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# The LOOPS Package

## Computing with quasigroups and loops in GAP

3.4.3

14 November 2022

Gábor Péter Nagy
Email: nagyg@math.u-szeged.hu
Homepage: http://www.math.u-szeged.hu/~nagyg/
Address:
Bolyai Institute, University of Szeged
6725 Szeged, Aradi vertanuk tere 1
Hungary

Petr Vojtěchovský
Email: petr@math.du.edu
Homepage: http://www.math.du.edu/~petr/
Address:
Department of Mathematics, University of Denver
2280 S. Vine Street
Denver, CO 80208
USA

### Abstract

The 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.

### Copyright

© 2005-2017 Gábor P. Nagy and Petr Vojtěchovský.

The LOOPS package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

### Acknowledgements

We 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.

The 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.

Gábor 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.

### Contents

1 Introduction
1.1 Installation
1.2 Documentation
1.3 Test Files
1.4 Memory Management
1.5 Feedback
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 Properties
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 Additional 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
References
Index
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