 DirectProduct
    4.12 [33X[0;0YOpposite Quasigroups and Loops[133X
      4.12-1 [33X[0;0YOpposite, OppositeQuasigroup and OppositeLoop[133X
  5 [33X[0;0YBasic Methods And Attributes[133X
    5.1 [33X[0;0YBasic Attributes[133X
      5.1-1 Elements
      5.1-2 CayleyTable
      5.1-3 One
      5.1-4 Size
      5.1-5 Exponent
    5.2 [33X[0;0YBasic Arithmetic Operations[133X
      5.2-1 [33X[0;0YLeftDivision and RightDivision[133X
      5.2-2 [33X[0;0YLeftDivisionCayleyTable and RightDivisionCayleyTable[133X
    5.3 [33X[0;0YPowers and Inverses[133X
      5.3-1 [33X[0;0YLeftInverse, RightInverse and Inverse[133X
    5.4 [33X[0;0YAssociators and Commutators[133X
      5.4-1 Associator
      5.4-2 Commutator
    5.5 [33X[0;0YGenerators[133X
      5.5-1 [33X[0;0YGeneratorsOfQuasigroup and GeneratorsOfLoop[133X
      5.5-2 GeneratorsSmallest
      5.5-3 SmallGeneratingSet
  6 [33X[0;0YMethods Based on Permutation Groups[133X
    6.1 [33X[0;0YParent of a Quasigroup[133X
      6.1-1 Parent
      6.1-2 Position
      6.1-3 PosInParent
    6.2 [33X[0;0YSubquasigroups and Subloops[133X
      6.2-1 Subquasigroup
      6.2-2 Subloop
      6.2-3 [33X[0;0YIsSubquasigroup and IsSubloop[133X
      6.2-4 AllSubquasigroups
      6.2-5 AllSubloops
      6.2-6 RightCosets
      6.2-7 RightTransversal
    6.3 [33X[0;0YTranslations and Sections[133X
      6.3-1 [33X[0;0YLeftTranslation and RightTranslation[133X
      6.3-2 [33X[0;0YLeftSection and RightSection[133X
    6.4 [33X[0;0YMultiplication Groups[133X
      6.4-1 [33X[0;0YLeftMutliplicationGroup, RightMultiplicationGroup and
      MultiplicationGroup[133X
      6.4-2 [33X[0;0YRelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup
      and RelativeMultiplicationGroup[133X
    6.5 [33X[0;0YInner Mapping Groups[133X
      6.5-1 [33X[0;0YLeftInnerMapping, RightInnerMapping, MiddleInnerMapping[133X
      6.5-2 [33X[0;0YLeftInnerMappingGroup, RightInnerMappingGroup,
      MiddleInnerMappingGroup[133X
      6.5-3 InnerMappingGroup
    6.6 [33X[0;0YNuclei, Commutant, Center, and Associator Subloop[133X
      6.6-1 [33X[0;0YLeftNucles, MiddleNucleus, and RightNucleus[133X
      6.6-2 [33X[0;0YNuc, NucleusOfQuasigroup and NucleusOfLoop[133X
      6.6-3 Commutant
      6.6-4 Center
      6.6-5 AssociatorSubloop
    6.7 [33X[0;0YNormal Subloops and Simple Loops[133X
      6.7-1 IsNormal
      6.7-2 NormalClosure
      6.7-3 IsSimple
    6.8 [33X[0;0YFactor Loops[133X
      6.8-1 FactorLoop
      6.8-2 NaturalHomomorphismByNormalSubloop
    6.9 [33X[0;0YNilpotency and Central Series[133X
      6.9-1 IsNilpotent
      6.9-2 NilpotencyClassOfLoop
      6.9-3 IsStronglyNilpotent
      6.9-4 UpperCentralSeries
      6.9-5 LowerCentralSeries
    6.10 [33X[0;0YSolvability, Derived Series and Frattini Subloop[133X
      6.10-1 IsSolvable
      6.10-2 DerivedSubloop
      6.10-3 DerivedLength
      6.10-4 [33X[0;0YFrattiniSubloop and FrattinifactorSize[133X
      6.10-5 FrattinifactorSize
    6.11 [33X[0;0YIsomorphisms and Automorphisms[133X
      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 [33X[0;0YIsotopisms[133X
      6.12-1 IsotopismLoops
      6.12-2 LoopsUpToIsotopism
  7 [33X[0;0YTesting Properties of Quasigroups and Loops[133X
    7.1 [33X[0;0YAssociativity, Commutativity and Generalizations[133X
      7.1-1 IsAssociative
      7.1-2 IsCommutative
      7.1-3 IsPowerAssociative
      7.1-4 IsDiassociative
    7.2 [33X[0;0YInverse Properties[133X
      7.2-1 [33X[0;0YHasLeftInverseProperty, HasRightInverseProperty and
      HasInverseProperty[133X
      7.2-2 HasTwosidedInverses
      7.2-3 HasWeakInverseProperty
      7.2-4 HasAutomorphicInverseProperty
      7.2-5 HasAntiautomorphicInverseProperty
    7.3 [33X[0;0YSome Properties of Quasigroups[133X
      7.3-1 IsSemisymmetric
      7.3-2 IsTotallySymmetric
      7.3-3 IsIdempotent
      7.3-4 IsSteinerQuasigroup
      7.3-5 IsUnipotent
      7.3-6 [33X[0;0YIsLeftDistributive, IsRightDistributive, IsDistributive[133X
      7.3-7 [33X[0;0YIsEntropic and IsMedial[133X
    7.4 [33X[0;0YLoops of Bol Moufang Type[133X
      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 [33X[0;0YPower Alternative Loops[133X
      7.5-1 [33X[0;0YIsLeftPowerAlternative, IsRightPowerAlternative and
      IsPowerAlternative[133X
    7.6 [33X[0;0YConjugacy Closed Loops and Related Properties[133X
      7.6-1 IsLCCLoop
      7.6-2 IsRCCLoop
      7.6-3 IsCCLoop
      7.6-4 IsOsbornLoop
    7.7 [33X[0;0YAutomorphic Loops[133X
      7.7-1 IsLeftAutomorphicLoop
      7.7-2 IsMiddleAutomorphicLoop
      7.7-3 IsRightAutomorphicLoop
      7.7-4 IsAutomorphicLoop
    7.8 [33X[0;0YAdditional Varieties of Loops[133X
      7.8-1 IsCodeLoop
      7.8-2 IsSteinerLoop
      7.8-3 [33X[0;0YIsLeftBruckLoop and IsLeftKLoop[133X
      7.8-4 [33X[0;0YIsRightBruckLoop and IsRightKLoop[133X
  8 [33X[0;0YSpecific Methods[133X
    8.1 [33X[0;0YCore Methods for Bol Loops[133X
      8.1-1 [33X[0;0YAssociatedLeftBruckLoop and AssociatedRightBruckLoop[133X
      8.1-2 IsExactGroupFactorization
      8.1-3 RightBolLoopByExactGroupFactorization
    8.2 [33X[0;0YMoufang Modifications[133X
      8.2-1 LoopByCyclicModification
      8.2-2 LoopByDihedralModification
      8.2-3 LoopMG2
    8.3 [33X[0;0YTriality for Moufang Loops[133X
      8.3-1 TrialityPermGroup
      8.3-2 TrialityPcGroup
    8.4 [33X[0;0YRealizing Groups as Multiplication Groups of Loops[133X
      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 [33X[0;0YLibraries of Loops[133X
    9.1 [33X[0;0YA Typical Library[133X
      9.1-1 LibraryLoop
      9.1-2 MyLibraryLoop
      9.1-3 DisplayLibraryInfo
    9.2 [33X[0;0YLeft Bol Loops and Right Bol Loops[133X
      9.2-1 LeftBolLoop
      9.2-2 RightBolLoop
    9.3 [33X[0;0YLeft Bruck Loops and Right Bruck Loops[133X
      9.3-1 LeftBruckLoop
      9.3-2 RightBruckLoop
    9.4 [33X[0;0YMoufang Loops[133X
      9.4-1 MoufangLoop
    9.5 [33X[0;0YCode Loops[133X
      9.5-1 CodeLoop
    9.6 [33X[0;0YSteiner Loops[133X
      9.6-1 SteinerLoop
    9.7 [33X[0;0YConjugacy Closed Loops[133X
      9.7-1 [33X[0;0YRCCLoop and RightConjugacyClosedLoop[133X
      9.7-2 [33X[0;0YLCCLoop and LeftConjugacyClosedLoop[133X
      9.7-3 [33X[0;0YCCLoop and ConjugacyClosedLoop[133X
    9.8 [33X[0;0YSmall Loops[133X
      9.8-1 SmallLoop
    9.9 [33X[0;0YPaige Loops[133X
      9.9-1 PaigeLoop
    9.10 [33X[0;0YNilpotent Loops[133X
      9.10-1 NilpotentLoop
    9.11 [33X[0;0YAutomorphic Loops[133X
      9.11-1 AutomorphicLoop
    9.12 [33X[0;0YInteresting Loops[133X
      9.12-1 InterestingLoop
    9.13 [33X[0;0YLibraries of Loops Up To Isotopism[133X
      9.13-1 ItpSmallLoop
  A [33X[0;0YFiles[133X
  B [33X[0;0YFilters[133X
  
  
  [32X
