C grpsupp.tex 1. Supportive functions for groups S 1.1. Predefined groups F 1.1. TWGroup F 1.1. IdTWGroup S 1.2. Operation tables for groups F 1.2. PrintTable S 1.3. Group endomorphisms F 1.3. Endomorphisms S 1.4. Group automorphisms F 1.4. Automorphisms S 1.5. Inner automorphisms of a group F 1.5. InnerAutomorphisms S 1.6. Isomorphic groups F 1.6. IsIsomorphicGroup S 1.7. Subgroups of a group F 1.7. Subgroups S 1.8. Normal subgroups generated by a single element F 1.8. OneGeneratedNormalSubgroups S 1.9. Invariant subgroups F 1.9. IsInvariantUnderMaps F 1.9. IsCharacteristicSubgroup F 1.9. IsCharacteristicInParent F 1.9. IsFullinvariant F 1.9. IsFullinvariantInParent S 1.10. Coset representatives F 1.10. RepresentativesModNormalSubgroup F 1.10. NontrivialRepresentativesModNormalSubgroup S 1.11. Nilpotency class F 1.11. NilpotencyClass S 1.12. Scott length F 1.12. ScottLength S 1.13. Other useful functions for groups F 1.13. AsPermGroup C nr.tex 2. Nearrings S 2.1. Defining a nearring multiplication F 2.1. IsNearRingMultiplication F 2.1. NearRingMultiplicationByOperationTable S 2.2. Construction of nearrings F 2.2. ExplicitMultiplicationNearRing F 2.2. ExplicitMultiplicationNearRingNC F 2.2. IsNearRing F 2.2. IsExplicitMultiplicationNearRing S 2.3. Direct products of nearrings F 2.3. DirectProductNearRing S 2.4. Operation tables for nearrings F 2.4. PrintTable!near rings S 2.5. Modified symbols for the operation tables F 2.5. SetSymbols F 2.5. SetSymbolsSupervised F 2.5. Symbols S 2.6. Accessing nearring elements F 2.6. AsNearRingElement F 2.6. AsGroupReductElement S 2.7. Nearring elements F 2.7. AsList!near rings F 2.7. AsSortedList!near rings F 2.7. Enumerator!near rings S 2.8. Random nearring elements F 2.8. Random!near ring element S 2.9. Nearring generators F 2.9. GeneratorsOfNearRing S 2.10. Size of a nearring F 2.10. Size!near rings S 2.11. The additive group of a nearring F 2.11. GroupReduct S 2.12. Nearring endomorphisms F 2.12. Endomorphisms!near rings S 2.13. Nearring automorphisms F 2.13. Automorphisms!near rings S 2.14. Isomorphic nearrings F 2.14. IsIsomorphicNearRing S 2.15. Subnearrings F 2.15. SubNearRings S 2.16. Invariant subnearrings F 2.16. InvariantSubNearRings S 2.17. Constructing subnearrings F 2.17. SubNearRingBySubgroupNC S 2.18. Intersection of nearrings F 2.18. Intersection!for nearrings S 2.19. Identity of a nearring F 2.19. Identity F 2.19. One F 2.19. IsNearRingWithOne S 2.20. Units of a nearring F 2.20. IsNearRingUnit F 2.20. NearRingUnits S 2.21. Distributivity in a nearring F 2.21. Distributors F 2.21. DistributiveElements F 2.21. IsDistributiveNearRing S 2.22. Elements of a nearring with special properties F 2.22. ZeroSymmetricElements F 2.22. IdempotentElements F 2.22. NilpotentElements F 2.22. QuasiregularElements F 2.22. RegularElements S 2.23. Special properties of a nearring F 2.23. IsAbelianNearRing F 2.23. IsAbstractAffineNearRing F 2.23. IsBooleanNearRing F 2.23. IsNilNearRing F 2.23. IsNilpotentNearRing F 2.23. IsNilpotentFreeNearRing F 2.23. IsCommutative F 2.23. IsDgNearRing F 2.23. IsIntegralNearRing F 2.23. IsPrimeNearRing F 2.23. IsQuasiregularNearRing F 2.23. IsRegularNearRing F 2.23. IsNearField F 2.23. IsPlanarNearRing F 2.23. IsWdNearRing C libnr.tex 3. The nearring library S 3.1. Extracting nearrings from the library F 3.1. LibraryNearRing F 3.1. NumberLibraryNearRings F 3.1. AllLibraryNearRings F 3.1. LibraryNearRingWithOne F 3.1. NumberLibraryNearRingsWithOne F 3.1. AllLibraryNearRingsWithOne S 3.2. Identifying nearrings F 3.2. IdLibraryNearRing F 3.2. IdLibraryNearRingWithOne S 3.3. IsLibraryNearRing F 3.3. IsLibraryNearRing S 3.4. Accessing the information about a nearring stored in the library F 3.4. LibraryNearRingInfo C tfms.tex 4. Arbitrary functions on groups: EndoMappings S 4.1. Defining endo mappings F 4.1. EndoMappingByPositionList F 4.1. EndoMappingByFunction F 4.1. AsEndoMapping F 4.1. AsGroupGeneralMappingByImages F 4.1. IsEndoMapping F 4.1. IdentityEndoMapping F 4.1. ConstantEndoMapping S 4.2. Properties of endo mappings F 4.2. IsIdentityEndoMapping F 4.2. IsConstantEndoMapping F 4.2. IsDistributiveEndoMapping S 4.3. Operations for endo mappings S 4.4. Nicer ways to print a mapping F 4.4. GraphOfMapping F 4.4. PrintAsTerm C tfmnr.tex 5. Transformation nearrings S 5.1. Constructing transformation nearrings F 5.1. TransformationNearRingByGenerators F 5.1. TransformationNearRingByAdditiveGenerators S 5.2. Nearrings of transformations F 5.2. MapNearRing F 5.2. TransformationNearRing F 5.2. IsFullTransformationNearRing F 5.2. PolynomialNearRing F 5.2. EndomorphismNearRing F 5.2. AutomorphismNearRing F 5.2. InnerAutomorphismNearRing F 5.2. CompatibleFunctionNearRing F 5.2. ZeroSymmetricCompatibleFunctionNearRing F 5.2. IsCompatibleEndoMapping F 5.2. Is1AffineComplete F 5.2. CentralizerNearRing F 5.2. RestrictedEndomorphismNearRing F 5.2. LocalInterpolationNearRing S 5.3. The group a transformation nearring acts on F 5.3. Gamma S 5.4. Transformation nearrings and other nearrings F 5.4. AsTransformationNearRing F 5.4. AsExplicitMultiplicationNearRing S 5.5. Noetherian quotients for transformation nearrings F 5.5. NoetherianQuotient!for transformation nearrings F 5.5. CongruenceNoetherianQuotient!for nearrings of polynomial functions F 5.5. CongruenceNoetherianQuotientForInnerAutomorphismNearRings !for inner automorphism nearrings S 5.6. Zerosymmetric mappings F 5.6. ZeroSymmetricPart!for transformation nearrings C ideals.tex 6. Nearring ideals S 6.1. Construction of nearring ideals F 6.1. NearRingIdealByGenerators F 6.1. NearRingLeftIdealByGenerators F 6.1. NearRingRightIdealByGenerators F 6.1. NearRingIdealBySubgroupNC F 6.1. NearRingLeftIdealBySubgroupNC F 6.1. NearRingRightIdealBySubgroupNC F 6.1. NearRingIdeals F 6.1. NearRingLeftIdeals F 6.1. NearRingRightIdeals S 6.2. Testing for ideal properties F 6.2. IsNRI F 6.2. IsNearRingLeftIdeal F 6.2. IsNearRingRightIdeal F 6.2. IsNearRingIdeal F 6.2. IsSubgroupNearRingLeftIdeal F 6.2. IsSubgroupNearRingRightIdeal S 6.3. Special ideal properties F 6.3. IsPrimeNearRingIdeal F 6.3. IsMaximalNearRingIdeal S 6.4. Generators of nearring ideals F 6.4. GeneratorsOfNearRingIdeal F 6.4. GeneratorsOfNearRingLeftIdeal F 6.4. GeneratorsOfNearRingRightIdeal S 6.5. Near-ring ideal elements F 6.5. AsList!near ring ideals F 6.5. AsSortedList!near ring ideals F 6.5. Enumerator!near ring ideals S 6.6. Random ideal elements F 6.6. Random!near ring ideal element S 6.7. Membership of an ideal F 6.7. in S 6.8. Size of ideals F 6.8. Size!near ring ideals S 6.9. Group reducts of ideals F 6.9. GroupReduct!near ring ideals S 6.10. Comparision of ideals F 6.10. = S 6.11. Operations with ideals F 6.11. Intersection!for nearring ideals F 6.11. Intersection F 6.11. ClosureNearRingLeftIdeal F 6.11. ClosureNearRingRightIdeal F 6.11. ClosureNearRingIdeal S 6.12. Commutators F 6.12. NearRingCommutator S 6.13. Simple nearrings F 6.13. IsSimpleNearRing S 6.14. Factor nearrings F 6.14. FactorNearRing F 6.14. / C xsonata.tex 7. Graphic ideal lattices (X-GAP only) F 7.0. GraphicIdealLattice C ngroups.tex 8. N-groups S 8.1. Construction of N-groups F 8.1. NGroup F 8.1. NGroupByNearRingMultiplication F 8.1. NGroupByApplication F 8.1. NGroupByRightIdealFactor S 8.2. Operation tables of N-groups F 8.2. PrintTable!for N-groups S 8.3. Functions for N-groups F 8.3. IsNGroup F 8.3. NearRingActingOnNGroup F 8.3. ActionOfNearRingOnNGroup S 8.4. N-subgroups F 8.4. NSubgroup F 8.4. NSubgroups F 8.4. IsNSubgroup S 8.5. N0-subgroups F 8.5. N0Subgroups S 8.6. Ideals of N-groups F 8.6. NIdeal F 8.6. NIdeals F 8.6. IsNIdeal F 8.6. IsSimpleNGroup F 8.6. IsN0SimpleNGroup S 8.7. Special properties of N-groups F 8.7. IsCompatible F 8.7. IsTameNGroup F 8.7. Is2TameNGroup F 8.7. Is3TameNGroup F 8.7. IsMonogenic F 8.7. IsStronglyMonogenic F 8.7. TypeOfNGroup S 8.8. Noetherian quotients F 8.8. NoetherianQuotient S 8.9. Nearring radicals F 8.9. NuRadical F 8.9. NuRadicals C fpf.tex 9. Fixed-point-free automorphism groups S 9.1. Fixed-point-free automorphism groups and Frobenius groups F 9.1. IsFpfAutomorphismGroup F 9.1. FpfAutomorphismGroupsMaxSize F 9.1. FrobeniusGroup S 9.2. Fixed-point-free representations F 9.2. IsFpfRepresentation F 9.2. DegreeOfIrredFpfRepCyclic F 9.2. DegreeOfIrredFpfRepMetacyclic F 9.2. DegreeOfIrredFpfRep2 F 9.2. DegreeOfIrredFpfRep3 F 9.2. DegreeOfIrredFpfRep4 F 9.2. FpfRepresentationsCyclic F 9.2. FpfRepresentationsMetacyclic F 9.2. FpfRepresentations2 F 9.2. FpfRepresentations3 F 9.2. FpfRepresentations4 S 9.3. Fixed-point-free automorphism groups F 9.3. FpfAutomorphismGroupsCyclic F 9.3. FpfAutomorphismGroupsMetacyclic F 9.3. FpfAutomorphismGroups2 F 9.3. FpfAutomorphismGroups3 F 9.3. FpfAutomorphismGroups4 C nfplwd.tex 10. Nearfields, planar nearrings and weakly divisible nearrings S 10.1. Dickson numbers F 10.1. IsPairOfDicksonNumbers S 10.2. Dickson nearfields F 10.2. DicksonNearFields F 10.2. NumberOfDicksonNearFields S 10.3. Exceptional nearfields F 10.3. ExceptionalNearFields F 10.3. AllExceptionalNearFields S 10.4. Planar nearrings F 10.4. PlanarNearRing F 10.4. OrbitRepresentativesForPlanarNearRing S 10.5. Weakly divisible nearrings F 10.5. WdNearRing C design.tex 11. Designs S 11.1. Constructing a design F 11.1. DesignFromPointsAndBlocks F 11.1. DesignFromIncidenceMat F 11.1. DesignFromPlanarNearRing F 11.1. DesignFromFerreroPair F 11.1. DesignFromWdNearRing S 11.2. Properties of a design F 11.2. PointsOfDesign F 11.2. BlocksOfDesign F 11.2. DesignParameter F 11.2. IncidenceMat F 11.2. PrintIncidenceMat F 11.2. BlockIntersectionNumbers F 11.2. BlockIntersectionNumbersK F 11.2. IsCircularDesign S 11.3. Working with the points and blocks of a design F 11.3. IsPointIncidentBlock F 11.3. PointsIncidentBlocks F 11.3. BlocksIncidentPoints