C intro.tex 1. Introduction
S 1.1. Installation
S 1.2. Documentation
S 1.3. Test files
S 1.4. Feedback
C mathback.tex 2. Mathematical background
S 2.1. Quasigroups and loops
I 2.1. groupoid
I 2.1. magma
I 2.1. semigroup
I 2.1. neutral element
I 2.1. identity element
I 2.1. monoid
I 2.1. two-sided inverse
I 2.1. group
I 2.1. quasigroup
I 2.1. Latin square
I 2.1. loop
S 2.2. Translations
I 2.2. left translation
I 2.2. right translation
I 2.2. left section
I 2.2. right section
I 2.2. left multiplication group
I 2.2. right multiplication group
I 2.2. multiplication group
S 2.3. Homomorphisms and homotopisms
I 2.3. homomorphism
I 2.3. isomorphism
I 2.3. homotopism
I 2.3. isotopism
I 2.3. principal isotopism
I 2.3. principal loop isotope
C how.tex 3. How the package works
S 3.1. Representing quasigroups
S 3.2. Conversions between magmas, quasigroups, loops and groups
F 3.2. AsLoop
F 3.2. AsQuasigroup
F 3.2. AsLoop
S 3.3. Calculating with quasigroups
S 3.4. Naming, viewing and printing quasigroups and their elements
F 3.4. SetQuasigroupElmName
F 3.4. SetLoopElmName
C basic.tex 4. Basic methods and attributes
S 4.1. About Cayley tables
I 4.1. Cayley table
I 4.1. multiplication table
I 4.1. quasigroup table
I 4.1. Latin square
I 4.1. loop table
S 4.2. Testing Cayley tables
F 4.2. IsQuasigroupTable
F 4.2. IsQuasigroupCayleyTable
F 4.2. IsLoopTable
F 4.2. IsLoopCayleyTable
S 4.3. Canonical and normalized Cayley tables
F 4.3. CanonicalCayleyTable
F 4.3. NormalizedQuasigroupTable
S 4.4. Creating quasigroups and loops manually
F 4.4. QuasigroupByCayleyTable
F 4.4. LoopByCayleyTable
S 4.5. Creating quasigroups and loops from a file
F 4.5. QuasigroupFromFile
F 4.5. LoopFromFile
S 4.6. Conversions
F 4.6. AsQuasigroup
F 4.6. AsLoop
F 4.6. PrincipalLoopIsotope
F 4.6. AsGroup
S 4.7. Products of loops
F 4.7. DirectProduct
S 4.8. Opposite quasigroups and loops
I 4.8. opposite quasigroup
F 4.8. Opposite
S 4.9. Basic attributes
F 4.9. Elements
F 4.9. CayleyTable
F 4.9. One
F 4.9. MultiplicativeNeutralElement
F 4.9. Size
I 4.9. power-associative loop
I 4.9. exponent
F 4.9. Exponent
S 4.10. Basic arithmetic operations
I 4.10. left division
I 4.10. right division
F 4.10. LeftDivision
F 4.10. RightDivision
F 4.10. LeftDivision
F 4.10. LeftDivision
F 4.10. RightDivision
F 4.10. RightDivision
S 4.11. Powers and inverses
I 4.11. power-associative
I 4.11. left inverse
I 4.11. right inverse
I 4.11. inverse
F 4.11. LeftInverse
F 4.11. RightInverse
F 4.11. Inverse
S 4.12. Associators and commutators
I 4.12. associator
I 4.12. commutator
F 4.12. Associator
F 4.12. Commutator
S 4.13. Generators
F 4.13. GeneratorsOfQuasigroup
F 4.13. GeneratorsOfLoop
F 4.13. GeneratorsSmallest
C perm.tex 5. Some methods based on permutation groups
S 5.1. Parent of a quasigroup
F 5.1. Parent
F 5.1. Position
F 5.1. PosInParent
S 5.2. Comparing quasigroups with common parent
S 5.3. Subquasigroups and subloops
F 5.3. Subquasigroup
F 5.3. Subloop
F 5.3. IsSubquasigroup
F 5.3. IsSubloop
S 5.4. Translations and sections
F 5.4. LeftTranslation
F 5.4. RightTranslation
F 5.4. LeftSection
F 5.4. RightSection
S 5.5. Multiplication groups
F 5.5. LeftMultiplicationGroup
F 5.5. RightMultiplicationGroup
F 5.5. MultiplicationGroup
I 5.5. relative left multiplication group
I 5.5. relative right multiplication group
I 5.5. relative multiplication group
F 5.5. RelativeLeftMultiplicationGroup
F 5.5. RelativeRightMultiplicationGroup
F 5.5. RelativeMultiplicationGroup
S 5.6. Inner mapping groups
I 5.6. inner mapping group
I 5.6. left inner mapping group
I 5.6. right inner mapping group
F 5.6. InnerMappingGroup
F 5.6. LeftInnerMappingGroup
F 5.6. RightInnerMappingGroup
S 5.7. Nuclei, commutant, center, and associator subloop
I 5.7. left nucleus
I 5.7. middle nucleus
I 5.7. right nucleus
I 5.7. nucleus
F 5.7. LeftNucleus
F 5.7. MiddleNucleus
F 5.7. RightNucleus
F 5.7. Nuc
F 5.7. NucleusOfLoop
F 5.7. NucleusOfQuasigroup
I 5.7. commutant
I 5.7. Moufang center
F 5.7. Commutant
F 5.7. Center
I 5.7. associator subloop
F 5.7. AssociatorSubloop
S 5.8. Normal subloops
I 5.8. normal subloop
F 5.8. IsNormal
I 5.8. normal closure
F 5.8. NormalClosure
I 5.8. simple loop
F 5.8. IsSimple
S 5.9. Factor loops
F 5.9. FactorLoop
F 5.9. NaturalHomomorphismByNormalSubloop
S 5.10. Nilpotency and central series
F 5.10. NilpotencyClassOfLoop
F 5.10. IsNilpotent
I 5.10. strongly nilpotent loop
F 5.10. IsStronglyNilpotent
I 5.10. iterated centers
I 5.10. upper central series
F 5.10. UpperCentralSeries
I 5.10. lower central series
F 5.10. LowerCentralSeries
S 5.11. Solvability
F 5.11. IsSolvable
F 5.11. DerivedSubloop
F 5.11. DerivedLength
F 5.11. FrattiniSubloop
F 5.11. FrattinifactorSize
S 5.12. Isomorphisms and automorphisms
F 5.12. IsomorphismLoops
F 5.12. AutomorphismGroup
S 5.13. How are isomorphisms computed
F 5.13. Discriminator
F 5.13. AreEqualDiscriminators
C testprop.tex 6. Testing properties of quasigroups and loops
S 6.1. Associativity, commutativity and generalizations
F 6.1. IsAssociative
F 6.1. IsCommutative
I 6.1. power-associative loop
I 6.1. diassociative loop
F 6.1. IsPowerAssociative
F 6.1. IsDiassociative
S 6.2. Inverse properties
I 6.2. left inverse property
I 6.2. right inverse property
I 6.2. inverse property
I 6.2. two-sided inverses
F 6.2. HasLeftInverseProperty
F 6.2. HasRightInverseProperty
F 6.2. HasInverseProperty
F 6.2. HasTwosidedInverses
I 6.2. weak inverse property
F 6.2. HasWeakInverseProperty
I 6.2. automorphic inverse property
I 6.2. antiautomorphic inverse property
F 6.2. HasAutomorphicInverseProperty
F 6.2. HasAntiautomorphicInverseProperty
S 6.3. Some properties of quasigroups
I 6.3. semisymmetric quasigroup
I 6.3. totally symmetric quasigroup
F 6.3. IsSemisymmetric
F 6.3. IsTotallySymmetric
I 6.3. idempotent quasigroup
I 6.3. Steiner quasigroup
I 6.3. unipotent quasigroup
F 6.3. IsIdempotent
F 6.3. IsSteinerQuasigroup
F 6.3. IsUnipotent
I 6.3. left distributive quasigroup
I 6.3. right distributive quasigroup
I 6.3. distributive quasigroup
I 6.3. entropic quasigroup
I 6.3. medial quasigroup
F 6.3. IsLeftDistributive
F 6.3. IsRightDistributive
F 6.3. IsDistributive
F 6.3. IsEntropic
F 6.3. IsMedial
F 6.3. IsLDistributive
F 6.3. IsRDistributive
S 6.4. Loops of Bol-Moufang type and related properties
I 6.4. loops of Bol-Moufang type
I 6.4. identity of Bol-Moufang type
I 6.4. left alternative loops
I 6.4. right alternative loops
I 6.4. left nuclear square loops
I 6.4. middle nuclear square loops
I 6.4. right nuclear square loops
I 6.4. flexible loops
I 6.4. left Bol loops
I 6.4. right Bol loops
I 6.4. LC-loops
I 6.4. RC-loops
I 6.4. Moufang loops
I 6.4. C-loops
I 6.4. extra loops
I 6.4. alternative loop
I 6.4. nuclear square loop
F 6.4. IsExtraLoop
F 6.4. IsMoufangLoop
F 6.4. IsCLoop
F 6.4. IsLeftBolLoop
F 6.4. IsRightBolLoop
F 6.4. IsLCLoop
F 6.4. IsRCLoop
F 6.4. IsLeftNuclearSquareLoop
F 6.4. IsMiddleNuclearSquareLoop
F 6.4. IsRightNuclearSquareLoop
F 6.4. IsNuclearSquareLoop
F 6.4. IsFlexible
F 6.4. IsLeftAlternative
F 6.4. IsRightAlternative
F 6.4. IsAlternative
S 6.5. Conjugacy closed loops and related properties
I 6.5. left conjugacy closed loop
I 6.5. right conjugacy closed loop
I 6.5. conjugacy closed loop
F 6.5. IsLCCLoop
F 6.5. IsRCCLoop
F 6.5. IsCCLoop
I 6.5. Osborn loop
F 6.5. IsOsbornLoop
S 6.6. Additional varieties of loops
I 6.6. left Bruck loop
I 6.6. right Bruck loop
I 6.6. K-loop
F 6.6. IsLeftBruckLoop
F 6.6. IsLeftKLoop
F 6.6. IsRightBruckLoop
F 6.6. IsRightKLoop
I 6.6. Steiner loop
F 6.6. IsSteinerLoop
C testprop.tex 7. Specific methods
S 7.1. Moufang modifications
I 7.1. Moufang modifications
I 7.1. cyclic modification
F 7.1. LoopByCyclicModification
I 7.1. dihedral modification
F 7.1. LoopByDihedralModification
F 7.1. LoopMG2
S 7.2. Triality for Moufang loops
I 7.2. group with triality
F 7.2. TrialityPermGroup
F 7.2. TrialityPcGroup
C lib.tex 8. Libraries of small loops
S 8.1. A typical library
F 8.1. MyLibraryLoop
F 8.1. LibraryLoop
F 8.1. DisplayLibraryInfo
S 8.2. Left Bol loops
F 8.2. LeftBolLoop
S 8.3. Small Moufang loops
F 8.3. MoufangLoop
I 8.3. octonion loop
I 8.3. octonions
F 8.3. IsomorphismTypeOfMoufangLoop
S 8.4. Steiner loops
F 8.4. SteinerLoop
S 8.5. CC-loops
I 8.5. conjugacy closed loop
F 8.5. CCLoop
S 8.6. Paige loops
I 8.6. Paige loop
F 8.6. PaigeLoop
S 8.7. Interesting loops
I 8.7. sedenions
F 8.7. InterestingLoop
C files.tex 9. Files
I 9.0. list of files
C filters.tex 10. Filters built into the package
