C intro.tex 1. Introduction S 1.1. Installation S 1.2. Documentation S 1.3. Test files S 1.4. Feedback S 1.5. Acknowledgement 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 S 2.4. Extensions I 2.4. extension of loops I 2.4. nuclear extension I 2.4. cocycle 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 I 3.3. Bol loop I 3.3. simple loop S 3.4. Naming, viewing and printing quasigroups and their elements F 3.4. SetQuasigroupElmName F 3.4. SetLoopElmName C create.tex 4. Creating quasigroups and loops 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. Creating quasigroups and loops by sections F 4.6. CayleyTableByPerms F 4.6. QuasigroupByLeftSection F 4.6. LoopByLeftSection F 4.6. QuasigroupByRightSection F 4.6. LoopByRightSection I 4.6. transversal F 4.6. QuasigroupByRightSection F 4.6. LoopByRightSection I 4.6. simple Bol loop S 4.7. Creating quasigroups and loops by extensions F 4.7. NuclearExtension F 4.7. LoopByExtension S 4.8. Conversions F 4.8. AsQuasigroup F 4.8. PrincipalLoopIsotope F 4.8. AsLoop F 4.8. AsGroup S 4.9. Products of loops F 4.9. DirectProduct S 4.10. Opposite quasigroups and loops I 4.10. opposite quasigroup F 4.10. Opposite C create.tex 5. Basic methods and attributes S 5.1. Basic attributes F 5.1. Elements F 5.1. CayleyTable F 5.1. One F 5.1. MultiplicativeNeutralElement F 5.1. Size I 5.1. power-associative loop I 5.1. exponent F 5.1. Exponent S 5.2. Basic arithmetic operations I 5.2. left division I 5.2. right division F 5.2. LeftDivision F 5.2. RightDivision F 5.2. LeftDivision F 5.2. LeftDivision F 5.2. RightDivision F 5.2. RightDivision F 5.2. LeftDivisionCayleyTable F 5.2. RightDivisionCayleyTable S 5.3. Powers and inverses I 5.3. power-associativity I 5.3. left inverse I 5.3. right inverse I 5.3. inverse F 5.3. LeftInverse F 5.3. RightInverse F 5.3. Inverse S 5.4. Associators and commutators I 5.4. associator I 5.4. commutator F 5.4. Associator F 5.4. Commutator S 5.5. Generators F 5.5. GeneratorsOfQuasigroup F 5.5. GeneratorsOfLoop F 5.5. GeneratorsSmallest C perm.tex 6. Methods based on permutation groups S 6.1. Parent of a quasigroup F 6.1. Parent F 6.1. Position F 6.1. PosInParent S 6.2. Comparing quasigroups with common parent S 6.3. Subquasigroups and subloops F 6.3. Subquasigroup F 6.3. Subloop F 6.3. IsSubquasigroup F 6.3. IsSubloop F 6.3. AllSubloops I 6.3. coset F 6.3. RightCosets I 6.3. transversal F 6.3. RightTransversal S 6.4. Translations and sections F 6.4. LeftTranslation F 6.4. RightTranslation F 6.4. LeftSection F 6.4. RightSection S 6.5. Multiplication groups F 6.5. LeftMultiplicationGroup F 6.5. RightMultiplicationGroup F 6.5. MultiplicationGroup I 6.5. relative left multiplication group I 6.5. relative right multiplication group I 6.5. relative multiplication group F 6.5. RelativeLeftMultiplicationGroup F 6.5. RelativeRightMultiplicationGroup F 6.5. RelativeMultiplicationGroup S 6.6. Inner mapping groups I 6.6. inner mapping group I 6.6. left inner mapping group I 6.6. right inner mapping group I 6.6. left inner mapping I 6.6. right inner mapping I 6.6. conjugation I 6.6. middle inner mapping I 6.6. middle inner mapping F 6.6. LeftInnerMapping F 6.6. MiddleInnerMapping F 6.6. RightInnerMapping F 6.6. LeftInnerMappingGroup F 6.6. MiddleInnerMappingGroup F 6.6. RightInnerMappingGroup F 6.6. InnerMappingGroup S 6.7. Nuclei, commutant, center, and associator subloop I 6.7. left nucleus I 6.7. middle nucleus I 6.7. right nucleus I 6.7. nucleus F 6.7. LeftNucleus F 6.7. MiddleNucleus F 6.7. RightNucleus F 6.7. Nuc F 6.7. NucleusOfLoop F 6.7. NucleusOfQuasigroup I 6.7. commutant I 6.7. Moufang center I 6.7. centrum F 6.7. Commutant F 6.7. Center I 6.7. associator subloop F 6.7. AssociatorSubloop S 6.8. Normal subloops I 6.8. normal subloop F 6.8. IsNormal I 6.8. normal closure F 6.8. NormalClosure I 6.8. simple loop F 6.8. IsSimple S 6.9. Factor loops F 6.9. FactorLoop F 6.9. NaturalHomomorphismByNormalSubloop S 6.10. Nilpotency and central series F 6.10. NilpotencyClassOfLoop F 6.10. IsNilpotent I 6.10. strongly nilpotent loop F 6.10. IsStronglyNilpotent I 6.10. iterated centers I 6.10. upper central series F 6.10. UpperCentralSeries I 6.10. lower central series F 6.10. LowerCentralSeries S 6.11. Solvability F 6.11. IsSolvable F 6.11. DerivedSubloop F 6.11. DerivedLength F 6.11. FrattiniSubloop F 6.11. FrattinifactorSize S 6.12. Isomorphisms and automorphisms F 6.12. IsomorphismLoops F 6.12. LoopsUpToIsomorphism F 6.12. AutomorphismGroup F 6.12. IsomorphicCopyByPerm F 6.12. IsomorphicCopyByNormalSubloop S 6.13. How are isomorphisms computed F 6.13. Discriminator F 6.13. AreEqualDiscriminators S 6.14. Isotopisms F 6.14. IsotopismLoops F 6.14. LoopsUpToIsotopism C testprop.tex 7. Testing properties of quasigroups and loops S 7.1. Associativity, commutativity and generalizations F 7.1. IsAssociative F 7.1. IsCommutative I 7.1. power-associative loop I 7.1. diassociative loop F 7.1. IsPowerAssociative F 7.1. IsDiassociative S 7.2. Inverse properties I 7.2. left inverse property I 7.2. right inverse property I 7.2. inverse property I 7.2. two-sided inverses loop F 7.2. HasLeftInverseProperty F 7.2. HasRightInverseProperty F 7.2. HasInverseProperty F 7.2. HasTwosidedInverses I 7.2. weak inverse property F 7.2. HasWeakInverseProperty I 7.2. automorphic inverse property I 7.2. antiautomorphic inverse property F 7.2. HasAutomorphicInverseProperty F 7.2. HasAntiautomorphicInverseProperty S 7.3. Some properties of quasigroups I 7.3. semisymmetric quasigroup I 7.3. totally symmetric quasigroup F 7.3. IsSemisymmetric F 7.3. IsTotallySymmetric I 7.3. idempotent quasigroup I 7.3. Steiner quasigroup I 7.3. unipotent quasigroup F 7.3. IsIdempotent F 7.3. IsSteinerQuasigroup F 7.3. IsUnipotent I 7.3. left distributive quasigroup I 7.3. right distributive quasigroup I 7.3. distributive quasigroup I 7.3. entropic quasigroup I 7.3. medial quasigroup F 7.3. IsLeftDistributive F 7.3. IsRightDistributive F 7.3. IsDistributive F 7.3. IsEntropic F 7.3. IsMedial F 7.3. IsLDistributive F 7.3. IsRDistributive S 7.4. Loops of Bol-Moufang type I 7.4. loops of Bol-Moufang type I 7.4. identity of Bol-Moufang type I 7.4. left alternative loop I 7.4. right alternative loop I 7.4. left nuclear square loop I 7.4. middle nuclear square loop I 7.4. right nuclear square loop I 7.4. flexible loop I 7.4. left Bol loop I 7.4. right Bol loop I 7.4. LC-loop I 7.4. RC-loop I 7.4. Moufang loop I 7.4. C-loop I 7.4. extra loop I 7.4. alternative loop I 7.4. nuclear square loop F 7.4. IsExtraLoop F 7.4. IsMoufangLoop F 7.4. IsCLoop F 7.4. IsLeftBolLoop F 7.4. IsRightBolLoop F 7.4. IsLCLoop F 7.4. IsRCLoop F 7.4. IsLeftNuclearSquareLoop F 7.4. IsMiddleNuclearSquareLoop F 7.4. IsRightNuclearSquareLoop F 7.4. IsNuclearSquareLoop F 7.4. IsFlexible F 7.4. IsLeftAlternative F 7.4. IsRightAlternative F 7.4. IsAlternative S 7.5. Power alternative loops I 7.5. left power alternative loop I 7.5. right power alternative loop I 7.5. power alternative loop F 7.5. IsLeftPowerAlternative F 7.5. IsRightPowerAlternative F 7.5. IsPowerAlternative S 7.6. Conjugacy closed loops and related properties I 7.6. left conjugacy closed loop I 7.6. right conjugacy closed loop I 7.6. conjugacy closed loop F 7.6. IsLCCLoop F 7.6. IsRCCLoop F 7.6. IsCCLoop I 7.6. Osborn loop F 7.6. IsOsbornLoop S 7.7. Additional varieties of loops I 7.7. code loop F 7.7. IsCodeLoop I 7.7. Steiner loop F 7.7. IsSteinerLoop I 7.7. left Bruck loop I 7.7. right Bruck loop I 7.7. K-loop F 7.7. IsLeftBruckLoop F 7.7. IsLeftKLoop F 7.7. IsRightBruckLoop F 7.7. IsRightKLoop I 7.7. left A-loop I 7.7. middle A-loop I 7.7. right A-loop I 7.7. A-loop F 7.7. IsLeftALoop F 7.7. IsMiddleALoop F 7.7. IsRightALoop F 7.7. IsALoop C specific.tex 8. Specific methods S 8.1. Core methods for Bol loops I 8.1. left Bol loop I 8.1. associated left Bruck loop F 8.1. AssoicatedLeftBruckLoop S 8.2. Moufang modifications I 8.2. Moufang modifications I 8.2. cyclic modification F 8.2. LoopByCyclicModification I 8.2. dihedral modification F 8.2. LoopByDihedralModification F 8.2. LoopMG2 S 8.3. Triality for Moufang loops I 8.3. group with triality F 8.3. TrialityPermGroup F 8.3. TrialityPcGroup C lib.tex 9. Libraries of small loops S 9.1. A typical library F 9.1. MyLibraryLoop F 9.1. LibraryLoop F 9.1. DisplayLibraryInfo S 9.2. Left Bol loops F 9.2. LeftBolLoop S 9.3. Moufang loops F 9.3. MoufangLoop I 9.3. octonion loop I 9.3. octonions S 9.4. Code loops F 9.4. CodeLoop S 9.5. Steiner loops F 9.5. SteinerLoop S 9.6. CC-loops I 9.6. conjugacy closed loop F 9.6. CCLoop S 9.7. Small loops F 9.7. SmallLoop S 9.8. Paige loops I 9.8. Paige loop F 9.8. PaigeLoop S 9.9. Interesting loops I 9.9. sedenions F 9.9. InterestingLoop S 9.10. Libraries of loops up to isotopism F 9.10. ItpSmallLoop C files.tex 10. Files I 10.0. list of files C filters.tex 11. Filters built into the package