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The LOOPS Package

Computing with quasigroups and loops in GAP

Version 3.4.0

Gábor P. Nagy
Email: nagyg@math.u-szeged.hu
Address:
Department of Mathematics, University of Szeged

Petr Vojtěchovský
Email: petr@math.du.edu
Address:
Department of Mathematics, University of Denver

Copyright

© 2017 Gábor P. Nagy and Petr Vojtěchovský.

Contents

1.2 Installation

1.3 Documentation

1.5 Memory Management

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

  5.5-1 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

9.6 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

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