C about.tex 1. About the GAP Reference Manual S 1.1. Manual Conventions S 1.2. Credit C help.tex 2. The Help System S 2.1. Invoking the Help F 2.1. getting help S 2.2. Browsing through the Sections F 2.2. browsing forward F 2.2. browsing backwards F 2.2. browsing forward one chapter F 2.2. browsing backwards one chapter F 2.2. browsing the previous section browsed F 2.2. browsing the next section browsed F 2.2. list of available books F 2.2. table of sections for help books F 2.2. table of chapters for help books F 2.2. redisplay a help section F 2.2. redisplay with next help viewer S 2.3. Changing the Help Viewer I 2.3. document formats (text, dvi, ps, pdf, HTML) F 2.3. SetHelpViewer S 2.4. The Pager Command F 2.4. Pager C run.tex 3. Running GAP I 3.0. options S 3.1. Command Line Options I 3.1. features!under UNIX I 3.1. UNIX!features I 3.1. options!under UNIX I 3.1. UNIX!options I 3.1. -h I 3.1. -b I 3.1. -q I 3.1. -e I 3.1. -f I 3.1. -n I 3.1. -x I 3.1. -y I 3.1. -g I 3.1. -g -g I 3.1. -m I 3.1. -o I 3.1. -K I 3.1. -l I 3.1. GAPInfo.RootPaths I 3.1. -r I 3.1. -L I 3.1. -R I 3.1. options!command line, filenames S 3.2. Advanced Features of GAP I 3.2. -a I 3.2. -A I 3.2. -B I 3.2. -D I 3.2. -M I 3.2. -N I 3.2. -O I 3.2. -T I 3.2. -X I 3.2. -Y I 3.2. -i I 3.2. options!command line, internal I 3.2. -C I 3.2. -U I 3.2. -P I 3.2. -W I 3.2. -z I 3.2. -p S 3.3. Running GAP under MacOS I 3.3. -z!on Macintosh I 3.3. -P!on Macintosh I 3.3. -W!on Macintosh I 3.3. -a!on Macintosh I 3.3. -f!on Macintosh I 3.3. -n!on Macintosh I 3.3. -e!on Macintosh I 3.3. -o!on Macintosh I 3.3. gap.rc S 3.4. The .gaprc file I 3.4. gap.rc I 3.4. .gaprc I 3.4. GAP3 S 3.5. Completion Files F 3.5. CreateCompletionFiles F 3.5. CreateCompletionFiles S 3.6. Testing for the System Architecture F 3.6. ARCH_IS_UNIX F 3.6. ARCH_IS_MAC F 3.6. ARCH_IS_WINDOWS S 3.7. The Compiler I 3.7. gac F 3.7. LoadDynamicModule F 3.7. LoadDynamicModule S 3.8. Suitability for Compilation S 3.9. Compiling Library Code S 3.10. CRC Numbers I 3.10. CRC I 3.10. CrcFile!example S 3.11. Saving and Loading a Workspace I 3.11. save F 3.11. SaveWorkspace F 3.11. loading a saved workspace S 3.12. Coloring the Prompt and Input F 3.12. ColorPrompt C language.tex 4. The Programming Language S 4.1. Language Overview S 4.2. Lexical Structure S 4.3. Symbols S 4.4. Whitespaces I 4.4. space I 4.4. blank I 4.4. tabulator I 4.4. newline I 4.4. comments S 4.5. Keywords S 4.6. Identifiers F 4.6. IsValidIdentifier S 4.7. Expressions I 4.7. evaluation I 4.7. operators S 4.8. Variables I 4.8. scope I 4.8. bound F 4.8. Unbind S 4.9. More About Global Variables F 4.9. IsReadOnlyGlobal F 4.9. MakeReadOnlyGlobal F 4.9. MakeReadWriteGlobal F 4.9. ValueGlobal F 4.9. IsBoundGlobal F 4.9. UnbindGlobal F 4.9. BindGlobal F 4.9. NamesGVars F 4.9. NamesSystemGVars F 4.9. NamesUserGVars F 4.9. TemporaryGlobalVarName S 4.10. Function Calls F 4.10. function call F 4.10. function call!with arguments I 4.10. functions!with a variable number of arguments I 4.10. arg!special function argument F 4.10. function call!with options S 4.11. Comparisons F 4.11. equality test F 4.11. inequality test F 4.11. smaller test F 4.11. larger test F 4.11. smaller or equal F 4.11. larger or equal I 4.11. operators!precedence S 4.12. Arithmetic Operators I 4.12. precedence I 4.12. associativity I 4.12. operators!arithmetic I 4.12. + I 4.12. - I 4.12. \* I 4.12. / I 4.12. {\accent 94 } I 4.12. mod!arithmetic operators I 4.12. modulo I 4.12. modulo!arithmetic operators F 4.12. positive number F 4.12. negative number F 4.12. addition F 4.12. subtraction F 4.12. multiplication F 4.12. division F 4.12. mod F 4.12. power I 4.12. mod!rationals I 4.12. modular remainder I 4.12. modular inverse I 4.12. coprime I 4.12. relatively prime I 4.12. operators!precedence I 4.12. operators!associativity S 4.13. Statements I 4.13. execution S 4.14. Assignments F 4.14. assignment!variable S 4.15. Procedure Calls F 4.15. procedure call F 4.15. procedure call with arguments S 4.16. If I 4.16. fi I 4.16. then I 4.16. else I 4.16. elif F 4.16. if statement S 4.17. While I 4.17. loop!while F 4.17. while loop S 4.18. Repeat I 4.18. loop!repeat I 4.18. until F 4.18. repeat loop S 4.19. For I 4.19. loop!for I 4.19. do I 4.19. od F 4.19. for loop F 4.19. loop over range F 4.19. loop over iterator F 4.19. loop over object S 4.20. Break I 4.20. loops!leaving F 4.20. break statement S 4.21. Continue I 4.21. loops!restarting F 4.21. continue statement S 4.22. Function I 4.22. functions!definition of I 4.22. end I 4.22. local I 4.22. recursion I 4.22. functions!recursive I 4.22. environment I 4.22. body F 4.22. function I 4.22. functions!with a variable number of arguments I 4.22. arg!special function argument I 4.22. functions!definition by arrow notation F 4.22. arrow notation for functions S 4.23. Return F 4.23. return!no value F 4.23. return!with value S 4.24. The Syntax in BNF I 4.24. BNF C function.tex 5. Functions I 5.0. functions S 5.1. Information about a function F 5.1. NameFunction F 5.1. NumberArgumentsFunction F 5.1. NamesLocalVariablesFunction S 5.2. Calling a function with a list argument that is interpreted as several arguments F 5.2. CallFuncList S 5.3. Functions that do nothing F 5.3. ReturnTrue F 5.3. ReturnFalse F 5.3. ReturnFail F 5.3. IdFunc S 5.4. Function Types F 5.4. IsFunction F 5.4. IsOperation F 5.4. FunctionsFamily C mloop.tex 6. Main Loop and Break Loop S 6.1. Main Loop I 6.1. read eval print loop I 6.1. loop!read eval print I 6.1. prompt I 6.1. prompt!partial I 6.1. syntax errors I 6.1. errors!syntax I 6.1. output!suppressing I 6.1. last I 6.1. previous result S 6.2. Special Rules for Input Lines S 6.3. View and Print F 6.3. View F 6.3. Print F 6.3. ViewObj F 6.3. PrintObj F 6.3. Display S 6.4. Break Loops F 6.4. quit I 6.4. return F 6.4. return from break loop F 6.4. OnBreak I 6.4. ErrorNoTraceBack I 6.4. Break loop message F 6.4. OnBreakMessage I 6.4. Backtrace!GAP3 name for Where F 6.4. Where S 6.5. Variable Access in a Break Loop F 6.5. DownEnv F 6.5. UpEnv S 6.6. Error F 6.6. Error S 6.7. ErrorCount F 6.7. ErrorCount S 6.8. Leaving GAP I 6.8. quit!in emergency I 6.8. exit I 6.8. at exit functions I 6.8. saving on exit F 6.8. QUIT!emergency quit F 6.8. InstallAtExit F 6.8. QUITTING F 6.8. SaveOnExitFile S 6.9. Line Editing S 6.10. Editing Files F 6.10. Edit S 6.11. Editor Support I 6.11. utilities for editing GAP files I 6.11. vi I 6.11. vim I 6.11. emacs S 6.12. SizeScreen F 6.12. SizeScreen F 6.12. SizeScreen C debug.tex 7. Debugging and Profiling Facilities S 7.1. Recovery from NoMethodFound-Errors F 7.1. ShowArguments F 7.1. ShowArgument F 7.1. ShowDetails F 7.1. ShowMethods F 7.1. ShowMethods F 7.1. ShowOtherMethods F 7.1. ShowOtherMethods S 7.2. ApplicableMethod F 7.2. ApplicableMethod F 7.2. ApplicableMethod F 7.2. ApplicableMethod F 7.2. ApplicableMethodTypes F 7.2. ApplicableMethodTypes F 7.2. ApplicableMethodTypes S 7.3. Tracing Methods F 7.3. TraceMethods F 7.3. UntraceMethods F 7.3. TraceImmediateMethods S 7.4. Info Functions F 7.4. NewInfoClass F 7.4. DeclareInfoClass F 7.4. SetInfoLevel F 7.4. InfoLevel F 7.4. Info F 7.4. InfoWarning S 7.5. Assertions F 7.5. SetAssertionLevel F 7.5. AssertionLevel F 7.5. Assert F 7.5. Assert S 7.6. Timing F 7.6. Runtimes F 7.6. Runtime F 7.6. time S 7.7. Profiling F 7.7. ProfileOperations F 7.7. ProfileOperationsAndMethods F 7.7. ProfileMethods F 7.7. UnprofileMethods F 7.7. ProfileFunctions F 7.7. UnprofileFunctions F 7.7. ProfileGlobalFunctions F 7.7. ProfileGlobalFunctions F 7.7. DisplayProfile F 7.7. DisplayProfile F 7.7. PROFILETHRESHOLD F 7.7. ClearProfile F 7.7. DisplayCacheStats F 7.7. ClearCacheStats S 7.8. Information about the version used F 7.8. DisplayRevision S 7.9. Test Files F 7.9. ReadTest S 7.10. Debugging Recursion F 7.10. SetRecursionTrapInterval S 7.11. Global Memory Information F 7.11. GasmanStatistics F 7.11. GasmanMessageStatus F 7.11. SetGasmanMessageStatus F 7.11. GasmanLimits C options.tex 8. Options Stack F 8.0. PushOptions F 8.0. PopOptions F 8.0. ResetOptionsStack F 8.0. ValueOption F 8.0. DisplayOptionsStack F 8.0. InfoOptions C files.tex 9. Files and Filenames S 9.1. Portability F 9.1. LastSystemError S 9.2. GAP Root Directory S 9.3. Directories F 9.3. Directory F 9.3. DirectoryTemporary F 9.3. DirectoryTemporary F 9.3. DirectoryCurrent F 9.3. DirectoriesLibrary F 9.3. DirectoriesLibrary F 9.3. DirectoriesSystemPrograms F 9.3. DirectoryContents S 9.4. Filename F 9.4. Filename F 9.4. Filename S 9.5. Special Filenames S 9.6. File Access F 9.6. IsExistingFile F 9.6. IsReadableFile F 9.6. IsWritableFile F 9.6. IsExecutableFile F 9.6. IsDirectoryPath S 9.7. File Operations F 9.7. Read F 9.7. ReadAsFunction F 9.7. PrintTo F 9.7. AppendTo F 9.7. LogTo F 9.7. LogTo!stop logging F 9.7. InputLogTo F 9.7. InputLogTo!stop logging input F 9.7. OutputLogTo F 9.7. OutputLogTo!stop logging output F 9.7. CrcFile F 9.7. RemoveFile F 9.7. Reread F 9.7. REREADING C streams.tex 10. Streams S 10.1. Categories for Streams and the StreamsFamily F 10.1. IsStream F 10.1. IsClosedStream F 10.1. IsInputStream F 10.1. IsInputTextStream F 10.1. IsInputTextNone F 10.1. IsOutputStream F 10.1. IsOutputTextStream F 10.1. IsOutputTextNone F 10.1. StreamsFamily S 10.2. Operations applicable to All Streams F 10.2. CloseStream F 10.2. FileDescriptorOfStream F 10.2. UNIXSelect S 10.3. Operations for Input Streams F 10.3. Read!for streams F 10.3. ReadAsFunction!for streams F 10.3. ReadTest!for streams F 10.3. ReadByte F 10.3. ReadLine F 10.3. ReadAll F 10.3. ReadAll F 10.3. IsEndOfStream F 10.3. PositionStream F 10.3. RewindStream F 10.3. SeekPositionStream S 10.4. Operations for Output Streams F 10.4. WriteByte F 10.4. WriteLine F 10.4. WriteAll F 10.4. PrintTo!for streams F 10.4. AppendTo!for streams F 10.4. LogTo!for streams F 10.4. InputLogTo!for streams F 10.4. OutputLogTo!for streams F 10.4. SetPrintFormattingStatus F 10.4. PrintFormattingStatus S 10.5. File Streams F 10.5. InputTextFile F 10.5. OutputTextFile S 10.6. User Streams F 10.6. InputTextUser F 10.6. OutputTextUser S 10.7. String Streams F 10.7. InputTextString F 10.7. OutputTextString S 10.8. Input-Output Streams F 10.8. IsInputOutputStream F 10.8. InputOutputLocalProcess F 10.8. ReadAllLine S 10.9. Dummy Streams F 10.9. InputTextNone F 10.9. OutputTextNone S 10.10. Handling of Streams in the Background F 10.10. InstallCharReadHookFunc F 10.10. UnInstallCharReadHookFunc C process.tex 11. Processes S 11.1. Process F 11.1. Process S 11.2. Exec F 11.2. Exec C objects.tex 12. Objects and Elements S 12.1. Objects F 12.1. IsObject S 12.2. Elements as equivalence classes I 12.2. elements!definition S 12.3. Sets S 12.4. Domains S 12.5. Identical Objects F 12.5. IsIdenticalObj F 12.5. IsNotIdenticalObj S 12.6. Mutability and Copyability F 12.6. IsCopyable F 12.6. IsMutable F 12.6. Immutable F 12.6. MakeImmutable S 12.7. Duplication of Objects I 12.7. Copy I 12.7. copy!an object I 12.7. clone!an object F 12.7. ShallowCopy F 12.7. StructuralCopy S 12.8. Other Operations Applicable to any Object F 12.8. SetName F 12.8. Name F 12.8. IsInternallyConsistent F 12.8. MemoryUsage C types.tex 13. Types of Objects S 13.1. Families F 13.1. FamilyObj S 13.2. Filters I 13.2. and!for filters F 13.2. RankFilter F 13.2. NamesFilter F 13.2. ShowImpliedFilters S 13.3. Categories F 13.3. CategoriesOfObject S 13.4. Representation F 13.4. RepresentationsOfObject S 13.5. Attributes I 13.5. system getter I 13.5. system setter F 13.5. KnownAttributesOfObject S 13.6. Setter and Tester for Attributes I 13.6. setter I 13.6. tester F 13.6. Tester F 13.6. Setter F 13.6. AttributeValueNotSet F 13.6. InfoAttributes F 13.6. DisableAttributeValueStoring F 13.6. EnableAttributeValueStoring S 13.7. Properties F 13.7. KnownPropertiesOfObject F 13.7. KnownTruePropertiesOfObject S 13.8. Other Filters S 13.9. Types F 13.9. TypeObj F 13.9. DataType C integers.tex 14. Integers F 14.0. Integers F 14.0. PositiveIntegers F 14.0. NonnegativeIntegers F 14.0. IsIntegers F 14.0. IsPositiveIntegers F 14.0. IsNonnegativeIntegers S 14.1. Elementary Operations for Integers F 14.1. IsInt F 14.1. IsPosInt F 14.1. Int F 14.1. IsEvenInt F 14.1. IsOddInt F 14.1. AbsInt I 14.1. absolute value of an integer F 14.1. SignInt I 14.1. sign!of an integer F 14.1. LogInt F 14.1. RootInt F 14.1. RootInt I 14.1. root!of an integer I 14.1. square root!of an integer F 14.1. SmallestRootInt I 14.1. root!of an integer, smallest F 14.1. Random!for integers S 14.2. Quotients and Remainders F 14.2. QuoInt I 14.2. integer part of a quotient F 14.2. BestQuoInt F 14.2. RemInt I 14.2. remainder of a quotient F 14.2. GcdInt F 14.2. Gcdex F 14.2. LcmInt F 14.2. CoefficientsQadic F 14.2. CoefficientsMultiadic F 14.2. ChineseRem I 14.2. Chinese remainder F 14.2. PowerModInt S 14.3. Prime Integers and Factorization F 14.3. Primes F 14.3. IsPrimeInt F 14.3. IsProbablyPrimeInt F 14.3. IsPrimePowerInt F 14.3. NextPrimeInt F 14.3. PrevPrimeInt F 14.3. FactorsInt F 14.3. FactorsInt F 14.3. PartialFactorization F 14.3. PartialFactorization I 14.3. CheapFactorsInt F 14.3. PrintFactorsInt F 14.3. PrimePowersInt F 14.3. DivisorsInt I 14.3. divisors!of an integer S 14.4. Residue Class Rings I 14.4. mod!residue class rings F 14.4. modulo!residue class rings F 14.4. ZmodnZ F 14.4. ZmodpZ F 14.4. ZmodpZNC I 14.4. mod!Integers F 14.4. ZmodnZObj F 14.4. ZmodnZObj F 14.4. IsZmodnZObj F 14.4. IsZmodnZObjNonprime F 14.4. IsZmodpZObj F 14.4. IsZmodpZObjSmall F 14.4. IsZmodpZObjLarge S 14.5. Random Sources F 14.5. IsRandomSource F 14.5. Random F 14.5. Random F 14.5. State F 14.5. Reset F 14.5. Reset F 14.5. Init F 14.5. Init F 14.5. IsGlobalRandomSource F 14.5. IsGAPRandomSource F 14.5. IsMersenneTwister F 14.5. GlobalRandomSource F 14.5. GlobalMersenneTwister F 14.5. RandomSource F 14.5. RandomSource C numtheor.tex 15. Number Theory I 15.0. prime residue group F 15.0. InfoNumtheor S 15.1. Prime Residues F 15.1. PrimeResidues!function I 15.1. prime residue group F 15.1. Phi I 15.1. order!of the prime residue group I 15.1. prime residue group!order I 15.1. Euler's totient function F 15.1. Lambda I 15.1. Carmichael's lambda function I 15.1. prime residue group!exponent I 15.1. exponent!of the prime residue group F 15.1. GeneratorsPrimeResidues S 15.2. Primitive Roots and Discrete Logarithms F 15.2. OrderMod I 15.2. multiplicative order of an integer F 15.2. LogMod F 15.2. LogModShanks I 15.2. logarithm!discrete F 15.2. PrimitiveRootMod I 15.2. primitive root modulo an integer I 15.2. prime residue group!generator I 15.2. generator!of the prime residue group F 15.2. IsPrimitiveRootMod I 15.2. test!for a primitive root I 15.2. prime residue group!generator I 15.2. generator!of the prime residue group S 15.3. Roots Modulo Integers F 15.3. Jacobi I 15.3. quadratic residue I 15.3. residue!quadratic F 15.3. Legendre I 15.3. quadratic residue I 15.3. residue!quadratic F 15.3. RootMod I 15.3. quadratic residue I 15.3. residue!quadratic I 15.3. root!of an integer modulo another F 15.3. RootsMod F 15.3. RootsUnityMod I 15.3. modular roots I 15.3. root!of 1 modulo an integer S 15.4. Multiplicative Arithmetic Functions F 15.4. Sigma F 15.4. Tau F 15.4. MoebiusMu S 15.5. Continued Fractions F 15.5. ContinuedFractionExpansionOfRoot F 15.5. ContinuedFractionApproximationOfRoot S 15.6. Miscellaneous F 15.6. TwoSquares I 15.6. representation!as a sum of two squares C rational.tex 16. Rational Numbers F 16.0. Rationals F 16.0. IsRationals S 16.1. Elementary Operations for Rationals F 16.1. IsRat I 16.1. test!for a rational F 16.1. IsPosRat F 16.1. IsNegRat F 16.1. NumeratorRat I 16.1. numerator!of a rational F 16.1. DenominatorRat I 16.1. denominator!of a rational F 16.1. Rat F 16.1. Random!for rationals C combinat.tex 17. Combinatorics S 17.1. Combinatorial Numbers F 17.1. Factorial F 17.1. Binomial I 17.1. coefficient!binomial I 17.1. number!binomial F 17.1. Bell I 17.1. number!Bell F 17.1. Bernoulli I 17.1. sequence!Bernoulli F 17.1. Stirling1 I 17.1. Stirling number of the first kind I 17.1. number!Stirling, of the first kind F 17.1. Stirling2 I 17.1. Stirling number of the second kind I 17.1. number!Stirling, of the second kind S 17.2. Combinations, Arrangements and Tuples F 17.2. Combinations F 17.2. NrCombinations I 17.2. powerset I 17.2. subsets F 17.2. Arrangements F 17.2. NrArrangements F 17.2. UnorderedTuples F 17.2. NrUnorderedTuples F 17.2. Tuples F 17.2. NrTuples F 17.2. PermutationsList F 17.2. NrPermutationsList F 17.2. Derangements F 17.2. NrDerangements F 17.2. PartitionsSet F 17.2. NrPartitionsSet F 17.2. Partitions F 17.2. NrPartitions F 17.2. OrderedPartitions F 17.2. NrOrderedPartitions I 17.2. partitions!ordered, of an integer I 17.2. partitions!improper, of an integer F 17.2. PartitionsGreatestLE F 17.2. PartitionsGreatestEQ F 17.2. RestrictedPartitions F 17.2. NrRestrictedPartitions I 17.2. partitions!restricted, of an integer F 17.2. SignPartition F 17.2. AssociatedPartition F 17.2. PowerPartition I 17.2. symmetric group!powermap F 17.2. PartitionTuples F 17.2. NrPartitionTuples S 17.3. Fibonacci and Lucas Sequences F 17.3. Fibonacci I 17.3. sequence!Fibonacci F 17.3. Lucas I 17.3. sequence!Lucas S 17.4. Permanent of a Matrix F 17.4. Permanent C cyclotom.tex 18. Cyclotomic Numbers I 18.0. type!cyclotomic I 18.0. irrationalities I 18.0. cyclotomic field elements S 18.1. Operations for Cyclotomics I 18.1. roots of unity F 18.1. E F 18.1. Cyclotomics F 18.1. IsCyclotomic F 18.1. IsCyc F 18.1. IsIntegralCyclotomic I 18.1. Int!for cyclotomics I 18.1. String!for cyclotomics F 18.1. Conductor F 18.1. Conductor F 18.1. AbsoluteValue F 18.1. RoundCyc F 18.1. CoeffsCyc I 18.1. coefficients!for cyclotomics F 18.1. DenominatorCyc F 18.1. ExtRepOfObj!external representation!for cyclotomics F 18.1. DescriptionOfRootOfUnity I 18.1. logarithm!of a root of unity F 18.1. IsGaussInt F 18.1. IsGaussRat I 18.1. DefaultField!for cyclotomics S 18.2. Infinity F 18.2. IsInfinity F 18.2. infinity S 18.3. Comparisons of Cyclotomics I 18.3. operators!for cyclotomics S 18.4. ATLAS Irrationalities I 18.4. atomic irrationalities I 18.4. b_N I 18.4. c_N I 18.4. d_N I 18.4. e_N I 18.4. f_N I 18.4. g_N I 18.4. h_N F 18.4. EB F 18.4. EC F 18.4. ED F 18.4. EE F 18.4. EF F 18.4. EG F 18.4. EH I 18.4. i_N I 18.4. r_N F 18.4. EI F 18.4. ER I 18.4. s_N I 18.4. t_N I 18.4. u_N I 18.4. v_N I 18.4. w_N I 18.4. x_N I 18.4. y_N F 18.4. EY F 18.4. EX F 18.4. EW F 18.4. EV F 18.4. EU F 18.4. ET F 18.4. ES I 18.4. j_N I 18.4. k_N I 18.4. l_N I 18.4. m_N F 18.4. EM F 18.4. EL F 18.4. EK F 18.4. EJ I 18.4. n_k F 18.4. NK F 18.4. AtlasIrrationality S 18.5. Galois Conjugacy of Cyclotomics F 18.5. GaloisCyc F 18.5. GaloisCyc F 18.5. ComplexConjugate F 18.5. RealPart F 18.5. ImaginaryPart F 18.5. StarCyc F 18.5. Quadratic F 18.5. GaloisMat F 18.5. RationalizedMat S 18.6. Internally Represented Cyclotomics C unknown.tex 19. Unknowns I 19.0. data type!unknown F 19.0. Unknown F 19.0. Unknown F 19.0. LargestUnknown F 19.0. IsUnknown C boolean.tex 20. Booleans I 20.0. type!boolean I 20.0. logical F 20.0. IsBool S 20.1. Fail F 20.1. fail S 20.2. Comparisons of Booleans F 20.2. comparisons!of booleans F 20.2. comparisons!of booleans F 20.2. ordering!booleans S 20.3. Operations for Booleans I 20.3. operations!for booleans I 20.3. logical operations F 20.3. or F 20.3. and F 20.3. and!for filters F 20.3. not C lists.tex 21. Lists I 21.0. Sets S 21.1. List Categories F 21.1. IsList F 21.1. IsDenseList F 21.1. IsHomogeneousList F 21.1. IsTable F 21.1. IsConstantTimeAccessList S 21.2. Basic Operations for Lists F 21.2. list element!operation F 21.2. list boundedness test!operation F 21.2. list assignment!operation F 21.2. list unbind!operation S 21.3. List Elements I 21.3. accessing!list elements F 21.3. list element!access F 21.3. sublist!access I 21.3. sublist F 21.3. sublist!operation S 21.4. List Assignment I 21.4. assignment!to a list F 21.4. list element!assignment F 21.4. sublist!assignment F 21.4. sublist assignment!operation F 21.4. Add F 21.4. Add F 21.4. Remove F 21.4. Remove F 21.4. COPY_LIST_ENTRIES F 21.4. Append S 21.5. IsBound and Unbind for Lists F 21.5. IsBound!for lists F 21.5. Unbind!for lists S 21.6. Identical Lists S 21.7. Duplication of Lists I 21.7. ShallowCopy!for lists I 21.7. StructuralCopy!for lists S 21.8. Membership Test for Lists I 21.8. in!for lists F 21.8. element test!for lists S 21.9. Enlarging Internally Represented Lists F 21.9. EmptyPlist F 21.9. ShrinkAllocationPlist S 21.10. Comparisons of Lists I 21.10. comparisons!of lists F 21.10. list equal!comparison F 21.10. list smaller!comparison S 21.11. Arithmetic for Lists I 21.11. operators!for lists S 21.12. Filters Controlling the Arithmetic Behaviour of Lists F 21.12. IsGeneralizedRowVector F 21.12. IsMultiplicativeGeneralizedRowVector F 21.12. IsListDefault F 21.12. NestingDepthA F 21.12. NestingDepthM S 21.13. Additive Arithmetic for Lists I 21.13. addition!list and non-list I 21.13. list and non-list!difference S 21.14. Multiplicative Arithmetic for Lists I 21.14. list and non-list!product I 21.14. list and non-list!quotient I 21.14. list and non-list!mod I 21.14. mod!lists I 21.14. list and non-list!left quotient S 21.15. Mutability Status and List Arithmetic F 21.15. ListWithIdenticalEntries S 21.16. Finding Positions in Lists F 21.16. Position F 21.16. Positions F 21.16. PositionsOp F 21.16. PositionCanonical F 21.16. PositionNthOccurrence F 21.16. PositionSorted F 21.16. PositionSorted F 21.16. PositionSet F 21.16. PositionSet F 21.16. PositionProperty F 21.16. PositionBound F 21.16. PositionNot F 21.16. PositionNonZero F 21.16. PositionSublist F 21.16. PositionSublist F 21.16. PositionFirstComponent S 21.17. Properties and Attributes for Lists F 21.17. IsMatchingSublist F 21.17. IsMatchingSublist F 21.17. IsDuplicateFree F 21.17. IsDuplicateFreeList I 21.17. duplicate free F 21.17. IsSortedList I 21.17. sorted list F 21.17. IsSSortedList F 21.17. IsSet I 21.17. strictly sorted list F 21.17. Length F 21.17. ConstantTimeAccessList S 21.18. Sorting Lists F 21.18. Sort F 21.18. Sort F 21.18. SortParallel F 21.18. SortParallel F 21.18. Sortex F 21.18. SortingPerm S 21.19. Sorted Lists and Sets I 21.19. sets I 21.19. multisets F 21.19. in!for strictly sorted lists F 21.19. IsEqualSet I 21.19. test!for set equality F 21.19. IsSubsetSet F 21.19. AddSet I 21.19. add!an element to a set F 21.19. RemoveSet I 21.19. remove!an element from a set F 21.19. UniteSet I 21.19. union!of sets F 21.19. IntersectSet I 21.19. intersection!of sets F 21.19. SubtractSet I 21.19. subtract!a set from another S 21.20. Operations for Lists F 21.20. Concatenation F 21.20. Concatenation I 21.20. concatenation!of lists F 21.20. Compacted F 21.20. Collected F 21.20. DuplicateFreeList F 21.20. Unique F 21.20. AsDuplicateFreeList F 21.20. Flat F 21.20. Reversed F 21.20. IsLexicographicallyLess F 21.20. Apply F 21.20. Perform F 21.20. PermListList F 21.20. Maximum F 21.20. Maximum F 21.20. Minimum F 21.20. Minimum F 21.20. MaximumList F 21.20. MinimumList F 21.20. Cartesian F 21.20. Cartesian F 21.20. Permuted F 21.20. List F 21.20. List F 21.20. List F 21.20. Filtered F 21.20. Filtered F 21.20. Number F 21.20. Number F 21.20. Number F 21.20. First F 21.20. ForAll F 21.20. ForAll F 21.20. ForAny F 21.20. ForAny F 21.20. Product F 21.20. Product F 21.20. Product F 21.20. Product F 21.20. Sum F 21.20. Sum F 21.20. Sum F 21.20. Sum F 21.20. Iterated F 21.20. ListN S 21.21. Advanced List Manipulations F 21.21. ListX F 21.21. SetX F 21.21. SumX F 21.21. ProductX S 21.22. Ranges I 21.22. range F 21.22. IsRange F 21.22. ConvertToRangeRep S 21.23. Enumerators F 21.23. IsQuickPositionList C blist.tex 22. Boolean Lists F 22.0. IsBlist S 22.1. Boolean Lists Representing Subsets F 22.1. BlistList F 22.1. ListBlist F 22.1. SizeBlist F 22.1. IsSubsetBlist S 22.2. Set Operations via Boolean Lists F 22.2. UnionBlist F 22.2. UnionBlist F 22.2. IntersectionBlist F 22.2. IntersectionBlist F 22.2. DifferenceBlist S 22.3. Function that Modify Boolean Lists F 22.3. UniteBlist F 22.3. UniteBlistList F 22.3. IntersectBlist F 22.3. SubtractBlist S 22.4. More about Boolean Lists C vector.tex 23. Row Vectors F 23.0. IsRowVector S 23.1. Operators for Row Vectors F 23.1. addition!vectors F 23.1. addition!scalar and vector F 23.1. addition!vector and scalar F 23.1. subtraction!vectors F 23.1. subtraction!scalar and vector F 23.1. subtraction!vector and scalar F 23.1. multiplication!scalar and vector F 23.1. multiplication!vector and scalar F 23.1. multiplication!vectors F 23.1. NormedRowVector S 23.2. Row Vectors over Finite Fields F 23.2. ConvertToVectorRep F 23.2. ConvertToVectorRep F 23.2. ConvertToVectorRep F 23.2. ConvertToVectorRepNC F 23.2. ConvertToVectorRepNC F 23.2. ConvertToVectorRepNC F 23.2. NumberFFVector S 23.3. Coefficient List Arithmetic F 23.3. AddRowVector F 23.3. AddCoeffs F 23.3. AddCoeffs F 23.3. AddCoeffs F 23.3. MultRowVector F 23.3. MultRowVector F 23.3. CoeffsMod S 23.4. Shifting and Trimming Coefficient Lists F 23.4. LeftShiftRowVector F 23.4. RightShiftRowVector F 23.4. ShrinkRowVector F 23.4. RemoveOuterCoeffs S 23.5. Functions for Coding Theory F 23.5. WeightVecFFE F 23.5. DistanceVecFFE F 23.5. DistancesDistributionVecFFEsVecFFE F 23.5. DistancesDistributionMatFFEVecFFE F 23.5. AClosestVectorCombinationsMatFFEVecFFE F 23.5. AClosestVectorCombinationsMatFFEVecFFECoords F 23.5. CosetLeadersMatFFE S 23.6. Vectors as coefficients of polynomials F 23.6. ValuePol F 23.6. ProductCoeffs F 23.6. ReduceCoeffs F 23.6. ReduceCoeffsMod F 23.6. PowerModCoeffs F 23.6. ShiftedCoeffs F 23.6. ShrinkCoeffs C matrix.tex 24. Matrices F 24.0. InfoMatrix S 24.1. Categories of Matrices F 24.1. IsMatrix F 24.1. IsOrdinaryMatrix F 24.1. IsLieMatrix S 24.2. Operators for Matrices F 24.2. addition!matrices F 24.2. addition!scalar and matrix F 24.2. addition!matrix and scalar F 24.2. subtraction!matrices F 24.2. subtraction!scalar and matrix F 24.2. subtraction!matrix and scalar F 24.2. multiplication!scalar and matrix F 24.2. multiplication!matrix and scalar F 24.2. multiplication!vector and matrix F 24.2. multiplication!matrix and vector F 24.2. multiplication!matrices F 24.2. inverse!matrix F 24.2. quotient!matrices F 24.2. quotient!scalar and matrix F 24.2. quotient!matrix and scalar F 24.2. quotient!vector and matrix F 24.2. power!matrix F 24.2. conjugate!matrix F 24.2. image!vector under matrix F 24.2. matrices!commutator F 24.2. addition!scalar and matrix list F 24.2. addition!scalar and matrix list F 24.2. subtraction!scalar and matrix list F 24.2. subtraction!scalar and matrix list F 24.2. multiplication!scalar and matrix list F 24.2. multiplication!scalar and matrix list F 24.2. quotient!scalar and matrix list F 24.2. multiplication!matrix and matrix list F 24.2. multiplication!matrix and matrix list F 24.2. quotient!matrix and matrix list F 24.2. multiplication!vector and matrix list S 24.3. Properties and Attributes of Matrices F 24.3. DimensionsMat F 24.3. DefaultFieldOfMatrix I 24.3. Trace!of a matrix F 24.3. TraceMat F 24.3. Trace F 24.3. DeterminantMat F 24.3. Determinant F 24.3. DeterminantMatDestructive F 24.3. DeterminantMatDivFree F 24.3. IsMonomialMatrix F 24.3. IsDiagonalMat F 24.3. IsUpperTriangularMat F 24.3. IsLowerTriangularMat S 24.4. Matrix Constructions F 24.4. IdentityMat F 24.4. NullMat F 24.4. EmptyMatrix F 24.4. DiagonalMat F 24.4. PermutationMat F 24.4. TransposedMatImmutable F 24.4. TransposedMatAttr F 24.4. TransposedMat F 24.4. TransposedMatMutable F 24.4. TransposedMatOp F 24.4. TransposedMatDestructive F 24.4. KroneckerProduct F 24.4. ReflectionMat F 24.4. ReflectionMat F 24.4. ReflectionMat F 24.4. ReflectionMat F 24.4. PrintArray F 24.4. MutableIdentityMat F 24.4. MutableNullMat F 24.4. MutableCopyMat S 24.5. Random Matrices F 24.5. RandomMat F 24.5. RandomInvertibleMat F 24.5. RandomUnimodularMat S 24.6. Matrices Representing Linear Equations and the Gaussian Algorithm I 24.6. Gaussian algorithm F 24.6. RankMat F 24.6. TriangulizeMat F 24.6. NullspaceMat F 24.6. TriangulizedNullspaceMat F 24.6. NullspaceMatDestructive F 24.6. TriangulizedNullspaceMatDestructive F 24.6. SolutionMat F 24.6. SolutionMatDestructive F 24.6. BaseFixedSpace S 24.7. Eigenvectors and eigenvalues F 24.7. GeneralisedEigenvalues F 24.7. GeneralizedEigenvalues F 24.7. GeneralisedEigenspaces F 24.7. GeneralizedEigenspaces F 24.7. Eigenvalues F 24.7. Eigenspaces F 24.7. Eigenvectors S 24.8. Elementary Divisors F 24.8. ElementaryDivisorsMat F 24.8. ElementaryDivisorsMatDestructive F 24.8. DiagonalizeMat S 24.9. Echelonized Matrices F 24.9. SemiEchelonMat F 24.9. SemiEchelonMatDestructive F 24.9. SemiEchelonMatTransformation F 24.9. SemiEchelonMats F 24.9. SemiEchelonMatsDestructive S 24.10. Matrices as Basis of a Row Space F 24.10. BaseMat F 24.10. BaseMatDestructive F 24.10. BaseOrthogonalSpaceMat F 24.10. SumIntersectionMat F 24.10. BaseSteinitzVectors S 24.11. Triangular Matrices F 24.11. DiagonalOfMat F 24.11. UpperSubdiagonal F 24.11. DepthOfUpperTriangularMatrix S 24.12. Matrices as Linear Mappings F 24.12. CharacteristicPolynomial F 24.12. CharacteristicPolynomial F 24.12. JordanDecomposition F 24.12. BlownUpMat F 24.12. BlownUpVector F 24.12. CompanionMat S 24.13. Matrices over Finite Fields F 24.13. ImmutableMatrix F 24.13. ConvertToMatrixRep F 24.13. ConvertToMatrixRep F 24.13. ConvertToMatrixRep F 24.13. ConvertToMatrixRepNC F 24.13. ConvertToMatrixRepNC F 24.13. ConvertToMatrixRepNC F 24.13. ProjectiveOrder F 24.13. SimultaneousEigenvalues F 24.13. InverseMatMod F 24.13. NullspaceModQ S 24.14. Special Multiplication Algorithms for Matrices over GF(2) F 24.14. PROD_GF2MAT_GF2MAT_SIMPLE F 24.14. PROD_GF2MAT_GF2MAT_ADVANCED S 24.15. Block Matrices I 24.15. IsBlockMatrixRep F 24.15. AsBlockMatrix F 24.15. BlockMatrix F 24.15. BlockMatrix F 24.15. MatrixByBlockMatrix C matint.tex 25. Integral matrices and lattices S 25.1. Linear equations over the integers and Integral Matrices F 25.1. NullspaceIntMat F 25.1. SolutionIntMat F 25.1. SolutionNullspaceIntMat F 25.1. BaseIntMat F 25.1. BaseIntersectionIntMats F 25.1. ComplementIntMat S 25.2. Normal Forms over the Integers F 25.2. TriangulizedIntegerMat F 25.2. TriangulizedIntegerMatTransform F 25.2. TriangulizeIntegerMat F 25.2. HermiteNormalFormIntegerMat F 25.2. HermiteNormalFormIntegerMatTransform F 25.2. SmithNormalFormIntegerMat F 25.2. SmithNormalFormIntegerMatTransforms F 25.2. DiagonalizeIntMat F 25.2. NormalFormIntMat F 25.2. AbelianInvariantsOfList S 25.3. Determinant of an integer matrix F 25.3. DeterminantIntMat S 25.4. Decompositions I 25.4. decomposition matrix I 25.4. DEC F 25.4. Decomposition F 25.4. Decomposition F 25.4. LinearIndependentColumns F 25.4. PadicCoefficients F 25.4. IntegralizedMat F 25.4. IntegralizedMat F 25.4. DecompositionInt S 25.5. Lattice Reduction I 25.5. LLL algorithm!for vectors I 25.5. short vectors spanning a lattice I 25.5. lattice base reduction F 25.5. LLLReducedBasis I 25.5. LLL algorithm!for Gram matrices I 25.5. lattice base reduction F 25.5. LLLReducedGramMat F 25.5. LLLReducedGramMat S 25.6. Orthogonal Embeddings F 25.6. OrthogonalEmbeddings F 25.6. ShortestVectors C string.tex 26. Strings and Characters I 26.0. type!strings I 26.0. doublequotes I 26.0. singlequotes F 26.0. IsChar F 26.0. IsCharCollection F 26.0. IsString S 26.1. Special Characters I 26.1. escaped characters I 26.1. special character sequences I 26.1. \\n I 26.1. newline character I 26.1. \\\" I 26.1. doublequote character I 26.1. \\' I 26.1. singlequote character I 26.1. \\\\ I 26.1. backslash character I 26.1. \\b I 26.1. backspace character I 26.1. \\r I 26.1. carriage return character I 26.1. \\c I 26.1. flush character I 26.1. \\XYZ I 26.1. octal character codes I 26.1. escaping non-special characters S 26.2. Internally Represented Strings I 26.2. convert!to a string F 26.2. IsStringRep F 26.2. ConvertToStringRep F 26.2. IsEmptyString F 26.2. EmptyString F 26.2. ShrinkAllocationString F 26.2. CharsFamily S 26.3. Recognizing Characters F 26.3. IsDigitChar F 26.3. IsLowerAlphaChar F 26.3. IsUpperAlphaChar F 26.3. IsAlphaChar S 26.4. Comparisons of Strings F 26.4. strings!equality of F 26.4. strings!inequality of F 26.4. strings!lexicographic ordering of S 26.5. Operations to Produce or Manipulate Strings F 26.5. String F 26.5. String F 26.5. HexStringInt F 26.5. StringPP F 26.5. WordAlp F 26.5. LowercaseString F 26.5. SplitString F 26.5. ReplacedString F 26.5. NormalizeWhitespace F 26.5. NormalizedWhitespace F 26.5. RemoveCharacters F 26.5. JoinStringsWithSeparator F 26.5. Chomp S 26.6. Character Conversion F 26.6. INT_CHAR F 26.6. CHAR_INT F 26.6. SINT_CHAR F 26.6. CHAR_SINT S 26.7. Operations to Evaluate Strings I 26.7. evaluation!strings F 26.7. Int!for strings F 26.7. Rat!for strings F 26.7. IntHexString F 26.7. Ordinal F 26.7. EvalString S 26.8. Calendar Arithmetic F 26.8. DaysInYear F 26.8. DaysInMonth F 26.8. DMYDay F 26.8. DayDMY F 26.8. WeekDay F 26.8. StringDate F 26.8. HMSMSec F 26.8. SecHMSM F 26.8. StringTime F 26.8. SecondsDMYhms F 26.8. DMYhmsSeconds C record.tex 27. Records I 27.0. type!records F 27.0. IsRecord F 27.0. IsRecordCollection F 27.0. IsRecordCollColl I 27.0. test!for records F 27.0. RecNames S 27.1. Accessing Record Elements I 27.1. accessing!record elements F 27.1. record!component access F 27.1. record!component variable S 27.2. Record Assignment I 27.2. assignment!to a record F 27.2. record!component assignment F 27.2. record!component variable assignment S 27.3. Identical Records S 27.4. Comparisons of Records F 27.4. equality!of records F 27.4. inequality!of records F 27.4. ordering!of records F 27.4. ordering!of records S 27.5. IsBound and Unbind for Records S 27.6. Record Access Operations F 27.6. NameRNam F 27.6. RNamObj F 27.6. RNamObj F 27.6. record component!operation F 27.6. record boundness test!operation F 27.6. record assignment!operation F 27.6. record unbind!operation C coll.tex 28. Collections F 28.0. IsCollection S 28.1. Collection Families F 28.1. CollectionsFamily F 28.1. IsCollectionFamily F 28.1. ElementsFamily F 28.1. CategoryCollections S 28.2. Lists and Collections I 28.2. Sorted Lists as Collections F 28.2. IsListOrCollection F 28.2. Enumerator F 28.2. Enumerator F 28.2. EnumeratorSorted F 28.2. EnumeratorSorted F 28.2. EnumeratorByFunctions F 28.2. EnumeratorByFunctions F 28.2. SortedList F 28.2. SortedList F 28.2. SSortedList F 28.2. SSortedList F 28.2. Set F 28.2. AsList F 28.2. AsList F 28.2. AsSortedList F 28.2. AsSortedList F 28.2. AsSSortedList F 28.2. AsSSortedList F 28.2. AsSet I 28.2. elements!of a list or collection F 28.2. Elements S 28.3. Attributes and Properties for Collections F 28.3. IsEmpty F 28.3. IsEmpty F 28.3. IsFinite I 28.3. finiteness test!for a list or collection F 28.3. IsTrivial F 28.3. IsNonTrivial F 28.3. IsWholeFamily F 28.3. Size F 28.3. Size I 28.3. size!of a list or collection I 28.3. order!of a list, collection or domain F 28.3. Representative I 28.3. representative!of a list or collection F 28.3. RepresentativeSmallest S 28.4. Operations for Collections F 28.4. IsSubset I 28.4. subset test!for collections F 28.4. Intersection F 28.4. Intersection F 28.4. Intersection2 I 28.4. intersection!of collections F 28.4. Union F 28.4. Union F 28.4. Union2 I 28.4. union!of collections F 28.4. Difference I 28.4. set difference!of collections S 28.5. Membership Test for Collections I 28.5. \\in!operation for testing membership F 28.5. in!for collections F 28.5. in!operation for S 28.6. Random Elements I 28.6. random element!of a list or collection F 28.6. Random![coll] F 28.6. Random![coll] F 28.6. StateRandom F 28.6. RestoreStateRandom F 28.6. PseudoRandom F 28.6. PseudoRandom F 28.6. RandomList I 28.6. random seed S 28.7. Iterators F 28.7. Iterator F 28.7. Iterator F 28.7. IteratorSorted F 28.7. IteratorSorted F 28.7. IsIterator F 28.7. IsDoneIterator F 28.7. NextIterator F 28.7. IteratorList F 28.7. TrivialIterator F 28.7. IteratorByFunctions C orders.tex 29. Orderings F 29.0. IsOrdering F 29.0. OrderingsFamily S 29.1. Building new orderings F 29.1. OrderingByLessThanFunctionNC F 29.1. OrderingByLessThanFunctionNC F 29.1. OrderingByLessThanOrEqualFunctionNC F 29.1. OrderingByLessThanOrEqualFunctionNC S 29.2. Properties and basic functionality F 29.2. IsWellFoundedOrdering F 29.2. IsTotalOrdering F 29.2. IsIncomparableUnder F 29.2. FamilyForOrdering F 29.2. LessThanFunction F 29.2. LessThanOrEqualFunction F 29.2. IsLessThanUnder F 29.2. IsLessThanOrEqualUnder S 29.3. Orderings on families of associative words F 29.3. IsOrderingOnFamilyOfAssocWords F 29.3. IsTranslationInvariantOrdering F 29.3. IsReductionOrdering F 29.3. OrderingOnGenerators F 29.3. LexicographicOrdering F 29.3. LexicographicOrdering F 29.3. LexicographicOrdering F 29.3. LexicographicOrdering F 29.3. LexicographicOrdering F 29.3. LexicographicOrdering F 29.3. ShortLexOrdering F 29.3. ShortLexOrdering F 29.3. ShortLexOrdering F 29.3. ShortLexOrdering F 29.3. ShortLexOrdering F 29.3. ShortLexOrdering F 29.3. IsShortLexOrdering F 29.3. WeightLexOrdering F 29.3. WeightLexOrdering F 29.3. WeightLexOrdering F 29.3. WeightLexOrdering F 29.3. IsWeightLexOrdering F 29.3. WeightOfGenerators F 29.3. BasicWreathProductOrdering F 29.3. BasicWreathProductOrdering F 29.3. BasicWreathProductOrdering F 29.3. BasicWreathProductOrdering F 29.3. BasicWreathProductOrdering F 29.3. BasicWreathProductOrdering F 29.3. IsBasicWreathProductOrdering F 29.3. WreathProductOrdering F 29.3. WreathProductOrdering F 29.3. WreathProductOrdering F 29.3. WreathProductOrdering F 29.3. WreathProductOrdering F 29.3. WreathProductOrdering F 29.3. IsWreathProductOrdering F 29.3. LevelsOfGenerators C domain.tex 30. Domains and their Elements S 30.1. Operational Structure of Domains S 30.2. Equality and Comparison of Domains S 30.3. Constructing Domains F 30.3. Struct F 30.3. IsGeneratorsOfStruct F 30.3. GeneratorsOfStruct F 30.3. StructByGenerators F 30.3. StructWithGenerators F 30.3. ClosureStruct S 30.4. Changing the Structure F 30.4. AsStruct S 30.5. Changing the Representation F 30.5. IsomorphismRepStruct S 30.6. Domain Categories F 30.6. IsStruct S 30.7. Parents F 30.7. Parent F 30.7. SetParent F 30.7. HasParent S 30.8. Constructing Subdomains I 30.8. Subdomains F 30.8. Substruct F 30.8. SubstructNC F 30.8. AsSubstruct F 30.8. IsSubstruct S 30.9. Operations for Domains F 30.9. IsGeneralizedDomain F 30.9. IsDomain F 30.9. GeneratorsOfDomain F 30.9. Domain F 30.9. DomainByGenerators S 30.10. Attributes and Properties of Elements F 30.10. Characteristic F 30.10. OneImmutable F 30.10. OneAttr F 30.10. One F 30.10. Identity F 30.10. OneMutable F 30.10. OneOp F 30.10. OneSameMutability F 30.10. OneSM F 30.10. ZeroImmutable F 30.10. ZeroAttr F 30.10. Zero F 30.10. ZeroMutable F 30.10. ZeroOp F 30.10. ZeroSameMutability F 30.10. ZeroSM F 30.10. MultiplicativeZeroOp F 30.10. IsOne F 30.10. IsZero F 30.10. IsIdempotent F 30.10. InverseImmutable F 30.10. InverseAttr F 30.10. Inverse F 30.10. InverseMutable F 30.10. InverseOp F 30.10. InverseSameMutability F 30.10. InverseSM F 30.10. AdditiveInverseImmutable F 30.10. AdditiveInverseAttr F 30.10. AdditiveInverse F 30.10. AdditiveInverseMutable F 30.10. AdditiveInverseOp F 30.10. AdditiveInverseSameMutability F 30.10. AdditiveInverseSM F 30.10. Order S 30.11. Comparison Operations for Elements F 30.11. equality!operation F 30.11. comparison!operation F 30.11. CanEasilyCompareElements F 30.11. CanEasilyCompareElementsFamily F 30.11. CanEasilySortElements F 30.11. CanEasilySortElementsFamily S 30.12. Arithmetic Operations for Elements F 30.12. addition!operation F 30.12. multiplication!operation F 30.12. division!operation F 30.12. exponentiation!operation F 30.12. remainder!operation F 30.12. LeftQuotient F 30.12. Comm F 30.12. LieBracket F 30.12. Sqrt S 30.13. Relations Between Domains F 30.13. UseSubsetRelation F 30.13. UseIsomorphismRelation F 30.13. UseFactorRelation F 30.13. InstallSubsetMaintenance F 30.13. InstallIsomorphismMaintenance F 30.13. InstallFactorMaintenance S 30.14. Useful Categories of Elements F 30.14. IsExtAElement F 30.14. IsNearAdditiveElement F 30.14. IsAdditiveElement F 30.14. IsNearAdditiveElementWithZero F 30.14. IsAdditiveElementWithZero F 30.14. IsNearAdditiveElementWithInverse F 30.14. IsAdditiveElementWithInverse F 30.14. IsExtLElement F 30.14. IsExtRElement F 30.14. IsMultiplicativeElement F 30.14. IsMultiplicativeElementWithOne F 30.14. IsMultiplicativeElementWithZero F 30.14. IsMultiplicativeElementWithInverse F 30.14. IsVector F 30.14. IsNearRingElement F 30.14. IsRingElement F 30.14. IsNearRingElementWithOne F 30.14. IsRingElementWithOne F 30.14. IsNearRingElementWithInverse F 30.14. IsRingElementWithInverse F 30.14. IsScalar S 30.15. Useful Categories for all Elements of a Family F 30.15. IsAssociativeElement F 30.15. IsAssociativeElementCollection F 30.15. IsAssociativeElementCollColl F 30.15. IsAdditivelyCommutativeElement F 30.15. IsAdditivelyCommutativeElementCollection F 30.15. IsAdditivelyCommutativeElementCollColl F 30.15. IsAdditivelyCommutativeElementFamily F 30.15. IsCommutativeElement F 30.15. IsCommutativeElementCollection F 30.15. IsCommutativeElementCollColl F 30.15. IsFiniteOrderElement F 30.15. IsFiniteOrderElementCollection F 30.15. IsFiniteOrderElementCollColl F 30.15. IsJacobianElement F 30.15. IsJacobianElementCollection F 30.15. IsJacobianElementCollColl F 30.15. IsZeroSquaredElement F 30.15. IsZeroSquaredElementCollection F 30.15. IsZeroSquaredElementCollColl C mapping.tex 31. Mappings I 31.0. functions I 31.0. relations F 31.0. IsTuple S 31.1. Creating Mappings F 31.1. GeneralMappingByElements F 31.1. MappingByFunction F 31.1. MappingByFunction F 31.1. MappingByFunction F 31.1. InverseGeneralMapping F 31.1. CompositionMapping F 31.1. CompositionMapping2 F 31.1. IsCompositionMappingRep F 31.1. ConstituentsCompositionMapping F 31.1. ZeroMapping F 31.1. IdentityMapping F 31.1. Embedding F 31.1. Embedding F 31.1. Projection F 31.1. Projection F 31.1. Projection F 31.1. RestrictedMapping S 31.2. Properties and Attributes of (General) Mappings F 31.2. IsTotal F 31.2. IsSingleValued F 31.2. IsMapping F 31.2. IsInjective F 31.2. IsSurjective F 31.2. IsBijective F 31.2. Range F 31.2. Source F 31.2. UnderlyingRelation F 31.2. UnderlyingGeneralMapping S 31.3. Images under Mappings F 31.3. ImagesSource F 31.3. ImagesRepresentative F 31.3. ImagesElm F 31.3. ImagesSet F 31.3. ImageElm F 31.3. Image F 31.3. Image F 31.3. Image F 31.3. Images F 31.3. Images F 31.3. Images S 31.4. Preimages under Mappings F 31.4. PreImagesRange F 31.4. PreImagesElm F 31.4. PreImageElm F 31.4. PreImagesRepresentative F 31.4. PreImagesSet F 31.4. PreImage F 31.4. PreImage F 31.4. PreImage F 31.4. PreImages F 31.4. PreImages F 31.4. PreImages S 31.5. Arithmetic Operations for General Mappings S 31.6. Mappings which are Compatible with Algebraic Structures S 31.7. Magma Homomorphisms F 31.7. IsMagmaHomomorphism F 31.7. MagmaHomomorphismByFunctionNC F 31.7. NaturalHomomorphismByGenerators S 31.8. Mappings that Respect Multiplication F 31.8. RespectsMultiplication F 31.8. RespectsOne F 31.8. RespectsInverses F 31.8. IsGroupGeneralMapping F 31.8. IsGroupHomomorphism F 31.8. KernelOfMultiplicativeGeneralMapping F 31.8. CoKernelOfMultiplicativeGeneralMapping S 31.9. Mappings that Respect Addition F 31.9. RespectsAddition F 31.9. RespectsAdditiveInverses F 31.9. RespectsZero F 31.9. IsAdditiveGroupGeneralMapping F 31.9. IsAdditiveGroupHomomorphism F 31.9. KernelOfAdditiveGeneralMapping F 31.9. CoKernelOfAdditiveGeneralMapping S 31.10. Linear Mappings F 31.10. RespectsScalarMultiplication F 31.10. IsLeftModuleGeneralMapping F 31.10. IsLeftModuleHomomorphism F 31.10. IsLinearMapping S 31.11. Ring Homomorphisms F 31.11. IsRingGeneralMapping F 31.11. IsRingHomomorphism F 31.11. IsRingWithOneGeneralMapping F 31.11. IsRingWithOneHomomorphism F 31.11. IsAlgebraGeneralMapping F 31.11. IsAlgebraHomomorphism F 31.11. IsAlgebraWithOneGeneralMapping F 31.11. IsAlgebraWithOneHomomorphism F 31.11. IsFieldHomomorphism S 31.12. General Mappings F 31.12. IsGeneralMapping F 31.12. IsConstantTimeAccessGeneralMapping F 31.12. IsEndoGeneralMapping S 31.13. Technical Matters Concerning General Mappings F 31.13. IsSPGeneralMapping F 31.13. IsNonSPGeneralMapping F 31.13. IsGeneralMappingFamily F 31.13. FamilyRange F 31.13. FamilySource F 31.13. FamiliesOfGeneralMappingsAndRanges F 31.13. GeneralMappingsFamily F 31.13. TypeOfDefaultGeneralMapping C relation.tex 32. Relations I 32.0. binary relation I 32.0. IsBinaryRelation!same as IsEndoGeneralMapping I 32.0. IsEndoGeneralMapping!same as IsBinaryRelation S 32.1. General Binary Relations F 32.1. IsBinaryRelation F 32.1. BinaryRelationByElements F 32.1. IdentityBinaryRelation F 32.1. IdentityBinaryRelation F 32.1. EmptyBinaryRelation F 32.1. EmptyBinaryRelation S 32.2. Properties and Attributes of Binary Relations F 32.2. IsReflexiveBinaryRelation I 32.2. reflexive relation F 32.2. IsSymmetricBinaryRelation I 32.2. symmetric relation F 32.2. IsTransitiveBinaryRelation I 32.2. transitive relation F 32.2. IsAntisymmetricBinaryRelation I 32.2. antisymmetric relation F 32.2. IsPreOrderBinaryRelation I 32.2. preorder F 32.2. IsPartialOrderBinaryRelation I 32.2. partial order F 32.2. IsHasseDiagram F 32.2. IsEquivalenceRelation I 32.2. equivalence relation F 32.2. Successors F 32.2. DegreeOfBinaryRelation F 32.2. PartialOrderOfHasseDiagram S 32.3. Binary Relations on Points F 32.3. BinaryRelationOnPoints F 32.3. BinaryRelationOnPointsNC F 32.3. RandomBinaryRelationOnPoints F 32.3. AsBinaryRelationOnPoints F 32.3. AsBinaryRelationOnPoints F 32.3. AsBinaryRelationOnPoints S 32.4. Closure Operations and Other Constructors F 32.4. ReflexiveClosureBinaryRelation F 32.4. SymmetricClosureBinaryRelation F 32.4. TransitiveClosureBinaryRelation F 32.4. HasseDiagramBinaryRelation F 32.4. StronglyConnectedComponents F 32.4. PartialOrderByOrderingFunction S 32.5. Equivalence Relations I 32.5. equivalence relation F 32.5. EquivalenceRelationByPartition F 32.5. EquivalenceRelationByPartitionNC F 32.5. EquivalenceRelationByRelation F 32.5. EquivalenceRelationByPairs F 32.5. EquivalenceRelationByPairsNC F 32.5. EquivalenceRelationByProperty S 32.6. Attributes of and Operations on Equivalence Relations F 32.6. EquivalenceRelationPartition F 32.6. GeneratorsOfEquivalenceRelationPartition F 32.6. JoinEquivalenceRelations F 32.6. MeetEquivalenceRelations S 32.7. Equivalence Classes F 32.7. IsEquivalenceClass I 32.7. equivalence class F 32.7. EquivalenceClassRelation F 32.7. EquivalenceClasses!attribute F 32.7. EquivalenceClassOfElement F 32.7. EquivalenceClassOfElementNC C magma.tex 33. Magmas S 33.1. Magma Categories F 33.1. IsMagma F 33.1. IsMagmaWithOne F 33.1. IsMagmaWithInversesIfNonzero F 33.1. IsMagmaWithInverses S 33.2. Magma Generation F 33.2. Magma F 33.2. Magma F 33.2. MagmaWithOne F 33.2. MagmaWithOne F 33.2. MagmaWithInverses F 33.2. MagmaWithInverses F 33.2. MagmaByGenerators F 33.2. MagmaByGenerators F 33.2. MagmaWithOneByGenerators F 33.2. MagmaWithOneByGenerators F 33.2. MagmaWithInversesByGenerators F 33.2. MagmaWithInversesByGenerators F 33.2. Submagma F 33.2. SubmagmaNC F 33.2. SubmagmaWithOne F 33.2. SubmagmaWithOneNC F 33.2. SubmagmaWithInverses F 33.2. SubmagmaWithInversesNC F 33.2. AsMagma F 33.2. AsSubmagma F 33.2. InjectionZeroMagma S 33.3. Magmas Defined by Multiplication Tables F 33.3. MagmaByMultiplicationTable F 33.3. MagmaWithOneByMultiplicationTable F 33.3. MagmaWithInversesByMultiplicationTable F 33.3. MagmaElement F 33.3. MultiplicationTable F 33.3. MultiplicationTable S 33.4. Attributes and Properties for Magmas F 33.4. GeneratorsOfMagma F 33.4. GeneratorsOfMagmaWithOne F 33.4. GeneratorsOfMagmaWithInverses I 33.4. centraliser I 33.4. center F 33.4. Centralizer F 33.4. Centralizer F 33.4. Centralizer F 33.4. Centre F 33.4. Center F 33.4. Idempotents F 33.4. IsAssociative F 33.4. IsCentral F 33.4. IsCommutative F 33.4. IsAbelian F 33.4. MultiplicativeNeutralElement F 33.4. MultiplicativeZero F 33.4. IsMultiplicativeZero F 33.4. SquareRoots F 33.4. TrivialSubmagmaWithOne C word.tex 34. Words S 34.1. Categories of Words and Nonassociative Words I 34.1. abstract word F 34.1. IsWord F 34.1. IsWordWithOne F 34.1. IsWordWithInverse F 34.1. IsWordCollection F 34.1. IsNonassocWord F 34.1. IsNonassocWordWithOne F 34.1. IsNonassocWordCollection F 34.1. IsNonassocWordWithOneCollection S 34.2. Comparison of Words F 34.2. equality!nonassociative words F 34.2. smaller!nonassociative words S 34.3. Operations for Words F 34.3. MappedWord S 34.4. Free Magmas F 34.4. FreeMagma F 34.4. FreeMagma F 34.4. FreeMagma F 34.4. FreeMagma F 34.4. FreeMagma F 34.4. FreeMagmaWithOne F 34.4. FreeMagmaWithOne F 34.4. FreeMagmaWithOne F 34.4. FreeMagmaWithOne F 34.4. FreeMagmaWithOne S 34.5. External Representation for Nonassociative Words C wordass.tex 35. Associative Words S 35.1. Categories of Associative Words F 35.1. IsAssocWord F 35.1. IsAssocWordWithOne F 35.1. IsAssocWordWithInverse S 35.2. Free Groups, Monoids and Semigroups F 35.2. FreeGroup F 35.2. FreeGroup F 35.2. FreeGroup F 35.2. FreeGroup F 35.2. FreeGroup F 35.2. IsFreeGroup F 35.2. FreeMonoid!with example F 35.2. FreeMonoid!with example F 35.2. FreeMonoid!with example F 35.2. FreeMonoid!with example F 35.2. FreeMonoid!with example F 35.2. FreeSemigroup F 35.2. FreeSemigroup F 35.2. FreeSemigroup F 35.2. FreeSemigroup F 35.2. FreeSemigroup F 35.2. AssignGeneratorVariables S 35.3. Comparison of Associative Words F 35.3. equality!associative words F 35.3. smaller!associative words F 35.3. IsShortLexLessThanOrEqual F 35.3. IsBasicWreathLessThanOrEqual S 35.4. Operations for Associative Words I 35.4. product!of words I 35.4. quotient!of words I 35.4. power!of words I 35.4. conjugate!of a word I 35.4. Comm!for words I 35.4. LeftQuotient!for words I 35.4. length!of a word F 35.4. Length!of an associative word F 35.4. ExponentSumWord F 35.4. Subword F 35.4. PositionWord F 35.4. SubstitutedWord F 35.4. SubstitutedWord F 35.4. EliminatedWord S 35.5. Operations for Associative Words by their Syllables F 35.5. NumberSyllables F 35.5. ExponentSyllable F 35.5. GeneratorSyllable F 35.5. SubSyllables S 35.6. Representations for Associative Words F 35.6. IsLetterAssocWordRep F 35.6. IsLetterWordsFamily F 35.6. IsBLetterAssocWordRep F 35.6. IsWLetterAssocWordRep F 35.6. IsBLetterWordsFamily F 35.6. IsWLetterWordsFamily F 35.6. IsSyllableAssocWordRep F 35.6. IsSyllableWordsFamily F 35.6. Is8BitsFamily F 35.6. Is16BitsFamily F 35.6. Is32BitsFamily F 35.6. IsInfBitsFamily F 35.6. LetterRepAssocWord F 35.6. LetterRepAssocWord F 35.6. AssocWordByLetterRep S 35.7. The External Representation for Associative Words S 35.8. Straight Line Programs F 35.8. IsStraightLineProgram F 35.8. StraightLineProgram F 35.8. StraightLineProgram F 35.8. StraightLineProgramNC F 35.8. StraightLineProgramNC F 35.8. LinesOfStraightLineProgram F 35.8. NrInputsOfStraightLineProgram F 35.8. ResultOfStraightLineProgram F 35.8. StringOfResultOfStraightLineProgram F 35.8. CompositionOfStraightLinePrograms F 35.8. IntegratedStraightLineProgram F 35.8. RestrictOutputsOfSLP F 35.8. IntermediateResultOfSLP F 35.8. IntermediateResultOfSLPWithoutOverwrite F 35.8. IntermediateResultsOfSLPWithoutOverwrite F 35.8. ProductOfStraightLinePrograms S 35.9. Straight Line Program Elements F 35.9. IsStraightLineProgElm F 35.9. StraightLineProgElm F 35.9. StraightLineProgGens F 35.9. EvalStraightLineProgElm F 35.9. StretchImportantSLPElement C rws.tex 36. Rewriting Systems S 36.1. Operations on rewriting systems F 36.1. IsRewritingSystem F 36.1. Rules F 36.1. OrderOfRewritingSystem F 36.1. OrderingOfRewritingSystem F 36.1. ReducedForm F 36.1. IsConfluent F 36.1. IsConfluent F 36.1. ConfluentRws F 36.1. IsReduced F 36.1. ReduceRules F 36.1. AddRule F 36.1. AddRuleReduced F 36.1. MakeConfluent F 36.1. GeneratorsOfRws F 36.1. AddGenerators S 36.2. Operations on elements of the algebra F 36.2. ReducedProduct F 36.2. ReducedSum F 36.2. ReducedOne F 36.2. ReducedAdditiveInverse F 36.2. ReducedComm F 36.2. ReducedConjugate F 36.2. ReducedDifference F 36.2. ReducedInverse F 36.2. ReducedLeftQuotient F 36.2. ReducedPower F 36.2. ReducedQuotient F 36.2. ReducedScalarProduct F 36.2. ReducedZero S 36.3. Properties of rewriting systems F 36.3. IsBuiltFromAdditiveMagmaWithInverses F 36.3. IsBuiltFromMagma F 36.3. IsBuiltFromMagmaWithOne F 36.3. IsBuiltFromMagmaWithInverses F 36.3. IsBuiltFromSemigroup F 36.3. IsBuiltFromGroup S 36.4. Rewriting in Groups and Monoids S 36.5. Developing rewriting systems C groups.tex 37. Groups S 37.1. Group Elements I 37.1. order! of a group S 37.2. Creating Groups F 37.2. Group F 37.2. Group F 37.2. Group F 37.2. GroupWithGenerators F 37.2. GroupWithGenerators F 37.2. GeneratorsOfGroup F 37.2. AsGroup F 37.2. ConjugateGroup F 37.2. IsGroup F 37.2. InfoGroup S 37.3. Subgroups F 37.3. Subgroup F 37.3. SubgroupNC F 37.3. Index F 37.3. IndexNC F 37.3. IndexInWholeGroup F 37.3. AsSubgroup F 37.3. IsSubgroup F 37.3. IsNormal F 37.3. IsCharacteristicSubgroup F 37.3. ConjugateSubgroup F 37.3. ConjugateSubgroups F 37.3. IsSubnormal F 37.3. SubgroupByProperty F 37.3. SubgroupShell S 37.4. Closures of (Sub)groups F 37.4. ClosureGroup F 37.4. ClosureGroupAddElm F 37.4. ClosureGroupCompare F 37.4. ClosureGroupIntest F 37.4. ClosureGroupDefault F 37.4. ClosureSubgroup F 37.4. ClosureSubgroupNC S 37.5. Expressing Group Elements as Words in Generators I 37.5. factorization I 37.5. words!in generators F 37.5. EpimorphismFromFreeGroup F 37.5. Factorization S 37.6. Structure Descriptions F 37.6. StructureDescription S 37.7. Cosets I 37.7. right cosets I 37.7. coset F 37.7. RightCoset F 37.7. RightCosets F 37.7. RightCosetsNC F 37.7. CanonicalRightCosetElement F 37.7. IsRightCoset I 37.7. left cosets S 37.8. Transversals F 37.8. RightTransversal S 37.9. Double Cosets F 37.9. DoubleCoset F 37.9. RepresentativesContainedRightCosets F 37.9. DoubleCosets!operation F 37.9. DoubleCosetsNC!operation F 37.9. IsDoubleCoset F 37.9. DoubleCosetRepsAndSizes F 37.9. InfoCoset S 37.10. Conjugacy Classes F 37.10. ConjugacyClass F 37.10. ConjugacyClasses!attribute F 37.10. ConjugacyClassesByRandomSearch F 37.10. ConjugacyClassesByOrbits F 37.10. NrConjugacyClasses F 37.10. RationalClass F 37.10. RationalClasses F 37.10. GaloisGroup!of rational class of a group F 37.10. IsConjugate F 37.10. IsConjugate S 37.11. Normal Structure I 37.11. normalizer F 37.11. Normalizer F 37.11. Normalizer F 37.11. Core F 37.11. PCore I 37.11. O_p(G)!see PCore F 37.11. NormalClosure F 37.11. NormalIntersection F 37.11. Complementclasses F 37.11. InfoComplement S 37.12. Specific and Parametrized Subgroups F 37.12. TrivialSubgroup F 37.12. CommutatorSubgroup F 37.12. DerivedSubgroup F 37.12. CommutatorLength F 37.12. FittingSubgroup F 37.12. FrattiniSubgroup F 37.12. PrefrattiniSubgroup F 37.12. PerfectResiduum F 37.12. RadicalGroup F 37.12. Socle F 37.12. SupersolvableResiduum F 37.12. PRump S 37.13. Sylow Subgroups and Hall Subgroups F 37.13. SylowSubgroup F 37.13. SylowComplement F 37.13. HallSubgroup F 37.13. SylowSystem F 37.13. ComplementSystem F 37.13. HallSystem S 37.14. Subgroups characterized by prime powers F 37.14. Omega F 37.14. Agemo S 37.15. Group Properties F 37.15. IsCyclic F 37.15. IsElementaryAbelian F 37.15. IsNilpotentGroup F 37.15. NilpotencyClassOfGroup F 37.15. IsPerfectGroup F 37.15. IsSolvableGroup F 37.15. IsPolycyclicGroup F 37.15. IsSupersolvableGroup F 37.15. IsMonomialGroup F 37.15. IsSimpleGroup F 37.15. IsomorphismTypeInfoFiniteSimpleGroup F 37.15. IsFinitelyGeneratedGroup F 37.15. IsSubsetLocallyFiniteGroup I 37.15. p-group F 37.15. IsPGroup F 37.15. PrimePGroup F 37.15. PClassPGroup F 37.15. RankPGroup F 37.15. IsPSolvable F 37.15. IsPNilpotent S 37.16. Numerical Group Attributes F 37.16. AbelianInvariants!for groups F 37.16. Exponent F 37.16. EulerianFunction S 37.17. Subgroup Series F 37.17. ChiefSeries F 37.17. ChiefSeriesThrough F 37.17. ChiefSeriesUnderAction F 37.17. SubnormalSeries F 37.17. CompositionSeries F 37.17. DisplayCompositionSeries F 37.17. DerivedSeriesOfGroup F 37.17. DerivedLength F 37.17. ElementaryAbelianSeries F 37.17. ElementaryAbelianSeriesLargeSteps F 37.17. ElementaryAbelianSeries F 37.17. InvariantElementaryAbelianSeries F 37.17. LowerCentralSeriesOfGroup F 37.17. UpperCentralSeriesOfGroup F 37.17. PCentralSeries F 37.17. JenningsSeries F 37.17. DimensionsLoewyFactors F 37.17. AscendingChain F 37.17. IntermediateGroup F 37.17. IntermediateSubgroups S 37.18. Factor Groups F 37.18. NaturalHomomorphismByNormalSubgroup F 37.18. NaturalHomomorphismByNormalSubgroupNC F 37.18. FactorGroup F 37.18. FactorGroupNC F 37.18. CommutatorFactorGroup F 37.18. MaximalAbelianQuotient F 37.18. HasAbelianFactorGroup F 37.18. HasElementaryAbelianFactorGroup F 37.18. CentralizerModulo S 37.19. Sets of Subgroups F 37.19. ConjugacyClassSubgroups F 37.19. IsConjugacyClassSubgroupsRep F 37.19. IsConjugacyClassSubgroupsByStabilizerRep F 37.19. ConjugacyClassesSubgroups F 37.19. ConjugacyClassesMaximalSubgroups F 37.19. MaximalSubgroupClassReps F 37.19. MaximalSubgroups F 37.19. NormalSubgroups F 37.19. MaximalNormalSubgroups F 37.19. MinimalNormalSubgroups S 37.20. Subgroup Lattice F 37.20. LatticeSubgroups F 37.20. ClassElementLattice F 37.20. MaximalSubgroupsLattice F 37.20. MinimalSupergroupsLattice F 37.20. RepresentativesPerfectSubgroups F 37.20. RepresentativesSimpleSubgroups F 37.20. ConjugacyClassesPerfectSubgroups F 37.20. Zuppos F 37.20. InfoLattice S 37.21. Specific Methods for Subgroup Lattice Computations F 37.21. LatticeByCyclicExtension F 37.21. InvariantSubgroupsElementaryAbelianGroup F 37.21. SubgroupsSolvableGroup F 37.21. SizeConsiderFunction F 37.21. ExactSizeConsiderFunction F 37.21. InfoPcSubgroup S 37.22. Special Generating Sets F 37.22. GeneratorsSmallest F 37.22. LargestElementGroup F 37.22. MinimalGeneratingSet F 37.22. SmallGeneratingSet F 37.22. IndependentGeneratorsOfAbelianGroup S 37.23. 1-Cohomology I 37.23. one cohomology I 37.23. cohomology I 37.23. cocycles F 37.23. OneCocycles F 37.23. OneCocycles F 37.23. OneCocycles F 37.23. OneCocycles F 37.23. OneCoboundaries F 37.23. OCOneCocycles F 37.23. ComplementclassesEA F 37.23. InfoCoh S 37.24. Schur Covers and Multipliers I 37.24. Darstellungsgruppe!see EpimorphismSchurCover F 37.24. EpimorphismSchurCover F 37.24. SchurCover F 37.24. AbelianInvariantsMultiplier I 37.24. Multiplier I 37.24. Schur multiplier F 37.24. Epicentre F 37.24. ExteriorCentre F 37.24. NonabelianExteriorSquare F 37.24. EpimorphismNonabelianExteriorSquare F 37.24. IsCentralFactor S 37.25. Tests for the Availability of Methods F 37.25. CanEasilyTestMembership F 37.25. CanComputeSize F 37.25. CanComputeSizeAnySubgroup F 37.25. CanComputeIndex F 37.25. CanComputeIsSubset F 37.25. KnowsHowToDecompose F 37.25. KnowsHowToDecompose C grphomom.tex 38. Group Homomorphisms S 38.1. Creating Group Homomorphisms F 38.1. GroupHomomorphismByImages F 38.1. GroupHomomorphismByImagesNC F 38.1. GroupGeneralMappingByImages F 38.1. GroupHomomorphismByFunction F 38.1. GroupHomomorphismByFunction F 38.1. GroupHomomorphismByFunction F 38.1. AsGroupGeneralMappingByImages S 38.2. Operations for Group Homomorphisms I 38.2. Inverse!group homomorphism S 38.3. Efficiency of Homomorphisms F 38.3. ImagesSmallestGenerators S 38.4. Homomorphism for very large groups S 38.5. Nice Monomorphisms F 38.5. IsHandledByNiceMonomorphism F 38.5. NiceMonomorphism F 38.5. NiceObject F 38.5. IsCanonicalNiceMonomorphism S 38.6. Group Automorphisms F 38.6. ConjugatorIsomorphism F 38.6. ConjugatorAutomorphism F 38.6. ConjugatorAutomorphismNC F 38.6. InnerAutomorphism F 38.6. InnerAutomorphismNC F 38.6. IsConjugatorIsomorphism F 38.6. IsConjugatorAutomorphism F 38.6. IsInnerAutomorphism F 38.6. ConjugatorOfConjugatorIsomorphism S 38.7. Groups of Automorphisms F 38.7. IsGroupOfAutomorphisms F 38.7. AutomorphismDomain F 38.7. AutomorphismGroup F 38.7. IsAutomorphismGroup F 38.7. InnerAutomorphismsAutomorphismGroup F 38.7. InducedAutomorphism S 38.8. Calculating with Group Automorphisms F 38.8. AssignNiceMonomorphismAutomorphismGroup F 38.8. NiceMonomorphismAutomGroup S 38.9. Searching for Homomorphisms I 38.9. homomorphisms!find all F 38.9. IsomorphismGroups I 38.9. isomorphisms!find all F 38.9. GQuotients I 38.9. epimorphisms!find all I 38.9. projections!find all F 38.9. IsomorphicSubgroups I 38.9. embeddings!find all I 38.9. monomorphisms!find all F 38.9. MorClassLoop S 38.10. Representations for Group Homomorphisms F 38.10. IsGroupGeneralMappingByImages F 38.10. IsGroupGeneralMappingByAsGroupGeneralMappingByImages F 38.10. IsPreimagesByAsGroupGeneralMappingByImages F 38.10. IsPermGroupGeneralMappingByImages F 38.10. IsPermGroupHomomorphismByImages F 38.10. IsToPermGroupGeneralMappingByImages F 38.10. IsToPermGroupHomomorphismByImages F 38.10. IsGroupGeneralMappingByPcgs F 38.10. IsPcGroupGeneralMappingByImages F 38.10. IsPcGroupHomomorphismByImages F 38.10. IsToPcGroupGeneralMappingByImages F 38.10. IsToPcGroupHomomorphismByImages F 38.10. IsFromFpGroupGeneralMappingByImages F 38.10. IsFromFpGroupHomomorphismByImages F 38.10. IsFromFpGroupStdGensGeneralMappingByImages F 38.10. IsFromFpGroupStdGensHomomorphismByImages C grpoper.tex 39. Group Actions I 39.0. group actions I 39.0. G-sets S 39.1. About Group Actions I 39.1. group actions!operations syntax S 39.2. Basic Actions I 39.2. group actions I 39.2. actions I 39.2. group operations F 39.2. OnPoints I 39.2. conjugation I 39.2. action!by conjugation F 39.2. OnRight F 39.2. OnLeftInverse F 39.2. OnSets I 39.2. action!on sets I 39.2. action!on blocks F 39.2. OnTuples F 39.2. OnPairs F 39.2. OnSetsSets F 39.2. OnSetsDisjointSets F 39.2. OnSetsTuples F 39.2. OnTuplesSets F 39.2. OnTuplesTuples F 39.2. OnLines F 39.2. OnIndeterminates!as a permutation action F 39.2. Permuted!as a permutation action F 39.2. OnSubspacesByCanonicalBasis S 39.3. Orbits F 39.3. Orbit F 39.3. Orbits!operation/attribute F 39.3. Orbits!operation/attribute F 39.3. OrbitsDomain F 39.3. OrbitsDomain F 39.3. OrbitLength F 39.3. OrbitLengths F 39.3. OrbitLengths F 39.3. OrbitLengthsDomain F 39.3. OrbitLengthsDomain S 39.4. Stabilizers I 39.4. point stabilizer I 39.4. set stabilizer I 39.4. tuple stabilizer F 39.4. OrbitStabilizer F 39.4. Stabilizer F 39.4. OrbitStabilizerAlgorithm S 39.5. Elements with Prescribed Images I 39.5. transporter F 39.5. RepresentativeAction S 39.6. The Permutation Image of an Action F 39.6. ActionHomomorphism F 39.6. ActionHomomorphism F 39.6. ActionHomomorphism F 39.6. Action F 39.6. Action I 39.6. regular action F 39.6. SparseActionHomomorphism F 39.6. SortedSparseActionHomomorphism S 39.7. Action of a group on itself F 39.7. FactorCosetAction F 39.7. RegularActionHomomorphism F 39.7. AbelianSubfactorAction S 39.8. Permutations Induced by Elements and Cycles F 39.8. Permutation F 39.8. Permutation F 39.8. PermutationCycle F 39.8. PermutationCycleOp F 39.8. Cycle F 39.8. CycleLength F 39.8. Cycles F 39.8. CycleLengths S 39.9. Tests for Actions F 39.9. IsTransitive!for group actions F 39.9. IsTransitive!for group actions I 39.9. transitive F 39.9. Transitivity!for group actions F 39.9. Transitivity!for group actions F 39.9. RankAction F 39.9. RankAction F 39.9. IsSemiRegular F 39.9. IsSemiRegular I 39.9. semiregular F 39.9. IsRegular F 39.9. IsRegular I 39.9. regular F 39.9. Earns F 39.9. Earns F 39.9. IsPrimitive F 39.9. IsPrimitive I 39.9. primitive S 39.10. Block Systems F 39.10. Blocks F 39.10. Blocks F 39.10. MaximalBlocks F 39.10. MaximalBlocks F 39.10. RepresentativesMinimalBlocks F 39.10. RepresentativesMinimalBlocks F 39.10. AllBlocks S 39.11. External Sets I 39.11. G-sets F 39.11. IsExternalSet F 39.11. ExternalSet F 39.11. ActingDomain F 39.11. FunctionAction F 39.11. HomeEnumerator F 39.11. IsExternalSubset F 39.11. ExternalSubset F 39.11. IsExternalOrbit F 39.11. ExternalOrbit F 39.11. StabilizerOfExternalSet F 39.11. ExternalOrbits F 39.11. ExternalOrbits F 39.11. ExternalOrbitsStabilizers F 39.11. ExternalOrbitsStabilizers F 39.11. CanonicalRepresentativeOfExternalSet F 39.11. CanonicalRepresentativeDeterminatorOfExternalSet F 39.11. ActorOfExternalSet F 39.11. UnderlyingExternalSet F 39.11. SurjectiveActionHomomorphismAttr C permutat.tex 40. Permutations F 40.0. IsPerm F 40.0. IsPermCollection F 40.0. IsPermCollColl F 40.0. PermutationsFamily S 40.1. Comparison of Permutations F 40.1. equality test!for permutations F 40.1. precedence test!for permutations F 40.1. SmallestGeneratorPerm S 40.2. Moved Points of Permutations F 40.2. SmallestMovedPoint F 40.2. SmallestMovedPoint F 40.2. LargestMovedPoint F 40.2. LargestMovedPoint F 40.2. MovedPoints F 40.2. MovedPoints F 40.2. NrMovedPoints F 40.2. NrMovedPoints S 40.3. Sign and Cycle Structure F 40.3. SignPerm F 40.3. CycleStructurePerm S 40.4. Creating Permutations F 40.4. ListPerm F 40.4. PermList F 40.4. MappingPermListList F 40.4. RestrictedPerm F 40.4. RestrictedPermNC C grpperm.tex 41. Permutation Groups F 41.0. IsPermGroup S 41.1. The Natural Action F 41.1. OrbitPerms F 41.1. OrbitsPerms S 41.2. Computing a Permutation Representation F 41.2. IsomorphismPermGroup F 41.2. SmallerDegreePermutationRepresentation S 41.3. Symmetric and Alternating Groups F 41.3. IsNaturalSymmetricGroup F 41.3. IsNaturalAlternatingGroup F 41.3. IsSymmetricGroup F 41.3. IsAlternatingGroup F 41.3. SymmetricParentGroup S 41.4. Primitive Groups F 41.4. ONanScottType F 41.4. SocleTypePrimitiveGroup S 41.5. Stabilizer Chains S 41.6. Randomized Methods for Permutation Groups I 41.6. Schreier-Sims!random S 41.7. Construction of Stabilizer Chains F 41.7. StabChain F 41.7. StabChain F 41.7. StabChainOp F 41.7. StabChainMutable F 41.7. StabChainMutable F 41.7. StabChainImmutable F 41.7. StabChainOptions F 41.7. DefaultStabChainOptions F 41.7. StabChainBaseStrongGenerators F 41.7. MinimalStabChain S 41.8. Stabilizer Chain Records S 41.9. Operations for Stabilizer Chains F 41.9. BaseStabChain F 41.9. BaseOfGroup F 41.9. SizeStabChain F 41.9. StrongGeneratorsStabChain F 41.9. GroupStabChain F 41.9. OrbitStabChain F 41.9. IndicesStabChain F 41.9. ListStabChain F 41.9. ElementsStabChain F 41.9. InverseRepresentative F 41.9. SiftedPermutation F 41.9. MinimalElementCosetStabChain F 41.9. LargestElementStabChain F 41.9. ApproximateSuborbitsStabilizerPermGroup S 41.10. Low Level Routines to Modify and Create Stabilizer Chains F 41.10. CopyStabChain F 41.10. CopyOptionsDefaults F 41.10. ChangeStabChain F 41.10. ExtendStabChain F 41.10. ReduceStabChain F 41.10. RemoveStabChain F 41.10. EmptyStabChain F 41.10. InsertTrivialStabilizer F 41.10. IsFixedStabilizer F 41.10. AddGeneratorsExtendSchreierTree S 41.11. Backtrack F 41.11. SubgroupProperty F 41.11. ElementProperty F 41.11. TwoClosure F 41.11. InfoBckt S 41.12. Working with large degree permutation groups C grpmat.tex 42. Matrix Groups F 42.0. IsMatrixGroup S 42.1. Attributes and Properties for Matrix Groups F 42.1. DimensionOfMatrixGroup F 42.1. DefaultFieldOfMatrixGroup F 42.1. FieldOfMatrixGroup F 42.1. TransposedMatrixGroup S 42.2. Actions of Matrix Groups F 42.2. ProjectiveActionOnFullSpace F 42.2. ProjectiveActionHomomorphismMatrixGroup F 42.2. BlowUpIsomorphism S 42.3. GL and SL F 42.3. IsGeneralLinearGroup F 42.3. IsGL F 42.3. IsNaturalGL F 42.3. IsSpecialLinearGroup F 42.3. IsSL F 42.3. IsNaturalSL F 42.3. IsSubgroupSL S 42.4. Invariant Forms F 42.4. InvariantBilinearForm F 42.4. IsFullSubgroupGLorSLRespectingBilinearForm F 42.4. InvariantSesquilinearForm F 42.4. IsFullSubgroupGLorSLRespectingSesquilinearForm F 42.4. InvariantQuadraticForm F 42.4. IsFullSubgroupGLorSLRespectingQuadraticForm S 42.5. Matrix Groups in Characteristic 0 F 42.5. IsCyclotomicMatrixGroup F 42.5. IsRationalMatrixGroup F 42.5. IsIntegerMatrixGroup F 42.5. IsNaturalGLnZ F 42.5. IsNaturalSLnZ F 42.5. InvariantLattice F 42.5. NormalizerInGLnZ F 42.5. CentralizerInGLnZ F 42.5. ZClassRepsQClass F 42.5. IsBravaisGroup F 42.5. BravaisGroup F 42.5. BravaisSubgroups F 42.5. BravaisSupergroups F 42.5. NormalizerInGLnZBravaisGroup S 42.6. Acting OnRight and OnLeft F 42.6. CrystGroupDefaultAction F 42.6. SetCrystGroupDefaultAction C pcgs.tex 43. Polycyclic Groups S 43.1. Polycyclic Generating Systems S 43.2. Computing a Pcgs F 43.2. Pcgs F 43.2. IsPcgs F 43.2. CanEasilyComputePcgs S 43.3. Defining a Pcgs Yourself F 43.3. PcgsByPcSequence F 43.3. PcgsByPcSequenceNC S 43.4. Elementary Operations for a Pcgs F 43.4. RelativeOrders!of a pcgs F 43.4. IsFiniteOrdersPcgs F 43.4. IsPrimeOrdersPcgs F 43.4. PcSeries F 43.4. GroupOfPcgs F 43.4. OneOfPcgs S 43.5. Elementary Operations for a Pcgs and an Element F 43.5. RelativeOrderOfPcElement F 43.5. ExponentOfPcElement F 43.5. ExponentsOfPcElement F 43.5. ExponentsOfPcElement F 43.5. DepthOfPcElement F 43.5. LeadingExponentOfPcElement F 43.5. PcElementByExponents F 43.5. PcElementByExponentsNC F 43.5. PcElementByExponentsNC F 43.5. LinearCombinationPcgs F 43.5. SiftedPcElement F 43.5. CanonicalPcElement F 43.5. ReducedPcElement F 43.5. CleanedTailPcElement F 43.5. HeadPcElementByNumber S 43.6. Exponents of Special Products F 43.6. ExponentsConjugateLayer F 43.6. ExponentsOfRelativePower F 43.6. ExponentsOfConjugate F 43.6. ExponentsOfCommutator S 43.7. Subgroups of Polycyclic Groups - Induced Pcgs F 43.7. IsInducedPcgs F 43.7. InducedPcgsByPcSequence F 43.7. InducedPcgsByPcSequenceNC F 43.7. InducedPcgsByPcSequenceNC F 43.7. ParentPcgs F 43.7. InducedPcgs F 43.7. InducedPcgsByGenerators F 43.7. InducedPcgsByGeneratorsNC F 43.7. InducedPcgsByPcSequenceAndGenerators F 43.7. LeadCoeffsIGS F 43.7. ExtendedPcgs F 43.7. SubgroupByPcgs S 43.8. Subgroups of Polycyclic Groups - Canonical Pcgs F 43.8. IsCanonicalPcgs F 43.8. CanonicalPcgs S 43.9. Factor Groups of Polycyclic Groups - Modulo Pcgs F 43.9. ModuloPcgs F 43.9. IsModuloPcgs F 43.9. NumeratorOfModuloPcgs F 43.9. DenominatorOfModuloPcgs F 43.9. modulo!for pcgs F 43.9. CorrespondingGeneratorsByModuloPcgs F 43.9. CanonicalPcgsByGeneratorsWithImages S 43.10. Factor Groups of Polycyclic Groups in their Own Representation F 43.10. ProjectedPcElement F 43.10. ProjectedInducedPcgs F 43.10. LiftedPcElement F 43.10. LiftedInducedPcgs S 43.11. Pcgs and Normal Series F 43.11. IsPcgsElementaryAbelianSeries F 43.11. PcgsElementaryAbelianSeries F 43.11. PcgsElementaryAbelianSeries F 43.11. IndicesEANormalSteps F 43.11. EANormalSeriesByPcgs F 43.11. IsPcgsCentralSeries F 43.11. PcgsCentralSeries F 43.11. IndicesCentralNormalSteps F 43.11. CentralNormalSeriesByPcgs F 43.11. IsPcgsPCentralSeriesPGroup F 43.11. PcgsPCentralSeriesPGroup F 43.11. IndicesPCentralNormalStepsPGroup F 43.11. PCentralNormalSeriesByPcgsPGroup F 43.11. IsPcgsChiefSeries F 43.11. PcgsChiefSeries F 43.11. IndicesChiefNormalSteps F 43.11. ChiefNormalSeriesByPcgs F 43.11. IndicesNormalSteps F 43.11. NormalSeriesByPcgs S 43.12. Sum and Intersection of Pcgs F 43.12. SumFactorizationFunctionPcgs S 43.13. Special Pcgs F 43.13. IsSpecialPcgs F 43.13. SpecialPcgs!attribute F 43.13. SpecialPcgs!attribute F 43.13. LGWeights F 43.13. LGLayers F 43.13. LGFirst F 43.13. LGLength F 43.13. IsInducedPcgsWrtSpecialPcgs F 43.13. InducedPcgsWrtSpecialPcgs S 43.14. Action on Subfactors Defined by a Pcgs F 43.14. VectorSpaceByPcgsOfElementaryAbelianGroup F 43.14. LinearOperation F 43.14. LinearAction F 43.14. LinearOperationLayer F 43.14. LinearActionLayer F 43.14. AffineOperation F 43.14. AffineAction F 43.14. AffineOperationLayer F 43.14. AffineActionLayer S 43.15. Orbit Stabilizer Methods for Polycyclic Groups F 43.15. StabilizerPcgs F 43.15. Pcgs_OrbitStabilizer S 43.16. Operations which have Special Methods for Groups with Pcgs I 43.16. IsNilpotent!for groups with pcgs I 43.16. IsSupersolvable!for groups with pcgs I 43.16. Size!for groups with pcgs I 43.16. CompositionSeries!for groups with pcgs I 43.16. ConjugacyClasses!for groups with pcgs I 43.16. Centralizer!for groups with pcgs I 43.16. FrattiniSubgroup!for groups with pcgs I 43.16. PrefrattiniSubgroup!for groups with pcgs I 43.16. MaximalSubgroups!for groups with pcgs I 43.16. HallSystem!for groups with pcgs I 43.16. MinimalGeneratingSet!for groups with pcgs I 43.16. Centre!for groups with pcgs I 43.16. Intersection!for groups with pcgs I 43.16. AutomorphismGroup!for groups with pcgs I 43.16. IrreducibleModules!for groups with pcgs S 43.17. Conjugacy Classes in Solvable Groups F 43.17. ClassesSolvableGroup F 43.17. CentralizerSizeLimitConsiderFunction C grppc.tex 44. Pc Groups S 44.1. The family pcgs F 44.1. FamilyPcgs F 44.1. IsFamilyPcgs F 44.1. InducedPcgsWrtFamilyPcgs F 44.1. IsParentPcgsFamilyPcgs S 44.2. Elements of pc groups F 44.2. equality!pcwords F 44.2. smaller!pcwords S 44.3. Pc groups versus fp groups F 44.3. IsPcGroup F 44.3. IsomorphismFpGroupByPcgs S 44.4. Constructing Pc Groups F 44.4. PcGroupFpGroup F 44.4. SingleCollector F 44.4. CombinatorialCollector F 44.4. SetConjugate F 44.4. SetCommutator F 44.4. SetPower F 44.4. GroupByRws F 44.4. GroupByRwsNC F 44.4. IsConfluent!for pc groups F 44.4. IsomorphismRefinedPcGroup I 44.4. isomorphic!pc group F 44.4. RefinedPcGroup S 44.5. Computing Pc Groups F 44.5. PcGroupWithPcgs F 44.5. IsomorphismPcGroup I 44.5. isomorphic!pc group F 44.5. IsomorphismSpecialPcGroup S 44.6. Saving a Pc Group F 44.6. GapInputPcGroup S 44.7. Operations for Pc Groups S 44.8. 2-Cohomology and Extensions F 44.8. TwoCoboundaries F 44.8. TwoCocycles F 44.8. TwoCohomology F 44.8. Extensions F 44.8. Extension F 44.8. ExtensionNC F 44.8. SplitExtension F 44.8. ModuleOfExtension F 44.8. CompatiblePairs F 44.8. ExtensionRepresentatives F 44.8. SplitExtensions S 44.9. Coding a Pc Presentation F 44.9. CodePcgs F 44.9. CodePcGroup F 44.9. PcGroupCode F 44.9. PcGroupCodeRec S 44.10. Random Isomorphism Testing F 44.10. RandomIsomorphismTest C grpfp.tex 45. Finitely Presented Groups F 45.0. IsSubgroupFpGroup F 45.0. IsFpGroup F 45.0. InfoFpGroup S 45.1. Creating Finitely Presented Groups F 45.1. quotient!for finitely presented groups F 45.1. FactorGroupFpGroupByRels S 45.2. Comparison of Elements of Finitely Presented Groups F 45.2. equality!elements of finitely presented groups F 45.2. smaller!elements of finitely presented groups F 45.2. FpElmComparisonMethod F 45.2. SetReducedMultiplication F 45.2. SetReducedMultiplication F 45.2. SetReducedMultiplication S 45.3. Preimages in the Free Group F 45.3. FreeGroupOfFpGroup F 45.3. FreeGeneratorsOfFpGroup F 45.3. FreeGeneratorsOfWholeGroup F 45.3. RelatorsOfFpGroup F 45.3. UnderlyingElement!fp group elements F 45.3. ElementOfFpGroup S 45.4. Operations for Finitely Presented Groups S 45.5. Coset Tables and Coset Enumeration F 45.5. CosetTable F 45.5. TracedCosetFpGroup F 45.5. FactorCosetAction!for fp groups F 45.5. FactorCosetOperation F 45.5. CosetTableBySubgroup F 45.5. CosetTableFromGensAndRels F 45.5. CosetTableDefaultMaxLimit F 45.5. CosetTableDefaultLimit F 45.5. MostFrequentGeneratorFpGroup F 45.5. IndicesInvolutaryGenerators S 45.6. Standardization of coset tables F 45.6. CosetTableStandard F 45.6. StandardizeTable S 45.7. Coset tables for subgroups in the whole group F 45.7. CosetTableInWholeGroup F 45.7. TryCosetTableInWholeGroup F 45.7. SubgroupOfWholeGroupByCosetTable S 45.8. Augmented Coset Tables and Rewriting F 45.8. AugmentedCosetTableInWholeGroup F 45.8. AugmentedCosetTableMtc F 45.8. AugmentedCosetTableRrs F 45.8. RewriteWord S 45.9. Low Index Subgroups I 45.9. iterator!for low index subgroups F 45.9. LowIndexSubgroupsFpGroupIterator F 45.9. LowIndexSubgroupsFpGroup S 45.10. Converting Groups to Finitely Presented Groups F 45.10. IsomorphismFpGroup F 45.10. IsomorphismFpGroupByGenerators F 45.10. IsomorphismFpGroupByGeneratorsNC S 45.11. New Presentations and Presentations for Subgroups I 45.11. IsomorphismFpGroup!for subgroups of fp groups F 45.11. IsomorphismSimplifiedFpGroup S 45.12. Preimages under Homomorphisms from an FpGroup F 45.12. SubgroupOfWholeGroupByQuotientSubgroup F 45.12. IsSubgroupOfWholeGroupByQuotientRep F 45.12. AsSubgroupOfWholeGroupByQuotient F 45.12. DefiningQuotientHomomorphism S 45.13. Quotient Methods F 45.13. PQuotient F 45.13. EpimorphismQuotientSystem F 45.13. EpimorphismPGroup F 45.13. EpimorphismPGroup F 45.13. EpimorphismNilpotentQuotient S 45.14. Abelian Invariants for Subgroups F 45.14. AbelianInvariantsSubgroupFpGroup F 45.14. AbelianInvariantsSubgroupFpGroupMtc F 45.14. AbelianInvariantsSubgroupFpGroupRrs F 45.14. AbelianInvariantsSubgroupFpGroupRrs F 45.14. AbelianInvariantsNormalClosureFpGroup F 45.14. AbelianInvariantsNormalClosureFpGroupRrs S 45.15. Testing Finiteness of Finitely Presented Groups F 45.15. NewmanInfinityCriterion C pres.tex 46. Presentations and Tietze Transformations S 46.1. Creating Presentations F 46.1. PresentationFpGroup F 46.1. TzSort F 46.1. GeneratorsOfPresentation F 46.1. FpGroupPresentation F 46.1. PresentationViaCosetTable F 46.1. PresentationViaCosetTable S 46.2. SimplifiedFpGroup F 46.2. SimplifiedFpGroup S 46.3. Subgroup Presentations I 46.3. Schreier F 46.3. PresentationSubgroup F 46.3. PresentationSubgroupRrs F 46.3. PresentationSubgroupRrs F 46.3. PrimaryGeneratorWords F 46.3. PresentationSubgroupMtc F 46.3. PresentationNormalClosureRrs F 46.3. PresentationNormalClosure S 46.4. Relators in a Presentation F 46.4. TietzeWordAbstractWord F 46.4. AbstractWordTietzeWord S 46.5. Printing Presentations F 46.5. TzPrintGenerators F 46.5. TzPrintRelators F 46.5. TzPrintLengths F 46.5. TzPrintStatus F 46.5. TzPrintPresentation F 46.5. TzPrint F 46.5. TzPrintPairs S 46.6. Changing Presentations F 46.6. AddGenerator F 46.6. TzNewGenerator F 46.6. AddRelator F 46.6. RemoveRelator S 46.7. Tietze Transformations F 46.7. TzGo F 46.7. SimplifyPresentation F 46.7. TzGoGo S 46.8. Elementary Tietze Transformations F 46.8. TzEliminate F 46.8. TzEliminate F 46.8. TzEliminate F 46.8. TzSearch F 46.8. TzSearchEqual F 46.8. TzFindCyclicJoins S 46.9. Tietze Transformations that introduce new Generators F 46.9. TzSubstitute F 46.9. TzSubstitute F 46.9. TzSubstituteCyclicJoins S 46.10. Tracing generator images through Tietze transformations F 46.10. TzInitGeneratorImages F 46.10. OldGeneratorsOfPresentation F 46.10. TzImagesOldGens F 46.10. TzPreImagesNewGens F 46.10. TzPrintGeneratorImages S 46.11. DecodeTree F 46.11. DecodeTree I 46.11. secondary subgroup generators I 46.11. primary subgroup generators I 46.11. subgroup generators tree S 46.12. Tietze Options F 46.12. TzOptions F 46.12. TzPrintOptions C grpprod.tex 47. Group Products S 47.1. Direct Products F 47.1. DirectProduct F 47.1. DirectProductOp I 47.1. Embedding!example for direct products I 47.1. Projection!example for direct products S 47.2. Semidirect Products F 47.2. SemidirectProduct F 47.2. SemidirectProduct I 47.2. Embedding!example for semidirect products I 47.2. Projection!example for semidirect products S 47.3. Subdirect Products F 47.3. SubdirectProduct I 47.3. Projection!example for subdirect products F 47.3. SubdirectProducts S 47.4. Wreath Products F 47.4. WreathProduct F 47.4. WreathProduct I 47.4. Embedding!example for wreath products I 47.4. Projection!example for wreath products F 47.4. WreathProductImprimitiveAction F 47.4. WreathProductProductAction F 47.4. KuKGenerators I 47.4. Krasner-Kaloujnine theorem I 47.4. Wreath product embedding S 47.5. Free Products F 47.5. FreeProduct F 47.5. FreeProduct S 47.6. Embeddings and Projections for Group Products F 47.6. Embedding!for group products F 47.6. Projection!for group products C grplib.tex 48. Group Libraries S 48.1. Basic Groups F 48.1. TrivialGroup F 48.1. CyclicGroup F 48.1. AbelianGroup F 48.1. ElementaryAbelianGroup F 48.1. DihedralGroup F 48.1. ExtraspecialGroup F 48.1. AlternatingGroup F 48.1. AlternatingGroup F 48.1. SymmetricGroup F 48.1. SymmetricGroup F 48.1. MathieuGroup F 48.1. SuzukiGroup F 48.1. Sz F 48.1. ReeGroup F 48.1. Ree S 48.2. Classical Groups F 48.2. GeneralLinearGroup F 48.2. GL F 48.2. GeneralLinearGroup F 48.2. GL F 48.2. SpecialLinearGroup F 48.2. SL F 48.2. SpecialLinearGroup F 48.2. SL I 48.2. OnLines!example F 48.2. GeneralUnitaryGroup F 48.2. GU F 48.2. SpecialUnitaryGroup F 48.2. SU F 48.2. SymplecticGroup F 48.2. Sp F 48.2. SP F 48.2. GeneralOrthogonalGroup F 48.2. GO F 48.2. SpecialOrthogonalGroup F 48.2. SO F 48.2. ProjectiveGeneralLinearGroup F 48.2. PGL F 48.2. ProjectiveSpecialLinearGroup F 48.2. PSL F 48.2. ProjectiveGeneralUnitaryGroup F 48.2. PGU F 48.2. ProjectiveSpecialUnitaryGroup F 48.2. PSU F 48.2. ProjectiveSymplecticGroup F 48.2. PSP F 48.2. PSp S 48.3. Conjugacy Classes in Classical Groups I 48.3. ConjugacyClasses!for linear groups F 48.3. NrConjugacyClassesGL F 48.3. NrConjugacyClassesGU F 48.3. NrConjugacyClassesSL F 48.3. NrConjugacyClassesSU F 48.3. NrConjugacyClassesPGL F 48.3. NrConjugacyClassesPGU F 48.3. NrConjugacyClassesPSL F 48.3. NrConjugacyClassesPSU F 48.3. NrConjugacyClassesSLIsogeneous F 48.3. NrConjugacyClassesSUIsogeneous S 48.4. Constructors for Basic Groups S 48.5. Selection Functions I 48.5. AllPrimitiveGroups I 48.5. AllTransitiveGroups F 48.5. AllLibraryGroups I 48.5. OnePrimitiveGroup I 48.5. OneTransitiveGroup F 48.5. OneLibraryGroup S 48.6. Transitive Permutation Groups F 48.6. TransitiveGroup F 48.6. NrTransitiveGroups F 48.6. TransitiveIdentification S 48.7. Small Groups I 48.7. TwoGroup library I 48.7. ThreeGroup library F 48.7. SmallGroup F 48.7. SmallGroup F 48.7. AllSmallGroups F 48.7. OneSmallGroup F 48.7. NumberSmallGroups F 48.7. IdSmallGroup F 48.7. IdGroup F 48.7. IdsOfAllSmallGroups F 48.7. IdGap3SolvableGroup F 48.7. Gap3CatalogueIdGroup F 48.7. SmallGroupsInformation F 48.7. UnloadSmallGroupsData S 48.8. Finite Perfect Groups I 48.8. perfect groups F 48.8. SizesPerfectGroups F 48.8. PerfectGroup F 48.8. PerfectGroup F 48.8. PerfectIdentification F 48.8. NumberPerfectGroups F 48.8. NumberPerfectLibraryGroups F 48.8. SizeNumbersPerfectGroups F 48.8. DisplayInformationPerfectGroups F 48.8. DisplayInformationPerfectGroups F 48.8. DisplayInformationPerfectGroups S 48.9. Primitive Permutation Groups F 48.9. PrimitiveGroup F 48.9. NrPrimitiveGroups F 48.9. PrimitiveGroupsIterator F 48.9. COHORTS_PRIMITIVE_GROUPS S 48.10. Index numbers of primitive groups F 48.10. PrimitiveIdentification F 48.10. SimsNo F 48.10. PRIMITIVE_INDICES_MAGMA S 48.11. Irreducible Solvable Matrix Groups F 48.11. IrreducibleSolvableGroupMS F 48.11. NumberIrreducibleSolvableGroups F 48.11. AllIrreducibleSolvableGroups F 48.11. OneIrreducibleSolvableGroup F 48.11. PrimitiveIndexIrreducibleSolvableGroup F 48.11. IrreducibleSolvableGroup S 48.12. Irreducible Maximal Finite Integral Matrix Groups F 48.12. ImfNumberQQClasses F 48.12. ImfNumberQClasses F 48.12. ImfNumberZClasses F 48.12. DisplayImfInvariants F 48.12. DisplayImfInvariants F 48.12. ImfInvariants F 48.12. ImfInvariants F 48.12. ImfMatrixGroup F 48.12. ImfMatrixGroup F 48.12. IsomorphismPermGroup!for Imf matrix groups F 48.12. IsomorphismPermGroupImfGroup C semigrp.tex 49. Semigroups F 49.0. IsSemigroup I 49.0. semigroup F 49.0. Semigroup F 49.0. Semigroup F 49.0. Subsemigroup F 49.0. SubsemigroupNC F 49.0. SemigroupByGenerators F 49.0. AsSemigroup F 49.0. AsSubsemigroup F 49.0. GeneratorsOfSemigroup F 49.0. FreeSemigroup!with examples F 49.0. FreeSemigroup!with examples F 49.0. FreeSemigroup!with examples F 49.0. FreeSemigroup!with examples F 49.0. FreeSemigroup!with examples F 49.0. SemigroupByMultiplicationTable F 49.0. IsRegularSemigroup F 49.0. IsRegularSemigroupElement F 49.0. IsSimpleSemigroup F 49.0. IsZeroSimpleSemigroup F 49.0. IsZeroGroup F 49.0. IsReesCongruenceSemigroup S 49.1. Making transformation semigroups F 49.1. IsTransformationSemigroup F 49.1. IsTransformationMonoid F 49.1. DegreeOfTransformationSemigroup F 49.1. IsomorphismTransformationSemigroup F 49.1. HomomorphismTransformationSemigroup F 49.1. IsFullTransformationSemigroup F 49.1. FullTransformationSemigroup S 49.2. Ideals of semigroups F 49.2. SemigroupIdealByGenerators F 49.2. ReesCongruenceOfSemigroupIdeal F 49.2. IsLeftSemigroupIdeal F 49.2. IsRightSemigroupIdeal F 49.2. IsSemigroupIdeal S 49.3. Congruences for semigroups F 49.3. IsSemigroupCongruence F 49.3. IsReesCongruence S 49.4. Quotients F 49.4. IsQuotientSemigroup F 49.4. HomomorphismQuotientSemigroup F 49.4. QuotientSemigroupPreimage F 49.4. QuotientSemigroupCongruence F 49.4. QuotientSemigroupHomomorphism S 49.5. Green's Relations F 49.5. GreensRRelation F 49.5. GreensLRelation F 49.5. GreensJRelation F 49.5. GreensDRelation F 49.5. GreensHRelation F 49.5. IsGreensRelation F 49.5. IsGreensRRelation F 49.5. IsGreensLRelation F 49.5. IsGreensJRelation F 49.5. IsGreensHRelation F 49.5. IsGreensDRelation F 49.5. IsGreensClass F 49.5. IsGreensRClass F 49.5. IsGreensLClass F 49.5. IsGreensJClass F 49.5. IsGreensHClass F 49.5. IsGreensDClass F 49.5. IsGreensLessThanOrEqual F 49.5. RClassOfHClass F 49.5. LClassOfHClass F 49.5. EggBoxOfDClass F 49.5. DisplayEggBoxOfDClass F 49.5. GreensRClassOfElement F 49.5. GreensLClassOfElement F 49.5. GreensDClassOfElement F 49.5. GreensJClassOfElement F 49.5. GreensHClassOfElement F 49.5. GreensRClasses F 49.5. GreensLClasses F 49.5. GreensJClasses F 49.5. GreensDClasses F 49.5. GreensHClasses F 49.5. GroupHClassOfGreensDClass F 49.5. IsGroupHClass F 49.5. IsRegularDClass S 49.6. Rees Matrix Semigroups F 49.6. ReesMatrixSemigroup F 49.6. ReesZeroMatrixSemigroup F 49.6. IsReesMatrixSemigroup F 49.6. IsReesZeroMatrixSemigroup F 49.6. ReesMatrixSemigroupElement F 49.6. ReesZeroMatrixSemigroupElement F 49.6. IsReesMatrixSemigroupElement F 49.6. IsReesZeroMatrixSemigroupElement F 49.6. SandwichMatrixOfReesMatrixSemigroup F 49.6. SandwichMatrixOfReesZeroMatrixSemigroup F 49.6. RowIndexOfReesMatrixSemigroupElement F 49.6. RowIndexOfReesZeroMatrixSemigroupElement F 49.6. ColumnIndexOfReesMatrixSemigroupElement F 49.6. ColumnIndexOfReesZeroMatrixSemigroupElement F 49.6. UnderlyingElementOfReesMatrixSemigroupElement F 49.6. UnderlyingElementOfReesZeroMatrixSemigroupElement F 49.6. ReesZeroMatrixSemigroupElementIsZero F 49.6. AssociatedReesMatrixSemigroupOfDClass F 49.6. IsomorphismReesMatrixSemigroup C monoid.tex 50. Monoids F 50.0. IsMonoid F 50.0. Monoid F 50.0. Monoid F 50.0. Monoid F 50.0. Submonoid F 50.0. SubmonoidNC F 50.0. MonoidByGenerators F 50.0. MonoidByGenerators F 50.0. AsMonoid F 50.0. AsSubmonoid F 50.0. GeneratorsOfMonoid F 50.0. TrivialSubmonoid F 50.0. FreeMonoid F 50.0. FreeMonoid F 50.0. FreeMonoid F 50.0. FreeMonoid F 50.0. FreeMonoid F 50.0. MonoidByMultiplicationTable C fpsemi.tex 51. Finitely Presented Semigroups and Monoids F 51.0. IsSubsemigroupFpSemigroup F 51.0. IsSubmonoidFpMonoid F 51.0. IsFpSemigroup F 51.0. IsFpMonoid F 51.0. IsElementOfFpSemigroup F 51.0. IsElementOfFpMonoid F 51.0. FpGrpMonSmgOfFpGrpMonSmgElement S 51.1. Creating Finitely Presented Semigroups F 51.1. quotient!of free semigroup F 51.1. FactorFreeSemigroupByRelations F 51.1. IsomorphismFpSemigroup S 51.2. Comparison of Elements of Finitely Presented Semigroups F 51.2. comparison!fp semigroup elements S 51.3. Preimages in the Free Semigroup F 51.3. FreeSemigroupOfFpSemigroup F 51.3. FreeGeneratorsOfFpSemigroup F 51.3. RelationsOfFpSemigroup F 51.3. UnderlyingElement!fp semigroup elements F 51.3. ElementOfFpSemigroup S 51.4. Finitely presented monoids F 51.4. quotient!of free monoid S 51.5. Rewriting Systems and the Knuth-Bendix Procedure F 51.5. ReducedConfluentRewritingSystem F 51.5. ReducedConfluentRewritingSystem F 51.5. KB_REW F 51.5. GAPKB_REW F 51.5. KnuthBendixRewritingSystem F 51.5. KnuthBendixRewritingSystem F 51.5. SemigroupOfRewritingSystem F 51.5. MonoidOfRewritingSystem F 51.5. FreeSemigroupOfRewritingSystem F 51.5. FreeMonoidOfRewritingSystem S 51.6. Todd-Coxeter Procedure F 51.6. CosetTableOfFpSemigroup C trans.tex 52. Transformations F 52.0. IsTransformation F 52.0. IsTransformationCollection F 52.0. TransformationFamily F 52.0. TransformationType F 52.0. TransformationData F 52.0. Transformation F 52.0. TransformationNC F 52.0. IdentityTransformation F 52.0. RandomTransformation F 52.0. DegreeOfTransformation F 52.0. ImageListOfTransformation F 52.0. ImageSetOfTransformation F 52.0. RankOfTransformation F 52.0. KernelOfTransformation F 52.0. PreimagesOfTransformation F 52.0. RestrictedTransformation F 52.0. AsTransformation F 52.0. AsTransformation F 52.0. AsTransformationNC F 52.0. PermLeftQuoTransformation F 52.0. BinaryRelationTransformation F 52.0. TransformationRelation C addmagma.tex 53. Additive Magmas (preliminary) S 53.1. (Near-)Additive Magma Categories F 53.1. IsNearAdditiveMagma F 53.1. IsNearAdditiveMagmaWithZero F 53.1. IsNearAdditiveGroup F 53.1. IsNearAdditiveMagmaWithInverses F 53.1. IsAdditiveMagma F 53.1. IsAdditiveMagmaWithZero F 53.1. IsAdditiveGroup F 53.1. IsAdditiveMagmaWithInverses S 53.2. (Near-)Additive Magma Generation F 53.2. NearAdditiveMagma F 53.2. NearAdditiveMagma F 53.2. NearAdditiveMagmaWithZero F 53.2. NearAdditiveMagmaWithZero F 53.2. NearAdditiveGroup F 53.2. NearAdditiveGroup F 53.2. NearAdditiveMagmaByGenerators F 53.2. NearAdditiveMagmaByGenerators F 53.2. NearAdditiveMagmaWithZeroByGenerators F 53.2. NearAdditiveMagmaWithZeroByGenerators F 53.2. NearAdditiveGroupByGenerators F 53.2. NearAdditiveGroupByGenerators F 53.2. SubnearAdditiveMagma F 53.2. SubnearAdditiveMagmaNC F 53.2. SubnearAdditiveMagmaWithZero F 53.2. SubnearAdditiveMagmaWithZeroNC F 53.2. SubnearAdditiveGroup F 53.2. SubnearAdditiveGroupNC S 53.3. Attributes and Properties for (Near-)Additive Magmas F 53.3. IsAdditivelyCommutative F 53.3. GeneratorsOfNearAdditiveMagma F 53.3. GeneratorsOfAdditiveMagma F 53.3. GeneratorsOfNearAdditiveMagmaWithZero F 53.3. GeneratorsOfAdditiveMagmaWithZero F 53.3. GeneratorsOfNearAdditiveGroup F 53.3. GeneratorsOfAdditiveGroup F 53.3. AdditiveNeutralElement F 53.3. TrivialSubnearAdditiveMagmaWithZero S 53.4. Operations for (Near-)Additive Magmas F 53.4. ClosureNearAdditiveGroup F 53.4. ClosureNearAdditiveGroup C rings.tex 54. Rings S 54.1. Generating Rings F 54.1. IsRing F 54.1. Ring F 54.1. Ring F 54.1. DefaultRing F 54.1. DefaultRing F 54.1. RingByGenerators F 54.1. DefaultRingByGenerators F 54.1. GeneratorsOfRing F 54.1. AsRing F 54.1. Subring F 54.1. SubringNC F 54.1. ClosureRing F 54.1. ClosureRing F 54.1. Quotient F 54.1. Quotient S 54.2. Ideals in Rings F 54.2. TwoSidedIdeal F 54.2. Ideal F 54.2. LeftIdeal F 54.2. RightIdeal F 54.2. TwoSidedIdealNC F 54.2. IdealNC F 54.2. LeftIdealNC F 54.2. RightIdealNC F 54.2. IsTwoSidedIdeal F 54.2. IsLeftIdeal F 54.2. IsRightIdeal F 54.2. IsTwoSidedIdealInParent F 54.2. IsLeftIdealInParent F 54.2. IsRightIdealInParent F 54.2. TwoSidedIdealByGenerators F 54.2. IdealByGenerators F 54.2. LeftIdealByGenerators F 54.2. RightIdealByGenerators F 54.2. GeneratorsOfTwoSidedIdeal F 54.2. GeneratorsOfIdeal F 54.2. GeneratorsOfLeftIdeal F 54.2. GeneratorsOfRightIdeal F 54.2. LeftActingRingOfIdeal F 54.2. RightActingRingOfIdeal F 54.2. AsLeftIdeal F 54.2. AsRightIdeal F 54.2. AsTwoSidedIdeal S 54.3. Rings With One F 54.3. IsRingWithOne F 54.3. RingWithOne F 54.3. RingWithOne F 54.3. RingWithOneByGenerators F 54.3. GeneratorsOfRingWithOne F 54.3. SubringWithOne F 54.3. SubringWithOneNC S 54.4. Properties of Rings F 54.4. IsIntegralRing F 54.4. IsUniqueFactorizationRing F 54.4. IsLDistributive F 54.4. IsRDistributive F 54.4. IsDistributive F 54.4. IsAnticommutative F 54.4. IsZeroSquaredRing F 54.4. IsJacobianRing S 54.5. Units and Factorizations F 54.5. IsUnit F 54.5. IsUnit F 54.5. Units F 54.5. IsAssociated F 54.5. IsAssociated F 54.5. Associates F 54.5. Associates F 54.5. StandardAssociate F 54.5. StandardAssociate F 54.5. IsIrreducibleRingElement F 54.5. IsIrreducibleRingElement F 54.5. IsPrime F 54.5. IsPrime F 54.5. Factors F 54.5. Factors F 54.5. PadicValuation S 54.6. Euclidean Rings F 54.6. IsEuclideanRing F 54.6. EuclideanDegree F 54.6. EuclideanDegree F 54.6. EuclideanQuotient F 54.6. EuclideanQuotient F 54.6. EuclideanRemainder F 54.6. EuclideanRemainder F 54.6. QuotientRemainder F 54.6. QuotientRemainder S 54.7. Gcd and Lcm F 54.7. Gcd F 54.7. Gcd F 54.7. Gcd F 54.7. Gcd F 54.7. GcdOp F 54.7. GcdOp F 54.7. GcdRepresentation F 54.7. GcdRepresentation F 54.7. GcdRepresentation F 54.7. GcdRepresentation F 54.7. GcdRepresentationOp F 54.7. GcdRepresentationOp F 54.7. Lcm F 54.7. Lcm F 54.7. Lcm F 54.7. Lcm F 54.7. LcmOp F 54.7. LcmOp F 54.7. QuotientMod F 54.7. QuotientMod F 54.7. PowerMod F 54.7. PowerMod F 54.7. InterpolatedPolynomial C module.tex 55. Modules (preliminary) S 55.1. Generating modules F 55.1. IsLeftOperatorAdditiveGroup F 55.1. IsLeftModule F 55.1. GeneratorsOfLeftOperatorAdditiveGroup F 55.1. GeneratorsOfLeftModule F 55.1. AsLeftModule F 55.1. IsRightOperatorAdditiveGroup F 55.1. IsRightModule F 55.1. GeneratorsOfRightOperatorAdditiveGroup F 55.1. GeneratorsOfRightModule F 55.1. LeftModuleByGenerators F 55.1. LeftModuleByGenerators F 55.1. LeftActingDomain S 55.2. Submodules F 55.2. Submodule F 55.2. Submodule F 55.2. SubmoduleNC F 55.2. SubmoduleNC F 55.2. ClosureLeftModule F 55.2. TrivialSubmodule S 55.3. Free Modules F 55.3. IsFreeLeftModule F 55.3. FreeLeftModule F 55.3. FreeLeftModule F 55.3. FreeLeftModule F 55.3. FreeLeftModule F 55.3. AsFreeLeftModule F 55.3. Dimension F 55.3. IsFiniteDimensional F 55.3. UseBasis F 55.3. IsRowModule F 55.3. IsMatrixModule F 55.3. IsFullRowModule F 55.3. FullRowModule F 55.3. IsFullMatrixModule F 55.3. FullMatrixModule F 55.3. IsHandledByNiceBasis C fields.tex 56. Fields and Division Rings I 56.0. fields I 56.0. division rings S 56.1. Generating Fields F 56.1. IsDivisionRing F 56.1. IsField F 56.1. Field F 56.1. Field F 56.1. Field F 56.1. DefaultField F 56.1. DefaultField F 56.1. DefaultFieldByGenerators F 56.1. GeneratorsOfDivisionRing F 56.1. GeneratorsOfField F 56.1. DivisionRingByGenerators F 56.1. DivisionRingByGenerators F 56.1. AsDivisionRing F 56.1. AsDivisionRing F 56.1. AsField F 56.1. AsField S 56.2. Subfields of Fields F 56.2. Subfield F 56.2. SubfieldNC F 56.2. FieldOverItselfByGenerators F 56.2. PrimitiveElement F 56.2. PrimeField F 56.2. IsPrimeField F 56.2. DegreeOverPrimeField F 56.2. DefiningPolynomial F 56.2. RootOfDefiningPolynomial F 56.2. FieldExtension F 56.2. Subfields S 56.3. Galois Action I 56.3. IsFieldControlledByGaloisGroup F 56.3. GaloisGroup!of field F 56.3. MinimalPolynomial!over a field F 56.3. TracePolynomial I 56.3. characteristic polynomial!for field elements F 56.3. Norm F 56.3. Norm F 56.3. Norm F 56.3. Trace!for field elements F 56.3. Trace!for field elements F 56.3. Trace!for field elements F 56.3. Trace!for field elements F 56.3. Conjugates F 56.3. Conjugates F 56.3. Conjugates F 56.3. NormalBase F 56.3. NormalBase C fieldfin.tex 57. Finite Fields S 57.1. Finite Field Elements F 57.1. IsFFE F 57.1. IsFFECollection F 57.1. IsFFECollColl F 57.1. Z F 57.1. Z F 57.1. IsLexOrderedFFE F 57.1. IsLogOrderedFFE S 57.2. Operations for Finite Field Elements F 57.2. DegreeFFE F 57.2. DegreeFFE F 57.2. DegreeFFE F 57.2. LogFFE F 57.2. IntFFE F 57.2. IntFFESymm F 57.2. IntFFESymm F 57.2. IntVecFFE S 57.3. Creating Finite Fields I 57.3. DefaultField!for finite field elements I 57.3. DefaultRing!for finite field elements F 57.3. GaloisField F 57.3. GF F 57.3. GaloisField F 57.3. GF F 57.3. GaloisField F 57.3. GF F 57.3. GaloisField F 57.3. GF F 57.3. GaloisField F 57.3. GF F 57.3. PrimitiveRoot S 57.4. FrobeniusAutomorphism I 57.4. homomorphisms!Frobenius, field I 57.4. field homomorphisms!Frobenius I 57.4. Image!for Frobenius automorphisms I 57.4. CompositionMapping!for Frobenius automorphisms F 57.4. FrobeniusAutomorphism I 57.4. Frobenius automorphism S 57.5. Conway Polynomials F 57.5. ConwayPolynomial F 57.5. IsCheapConwayPolynomial F 57.5. RandomPrimitivePolynomial S 57.6. Printing, Viewing and Displaying Finite Field Elements C fldabnum.tex 58. Abelian Number Fields S 58.1. Construction of Abelian Number Fields F 58.1. CyclotomicField F 58.1. CyclotomicField F 58.1. CyclotomicField F 58.1. CyclotomicField I 58.1. CF F 58.1. AbelianNumberField I 58.1. NF F 58.1. GaussianRationals F 58.1. IsGaussianRationals S 58.2. Operations for Abelian Number Fields I 58.2. cyclotomics!DefaultField I 58.2. polynomials over abelian number fields!Factors F 58.2. IsNumberField I 58.2. number field F 58.2. IsAbelianNumberField I 58.2. abelian number field F 58.2. IsCyclotomicField F 58.2. GaloisStabilizer S 58.3. Integral Bases of Abelian Number Fields I 58.3. cyclotomic fields!CanonicalBasis I 58.3. abelian number fields!CanonicalBasis F 58.3. ZumbroichBase F 58.3. LenstraBase S 58.4. Galois Groups of Abelian Number Fields I 58.4. abelian number fields!Galois group I 58.4. number fields!Galois group I 58.4. automorphism group!of number fields F 58.4. ANFAutomorphism S 58.5. Gaussians F 58.5. GaussianIntegers F 58.5. IsGaussianIntegers C vspc.tex 59. Vector Spaces F 59.0. IsLeftVectorSpace F 59.0. IsVectorSpace S 59.1. Constructing Vector Spaces F 59.1. VectorSpace F 59.1. Subspace F 59.1. SubspaceNC F 59.1. AsVectorSpace F 59.1. AsSubspace S 59.2. Operations and Attributes for Vector Spaces F 59.2. GeneratorsOfLeftVectorSpace F 59.2. GeneratorsOfVectorSpace F 59.2. TrivialSubspace S 59.3. Domains of Subspaces of Vector Spaces F 59.3. Subspaces F 59.3. Subspaces F 59.3. IsSubspacesVectorSpace S 59.4. Bases of Vector Spaces F 59.4. IsBasis F 59.4. Basis F 59.4. Basis F 59.4. BasisNC F 59.4. CanonicalBasis F 59.4. RelativeBasis F 59.4. RelativeBasisNC S 59.5. Operations for Vector Space Bases F 59.5. BasisVectors F 59.5. UnderlyingLeftModule F 59.5. Coefficients F 59.5. LinearCombination F 59.5. LinearCombination F 59.5. EnumeratorByBasis F 59.5. IteratorByBasis S 59.6. Operations for Special Kinds of Bases F 59.6. IsCanonicalBasis F 59.6. IsIntegralBasis F 59.6. IsNormalBasis F 59.6. StructureConstantsTable S 59.7. Mutable Bases F 59.7. IsMutableBasis F 59.7. MutableBasis F 59.7. NrBasisVectors F 59.7. ImmutableBasis F 59.7. IsContainedInSpan F 59.7. CloseMutableBasis S 59.8. Row and Matrix Spaces I 59.8. row spaces I 59.8. matrix spaces F 59.8. IsRowSpace F 59.8. IsMatrixSpace F 59.8. IsGaussianSpace F 59.8. FullRowSpace F 59.8. FullMatrixSpace F 59.8. DimensionOfVectors F 59.8. IsSemiEchelonized F 59.8. SemiEchelonBasis F 59.8. SemiEchelonBasis F 59.8. SemiEchelonBasisNC I 59.8. canonical basis!for row spaces F 59.8. IsCanonicalBasisFullRowModule I 59.8. canonical basis!for matrix spaces F 59.8. IsCanonicalBasisFullMatrixModule F 59.8. NormedRowVectors F 59.8. SiftedVector S 59.9. Vector Space Homomorphisms F 59.9. LeftModuleGeneralMappingByImages F 59.9. LeftModuleHomomorphismByImages F 59.9. LeftModuleHomomorphismByImagesNC F 59.9. LeftModuleHomomorphismByMatrix F 59.9. NaturalHomomorphismBySubspace F 59.9. Hom F 59.9. End F 59.9. IsFullHomModule F 59.9. IsPseudoCanonicalBasisFullHomModule F 59.9. IsLinearMappingsModule S 59.10. Vector Spaces Handled By Nice Bases F 59.10. NiceFreeLeftModule F 59.10. NiceVector F 59.10. UglyVector F 59.10. NiceFreeLeftModuleInfo F 59.10. NiceBasis F 59.10. IsBasisByNiceBasis F 59.10. IsHandledByNiceBasis!for vector spaces S 59.11. How to Implement New Kinds of Vector Spaces F 59.11. DeclareHandlingByNiceBasis F 59.11. InstallHandlingByNiceBasis F 59.11. NiceBasisFiltersInfo F 59.11. CheckForHandlingByNiceBasis C algebra.tex 60. Algebras F 60.0. InfoAlgebra S 60.1. Constructing Algebras by Generators F 60.1. Algebra F 60.1. Algebra F 60.1. Algebra F 60.1. Algebra F 60.1. AlgebraWithOne F 60.1. AlgebraWithOne F 60.1. AlgebraWithOne F 60.1. AlgebraWithOne S 60.2. Constructing Algebras as Free Algebras F 60.2. FreeAlgebra F 60.2. FreeAlgebra F 60.2. FreeAlgebra F 60.2. FreeAlgebraWithOne F 60.2. FreeAlgebraWithOne F 60.2. FreeAlgebraWithOne F 60.2. FreeAssociativeAlgebra F 60.2. FreeAssociativeAlgebra F 60.2. FreeAssociativeAlgebra F 60.2. FreeAssociativeAlgebraWithOne F 60.2. FreeAssociativeAlgebraWithOne F 60.2. FreeAssociativeAlgebraWithOne S 60.3. Constructing Algebras by Structure Constants F 60.3. EmptySCTable F 60.3. EmptySCTable F 60.3. EmptySCTable F 60.3. SetEntrySCTable F 60.3. GapInputSCTable F 60.3. TestJacobi F 60.3. AlgebraByStructureConstants F 60.3. AlgebraByStructureConstants F 60.3. AlgebraByStructureConstants F 60.3. AlgebraByStructureConstants F 60.3. IdentityFromSCTable F 60.3. QuotientFromSCTable S 60.4. Some Special Algebras F 60.4. QuaternionAlgebra F 60.4. ComplexificationQuat F 60.4. ComplexificationQuat F 60.4. OctaveAlgebra F 60.4. FullMatrixAlgebra F 60.4. MatrixAlgebra F 60.4. MatAlgebra F 60.4. NullAlgebra S 60.5. Subalgebras F 60.5. Subalgebra F 60.5. Subalgebra F 60.5. SubalgebraNC F 60.5. SubalgebraNC F 60.5. SubalgebraWithOne F 60.5. SubalgebraWithOne F 60.5. SubalgebraWithOneNC F 60.5. SubalgebraWithOneNC F 60.5. TrivialSubalgebra S 60.6. Ideals S 60.7. Categories and Properties of Algebras F 60.7. IsFLMLOR F 60.7. IsFLMLORWithOne F 60.7. IsAlgebra F 60.7. IsAlgebraWithOne F 60.7. IsLieAlgebra F 60.7. IsSimpleAlgebra F 60.7. IsFiniteDimensional!for matrix algebras F 60.7. IsQuaternion F 60.7. IsQuaternionCollection F 60.7. IsQuaternionCollColl S 60.8. Attributes and Operations for Algebras F 60.8. GeneratorsOfAlgebra F 60.8. GeneratorsOfAlgebraWithOne F 60.8. ProductSpace F 60.8. PowerSubalgebraSeries F 60.8. AdjointBasis F 60.8. IndicesOfAdjointBasis F 60.8. AsAlgebra F 60.8. AsAlgebraWithOne F 60.8. AsSubalgebra F 60.8. AsSubalgebraWithOne F 60.8. MutableBasisOfClosureUnderAction F 60.8. MutableBasisOfNonassociativeAlgebra F 60.8. MutableBasisOfIdealInNonassociativeAlgebra F 60.8. DirectSumOfAlgebras F 60.8. DirectSumOfAlgebras F 60.8. FullMatrixAlgebraCentralizer F 60.8. RadicalOfAlgebra F 60.8. CentralIdempotentsOfAlgebra F 60.8. DirectSumDecomposition F 60.8. LeviMalcevDecomposition F 60.8. Grading S 60.9. Homomorphisms of Algebras F 60.9. AlgebraGeneralMappingByImages F 60.9. AlgebraHomomorphismByImages F 60.9. AlgebraHomomorphismByImagesNC F 60.9. AlgebraWithOneGeneralMappingByImages F 60.9. AlgebraWithOneHomomorphismByImages F 60.9. AlgebraWithOneHomomorphismByImagesNC F 60.9. NaturalHomomorphismByIdeal F 60.9. OperationAlgebraHomomorphism F 60.9. OperationAlgebraHomomorphism F 60.9. IsomorphismFpAlgebra F 60.9. IsomorphismMatrixAlgebra F 60.9. IsomorphismSCAlgebra F 60.9. IsomorphismSCAlgebra F 60.9. RepresentativeLinearOperation S 60.10. Representations of Algebras F 60.10. LeftAlgebraModuleByGenerators F 60.10. RightAlgebraModuleByGenerators F 60.10. BiAlgebraModuleByGenerators F 60.10. LeftAlgebraModule F 60.10. RightAlgebraModule F 60.10. BiAlgebraModule F 60.10. GeneratorsOfAlgebraModule F 60.10. IsAlgebraModuleElement F 60.10. IsAlgebraModuleElementCollection F 60.10. IsAlgebraModuleElementFamily F 60.10. IsLeftAlgebraModuleElement F 60.10. IsLeftAlgebraModuleElementCollection F 60.10. IsRightAlgebraModuleElement F 60.10. IsRightAlgebraModuleElementCollection F 60.10. LeftActingAlgebra F 60.10. RightActingAlgebra F 60.10. ActingAlgebra F 60.10. IsBasisOfAlgebraModuleElementSpace F 60.10. MatrixOfAction F 60.10. MatrixOfAction F 60.10. SubAlgebraModule F 60.10. LeftModuleByHomomorphismToMatAlg F 60.10. RightModuleByHomomorphismToMatAlg F 60.10. AdjointModule F 60.10. FaithfulModule F 60.10. ModuleByRestriction F 60.10. ModuleByRestriction F 60.10. NaturalHomomorphismBySubAlgebraModule F 60.10. DirectSumOfAlgebraModules F 60.10. DirectSumOfAlgebraModules F 60.10. TranslatorSubalgebra C alglie.tex 61. Lie Algebras S 61.1. Lie objects F 61.1. LieObject F 61.1. IsLieObject F 61.1. IsLieObjectCollection F 61.1. LieFamily I 61.1. Embedding!for Lie algebras F 61.1. UnderlyingFamily S 61.2. Constructing Lie algebras F 61.2. LieAlgebraByStructureConstants F 61.2. LieAlgebraByStructureConstants F 61.2. LieAlgebraByStructureConstants F 61.2. LieAlgebra F 61.2. LieAlgebra F 61.2. LieAlgebra F 61.2. LieAlgebra F 61.2. LieAlgebra F 61.2. FreeLieAlgebra F 61.2. FreeLieAlgebra F 61.2. FreeLieAlgebra F 61.2. FullMatrixLieAlgebra F 61.2. MatrixLieAlgebra F 61.2. MatLieAlgebra F 61.2. RightDerivations F 61.2. LeftDerivations F 61.2. Derivations F 61.2. SimpleLieAlgebra S 61.3. Distinguished Subalgebras F 61.3. LieCentre F 61.3. LieCenter F 61.3. LieCentralizer F 61.3. LieNormalizer F 61.3. LieDerivedSubalgebra F 61.3. LieNilRadical F 61.3. LieSolvableRadical F 61.3. CartanSubalgebra S 61.4. Series of Ideals F 61.4. LieDerivedSeries F 61.4. LieLowerCentralSeries F 61.4. LieUpperCentralSeries S 61.5. Properties of a Lie Algebra F 61.5. IsLieAbelian F 61.5. IsLieNilpotent F 61.5. IsLieSolvable S 61.6. Direct Sum Decompositions F 61.6. LeviMalcevDecomposition!for Lie algebras F 61.6. DirectSumDecomposition!for Lie algebras S 61.7. Semisimple Lie Algebras and Root Systems F 61.7. SemiSimpleType F 61.7. ChevalleyBasis F 61.7. IsRootSystem F 61.7. IsRootSystemFromLieAlgebra F 61.7. RootSystem F 61.7. UnderlyingLieAlgebra F 61.7. PositiveRoots F 61.7. NegativeRoots F 61.7. PositiveRootVectors F 61.7. NegativeRootVectors F 61.7. SimpleSystem F 61.7. CartanMatrix F 61.7. BilinearFormMat F 61.7. CanonicalGenerators F 61.7. IsWeylGroup F 61.7. SparseCartanMatrix F 61.7. WeylGroup F 61.7. ApplySimpleReflection F 61.7. LongestWeylWordPerm F 61.7. ConjugateDominantWeight F 61.7. ConjugateDominantWeightWithWord F 61.7. WeylOrbitIterator S 61.8. Restricted Lie algebras F 61.8. IsRestrictedLieAlgebra F 61.8. PthPowerImages F 61.8. PthPowerImage F 61.8. JenningsLieAlgebra F 61.8. PCentralLieAlgebra S 61.9. The Adjoint Representation F 61.9. AdjointMatrix F 61.9. AdjointAssociativeAlgebra F 61.9. KillingMatrix F 61.9. KappaPerp F 61.9. IsNilpotentElement F 61.9. NonNilpotentElement F 61.9. FindSl2 S 61.10. Universal Enveloping Algebras F 61.10. UniversalEnvelopingAlgebra F 61.10. UniversalEnvelopingAlgebra S 61.11. Finitely Presented Lie Algebras F 61.11. FpLieAlgebraByCartanMatrix F 61.11. NilpotentQuotientOfFpLieAlgebra F 61.11. NilpotentQuotientOfFpLieAlgebra S 61.12. Modules over Lie Algebras and Their Cohomology F 61.12. FaithfulModule!for Lie algebras F 61.12. IsCochain F 61.12. IsCochainCollection F 61.12. Cochain F 61.12. CochainSpace F 61.12. ValueCochain F 61.12. LieCoboundaryOperator F 61.12. Cocycles F 61.12. Coboundaries S 61.13. Modules over Semisimple Lie Algebras F 61.13. DominantWeights F 61.13. DominantCharacter F 61.13. DominantCharacter F 61.13. DecomposeTensorProduct F 61.13. DimensionOfHighestWeightModule F 61.13. IsUEALatticeElement F 61.13. IsUEALatticeElementCollection F 61.13. IsUEALatticeElementFamily F 61.13. LatticeGeneratorsInUEA F 61.13. ObjByExtRep F 61.13. IsWeightRepElement F 61.13. IsWeightRepElementCollection F 61.13. IsWeightRepElementFamily F 61.13. HighestWeightModule S 61.14. Tensor Products and Exterior and Symmetric Powers F 61.14. TensorProductOfAlgebraModules F 61.14. TensorProductOfAlgebraModules F 61.14. ExteriorPowerOfAlgebraModule F 61.14. SymmetricPowerOfAlgebraModule F 61.14. DirectSumOfAlgebraModules!for Lie algebras F 61.14. DirectSumOfAlgebraModules!for Lie algebras C algfp.tex 62. Finitely Presented Algebras C mgmring.tex 63. Magma Rings I 63.0. group algebra I 63.0. group ring S 63.1. Free Magma Rings F 63.1. FreeMagmaRing F 63.1. GroupRing F 63.1. IsFreeMagmaRing F 63.1. IsFreeMagmaRingWithOne F 63.1. IsGroupRing F 63.1. UnderlyingMagma F 63.1. AugmentationIdeal S 63.2. Elements of Free Magma Rings F 63.2. IsElementOfFreeMagmaRing F 63.2. IsElementOfFreeMagmaRingCollection F 63.2. IsElementOfFreeMagmaRingFamily F 63.2. CoefficientsAndMagmaElements F 63.2. ZeroCoefficient F 63.2. ElementOfMagmaRing S 63.3. Natural Embeddings related to Magma Rings I 63.3. Embedding!for magma rings S 63.4. Magma Rings modulo Relations F 63.4. IsElementOfMagmaRingModuloRelations F 63.4. IsElementOfMagmaRingModuloRelationsCollection F 63.4. IsElementOfMagmaRingModuloRelationsFamily F 63.4. NormalizedElementOfMagmaRingModuloRelations F 63.4. IsMagmaRingModuloRelations S 63.5. Magma Rings modulo the Span of a Zero Element F 63.5. IsElementOfMagmaRingModuloSpanOfZeroFamily F 63.5. IsMagmaRingModuloSpanOfZero F 63.5. MagmaRingModuloSpanOfZero S 63.6. Technical Details about the Implementation of Magma Rings C ratfun.tex 64. Polynomials and Rational Functions S 64.1. Indeterminates F 64.1. Indeterminate F 64.1. Indeterminate F 64.1. Indeterminate F 64.1. Indeterminate F 64.1. IndeterminateNumberOfUnivariateRationalFunction F 64.1. IndeterminateOfUnivariateRationalFunction F 64.1. IndeterminateName F 64.1. HasIndeterminateName F 64.1. SetIndeterminateName F 64.1. CIUnivPols S 64.2. Operations for Rational Functions F 64.2. addition!rational functions F 64.2. subtraction!rational functions F 64.2. product!rational functions F 64.2. quotient!rational functions F 64.2. mod!Laurent polynomials S 64.3. Comparison of Rational Functions F 64.3. comparison!rational functions F 64.3. smaller!rational functions S 64.4. Properties and Attributes of Rational Functions F 64.4. IsPolynomialFunction F 64.4. IsRationalFunction F 64.4. NumeratorOfRationalFunction F 64.4. DenominatorOfRationalFunction F 64.4. IsPolynomial F 64.4. AsPolynomial F 64.4. IsUnivariateRationalFunction F 64.4. CoefficientsOfUnivariateRationalFunction F 64.4. IsUnivariatePolynomial F 64.4. CoefficientsOfUnivariatePolynomial F 64.4. IsLaurentPolynomial F 64.4. IsConstantRationalFunction F 64.4. IsPrimitivePolynomial F 64.4. SplittingField S 64.5. Univariate Polynomials F 64.5. UnivariatePolynomial F 64.5. UnivariatePolynomialByCoefficients F 64.5. DegreeOfLaurentPolynomial F 64.5. RootsOfUPol F 64.5. RootsOfUPol F 64.5. RootsOfUPol F 64.5. UnivariatenessTestRationalFunction S 64.6. Polynomials as Univariate Polynomials in one Indeterminate F 64.6. DegreeIndeterminate F 64.6. DegreeIndeterminate F 64.6. PolynomialCoefficientsOfPolynomial F 64.6. PolynomialCoefficientsOfPolynomial F 64.6. LeadingCoefficient F 64.6. LeadingMonomial F 64.6. Derivative F 64.6. Derivative F 64.6. Derivative F 64.6. Discriminant F 64.6. Discriminant F 64.6. Discriminant F 64.6. Resultant F 64.6. Resultant S 64.7. Multivariate Polynomials F 64.7. Value F 64.7. Value F 64.7. OnIndeterminates S 64.8. Minimal Polynomials I 64.8. MinimalPolynomial!over a ring F 64.8. MinimalPolynomial S 64.9. Cyclotomic Polynomials F 64.9. CyclotomicPolynomial S 64.10. Polynomial Factorization F 64.10. Factors!of univariate polynomial F 64.10. FactorsSquarefree S 64.11. Polynomials over the Rationals F 64.11. PrimitivePolynomial F 64.11. PolynomialModP F 64.11. GaloisType F 64.11. ProbabilityShapes F 64.11. BombieriNorm F 64.11. MinimizedBombieriNorm F 64.11. HenselBound F 64.11. OneFactorBound S 64.12. Laurent Polynomials F 64.12. LaurentPolynomialByCoefficients F 64.12. CoefficientsOfLaurentPolynomial F 64.12. IndeterminateNumberOfLaurentPolynomial F 64.12. QuotRemLaurpols S 64.13. Univariate Rational Functions F 64.13. UnivariateRationalFunctionByCoefficients S 64.14. Polynomial Rings F 64.14. PolynomialRing F 64.14. PolynomialRing F 64.14. PolynomialRing F 64.14. PolynomialRing F 64.14. IndeterminatesOfPolynomialRing F 64.14. CoefficientsRing F 64.14. IsPolynomialRing F 64.14. IsFiniteFieldPolynomialRing F 64.14. IsAbelianNumberFieldPolynomialRing F 64.14. IsRationalsPolynomialRing S 64.15. Univariate Polynomial Rings F 64.15. UnivariatePolynomialRing F 64.15. UnivariatePolynomialRing F 64.15. UnivariatePolynomialRing F 64.15. IsUnivariatePolynomialRing S 64.16. Monomial Orderings F 64.16. IsMonomialOrdering F 64.16. LeadingMonomialOfPolynomial F 64.16. LeadingTermOfPolynomial F 64.16. LeadingCoefficientOfPolynomial F 64.16. MonomialComparisonFunction F 64.16. MonomialExtrepComparisonFun F 64.16. MonomialLexOrdering F 64.16. MonomialLexOrdering F 64.16. MonomialGrlexOrdering F 64.16. MonomialGrlexOrdering F 64.16. MonomialGrevlexOrdering F 64.16. MonomialGrevlexOrdering F 64.16. EliminationOrdering F 64.16. EliminationOrdering F 64.16. PolynomialReduction F 64.16. PolynomialReducedRemainder F 64.16. PolynomialDivisionAlgorithm F 64.16. MonomialExtGrlexLess S 64.17. Groebner Bases F 64.17. GroebnerBasis F 64.17. GroebnerBasis F 64.17. GroebnerBasisNC F 64.17. ReducedGroebnerBasis F 64.17. ReducedGroebnerBasis F 64.17. StoredGroebnerBasis F 64.17. InfoGroebner S 64.18. Rational Function Families F 64.18. RationalFunctionsFamily F 64.18. IsPolynomialFunctionsFamily F 64.18. IsRationalFunctionsFamily F 64.18. CoefficientsFamily S 64.19. The Representations of Rational Functions S 64.20. The Defining Attributes of Rational Functions I 64.20. Expanded form of monomials I 64.20. External representation of polynomials F 64.20. IsRationalFunctionDefaultRep F 64.20. ExtRepNumeratorRatFun F 64.20. ExtRepDenominatorRatFun F 64.20. ZeroCoefficientRatFun F 64.20. IsPolynomialDefaultRep F 64.20. ExtRepPolynomialRatFun F 64.20. IsLaurentPolynomialDefaultRep S 64.21. Creation of Rational Functions F 64.21. RationalFunctionByExtRep F 64.21. RationalFunctionByExtRepNC F 64.21. PolynomialByExtRep F 64.21. PolynomialByExtRepNC F 64.21. LaurentPolynomialByExtRep F 64.21. LaurentPolynomialByExtRepNC S 64.22. Arithmetic for External Representations of Polynomials F 64.22. ZippedSum F 64.22. ZippedProduct F 64.22. QuotientPolynomialsExtRep S 64.23. Cancellation Tests for Rational Functions F 64.23. RationalFunctionByExtRepWithCancellation F 64.23. TryGcdCancelExtRepPolynomials F 64.23. HeuristicCancelPolynomials C algfld.tex 65. Algebraic extensions of fields S 65.1. Creation of Algebraic Extensions F 65.1. AlgebraicExtension F 65.1. IsAlgebraicExtension S 65.2. Elements in Algebraic Extensions I 65.2. Operations for algebraic elements F 65.2. IsAlgebraicElement C padics.tex 66. p-adic Numbers (preliminary) S 66.1. Pure p-adic Numbers F 66.1. PurePadicNumberFamily F 66.1. PadicNumber!for pure padics F 66.1. Valuation F 66.1. ShiftedPadicNumber F 66.1. IsPurePadicNumber F 66.1. IsPurePadicNumberFamily S 66.2. Extensions of the p-adic Numbers F 66.2. PadicExtensionNumberFamily F 66.2. PadicNumber F 66.2. PadicNumber F 66.2. PadicNumber F 66.2. IsPadicExtensionNumber F 66.2. IsPadicExtensionNumberFamily C meataxe.tex 67. The MeatAxe S 67.1. MeatAxe Modules F 67.1. GModuleByMats F 67.1. GModuleByMats S 67.2. Module Constructions F 67.2. PermutationGModule F 67.2. TensorProductGModule F 67.2. WedgeGModule S 67.3. Selecting a Different MeatAxe S 67.4. Accessing a Module F 67.4. MTX.Generators F 67.4. MTX.Dimension F 67.4. MTX.Field S 67.5. Irreducibility Tests F 67.5. MTX.IsIrreducible F 67.5. MTX.IsAbsolutelyIrreducible F 67.5. MTX.DegreeSplittingField S 67.6. Finding Submodules F 67.6. MTX.SubmoduleGModule F 67.6. MTX.SubGModule F 67.6. MTX.ProperSubmoduleBasis F 67.6. MTX.BasesSubmodules F 67.6. MTX.BasesMinimalSubmodules F 67.6. MTX.BasesMaximalSubmodules F 67.6. MTX.BasisRadical F 67.6. MTX.BasisSocle F 67.6. MTX.BasesMinimalSupermodules F 67.6. MTX.BasesCompositionSeries F 67.6. MTX.CompositionFactors F 67.6. MTX.CollectedFactors S 67.7. Induced Actions F 67.7. MTX.NormedBasisAndBaseChange F 67.7. MTX.InducedActionSubmodule F 67.7. MTX.InducedActionSubmoduleNB F 67.7. MTX.InducedActionFactorModule F 67.7. MTX.InducedActionMatrix F 67.7. MTX.InducedActionMatrixNB F 67.7. MTX.InducedActionFactorMatrix F 67.7. MTX.InducedAction S 67.8. Module Homomorphisms F 67.8. MTX.IsEquivalent F 67.8. MTX.Isomorphism F 67.8. MTX.Homomorphism F 67.8. MTX.Homomorphisms F 67.8. MTX.Distinguish S 67.9. Invariant Forms F 67.9. MTX.InvariantBilinearForm F 67.9. MTX.InvariantSesquilinearForm F 67.9. MTX.InvariantQuadraticForm F 67.9. MTX.BasisInOrbit F 67.9. MTX.OrthogonalSign S 67.10. The Smash MeatAxe F 67.10. SMTX.RandomIrreducibleSubGModule F 67.10. SMTX.GoodElementGModule F 67.10. SMTX.SortHomGModule F 67.10. SMTX.MinimalSubGModules F 67.10. SMTX.Setter F 67.10. SMTX.Getter F 67.10. SMTX.IrreducibilityTest F 67.10. SMTX.AbsoluteIrreducibilityTest F 67.10. SMTX.MinimalSubGModule F 67.10. SMTX.MatrixSum F 67.10. SMTX.CompleteBasis S 67.11. Smash MeatAxe Flags F 67.11. SMTX.Subbasis F 67.11. SMTX.AlgEl F 67.11. SMTX.AlgElMat F 67.11. SMTX.AlgElCharPol F 67.11. SMTX.AlgElCharPolFac F 67.11. SMTX.AlgElNullspaceVec F 67.11. SMTX.AlgElNullspaceDimension F 67.11. SMTX.CentMat F 67.11. SMTX.CentMatMinPoly C tom.tex 68. Tables of Marks S 68.1. More about Tables of Marks S 68.2. Table of Marks Objects in GAP S 68.3. Constructing Tables of Marks F 68.3. TableOfMarks F 68.3. TableOfMarks F 68.3. TableOfMarks F 68.3. TableOfMarksByLattice F 68.3. LatticeSubgroupsByTom S 68.4. Printing Tables of Marks I 68.4. ViewObj!for tables of marks I 68.4. PrintObj!for tables of marks I 68.4. Display!for tables of marks S 68.5. Sorting Tables of Marks F 68.5. SortedTom F 68.5. PermutationTom S 68.6. Technical Details about Tables of Marks F 68.6. InfoTom F 68.6. IsTableOfMarks F 68.6. TableOfMarksFamily F 68.6. TableOfMarksComponents F 68.6. ConvertToTableOfMarks S 68.7. Attributes of Tables of Marks F 68.7. MarksTom F 68.7. SubsTom F 68.7. NrSubsTom F 68.7. OrdersTom F 68.7. LengthsTom F 68.7. ClassTypesTom F 68.7. ClassNamesTom F 68.7. FusionsTom F 68.7. UnderlyingGroup!for tables of marks F 68.7. IdempotentsTom F 68.7. IdempotentsTomInfo F 68.7. Identifier!for tables of marks F 68.7. MatTom F 68.7. MoebiusTom F 68.7. WeightsTom S 68.8. Properties of Tables of Marks F 68.8. IsAbelianTom F 68.8. IsCyclicTom F 68.8. IsNilpotentTom F 68.8. IsPerfectTom F 68.8. IsSolvableTom S 68.9. Other Operations for Tables of Marks F 68.9. IsInternallyConsistent!for tables of marks F 68.9. DerivedSubgroupTom F 68.9. DerivedSubgroupsTom F 68.9. DerivedSubgroupsTomPossible F 68.9. DerivedSubgroupsTomUnique F 68.9. NormalizerTom F 68.9. NormalizersTom F 68.9. ContainedTom F 68.9. ContainingTom F 68.9. CyclicExtensionsTom F 68.9. CyclicExtensionsTom F 68.9. CyclicExtensionsTom F 68.9. DecomposedFixedPointVector F 68.9. EulerianFunctionByTom F 68.9. IntersectionsTom F 68.9. FactorGroupTom F 68.9. MaximalSubgroupsTom F 68.9. MaximalSubgroupsTom F 68.9. MinimalSupergroupsTom S 68.10. Standard Generators of Groups F 68.10. StandardGeneratorsInfo!for groups F 68.10. HumanReadableDefinition F 68.10. ScriptFromString F 68.10. StandardGeneratorsFunctions F 68.10. IsStandardGeneratorsOfGroup F 68.10. StandardGeneratorsOfGroup S 68.11. Accessing Subgroups via Tables of Marks F 68.11. GeneratorsSubgroupsTom F 68.11. StraightLineProgramsTom F 68.11. IsTableOfMarksWithGens F 68.11. RepresentativeTom F 68.11. RepresentativeTomByGenerators F 68.11. RepresentativeTomByGeneratorsNC F 68.11. StandardGeneratorsInfo!for tables of marks S 68.12. The Interface between Tables of Marks and Character Tables F 68.12. FusionCharTableTom F 68.12. PossibleFusionsCharTableTom F 68.12. PermCharsTom F 68.12. PermCharsTom S 68.13. Generic Construction of Tables of Marks F 68.13. TableOfMarksCyclic F 68.13. TableOfMarksDihedral F 68.13. TableOfMarksFrobenius S 68.14. The Library of Tables of Marks C ctbl.tex 69. Character Tables I 69.0. tables S 69.1. Some Remarks about Character Theory in GAP S 69.2. History of Character Theory Stuff in GAP S 69.3. Creating Character Tables I 69.3. tables I 69.3. character tables I 69.3. library tables I 69.3. character tables!access to I 69.3. character tables!calculate I 69.3. character tables!of groups F 69.3. CharacterTable F 69.3. CharacterTable F 69.3. CharacterTable F 69.3. CharacterTable F 69.3. BrauerTable F 69.3. BrauerTable F 69.3. BrauerTableOp F 69.3. ComputedBrauerTables F 69.3. CharacterTableRegular F 69.3. SupportedCharacterTableInfo F 69.3. ConvertToCharacterTable F 69.3. ConvertToCharacterTableNC S 69.4. Character Table Categories F 69.4. IsNearlyCharacterTable F 69.4. IsCharacterTable F 69.4. IsOrdinaryTable F 69.4. IsBrauerTable F 69.4. IsCharacterTableInProgress F 69.4. InfoCharacterTable F 69.4. NearlyCharacterTablesFamily S 69.5. Conventions for Character Tables S 69.6. The Interface between Character Tables and Groups F 69.6. UnderlyingGroup!for character tables F 69.6. ConjugacyClasses!for character tables F 69.6. IdentificationOfConjugacyClasses F 69.6. ConnectGroupAndCharacterTable F 69.6. ConnectGroupAndCharacterTable F 69.6. CompatibleConjugacyClasses F 69.6. CompatibleConjugacyClasses S 69.7. Operators for Character Tables I 69.7. \*!for character tables I 69.7. /!for character tables I 69.7. mod!for character tables I 69.7. character tables!infix operators S 69.8. Attributes and Properties of Character Tables F 69.8. CharacterDegrees F 69.8. CharacterDegrees F 69.8. CharacterDegrees F 69.8. Irr F 69.8. Irr F 69.8. Irr F 69.8. LinearCharacters F 69.8. LinearCharacters F 69.8. LinearCharacters F 69.8. OrdinaryCharacterTable F 69.8. OrdinaryCharacterTable I 69.8. AbelianInvariants!for character tables I 69.8. CommutatorLength!for character tables I 69.8. Exponent!for character tables I 69.8. IsAbelian!for character tables I 69.8. IsCyclic!for character tables I 69.8. IsElementaryAbelian!for character tables I 69.8. IsFinite!for character tables I 69.8. IsMonomial!for character tables I 69.8. IsNilpotent!for character tables I 69.8. IsPerfect!for character tables I 69.8. IsSimple!for character tables I 69.8. IsSolvable!for character tables I 69.8. IsSporadicSimple!for character tables I 69.8. IsSupersolvable!for character tables I 69.8. NrConjugacyClasses!for character tables I 69.8. Size!for character tables F 69.8. OrdersClassRepresentatives F 69.8. SizesCentralizers F 69.8. SizesConjugacyClasses F 69.8. AutomorphismsOfTable F 69.8. UnderlyingCharacteristic F 69.8. UnderlyingCharacteristic F 69.8. ClassNames F 69.8. ClassNames F 69.8. CharacterNames F 69.8. ClassParameters F 69.8. CharacterParameters F 69.8. Identifier!for character tables F 69.8. InfoText F 69.8. InverseClasses F 69.8. RealClasses I 69.8. classes!real F 69.8. ClassOrbit F 69.8. ClassRoots F 69.8. ClassPositionsOfNormalSubgroups F 69.8. ClassPositionsOfMaximalNormalSubgroups F 69.8. ClassPositionsOfMinimalNormalSubgroups F 69.8. ClassPositionsOfAgemo F 69.8. ClassPositionsOfCentre!for character tables F 69.8. ClassPositionsOfDirectProductDecompositions F 69.8. ClassPositionsOfDirectProductDecompositions F 69.8. ClassPositionsOfDerivedSubgroup F 69.8. ClassPositionsOfElementaryAbelianSeries F 69.8. ClassPositionsOfFittingSubgroup F 69.8. ClassPositionsOfLowerCentralSeries F 69.8. ClassPositionsOfUpperCentralSeries F 69.8. ClassPositionsOfSupersolvableResiduum F 69.8. ClassPositionsOfNormalClosure S 69.9. Operations Concerning Blocks F 69.9. PrimeBlocks F 69.9. PrimeBlocksOp F 69.9. ComputedPrimeBlockss F 69.9. SameBlock F 69.9. BlocksInfo F 69.9. DecompositionMatrix F 69.9. DecompositionMatrix F 69.9. LaTeXStringDecompositionMatrix S 69.10. Other Operations for Character Tables F 69.10. IsInternallyConsistent!for character tables F 69.10. IsPSolvableCharacterTable F 69.10. IsPSolvableCharacterTableOp F 69.10. ComputedIsPSolvableCharacterTables F 69.10. IsClassFusionOfNormalSubgroup F 69.10. Indicator F 69.10. Indicator F 69.10. Indicator F 69.10. IndicatorOp F 69.10. ComputedIndicators F 69.10. NrPolyhedralSubgroups I 69.10. subgroups!polyhedral F 69.10. ClassMultiplicationCoefficient!for character tables I 69.10. class multiplication coefficient I 69.10. structure constant F 69.10. ClassStructureCharTable I 69.10. class multiplication coefficient I 69.10. structure constant F 69.10. MatClassMultCoeffsCharTable I 69.10. structure constant I 69.10. class multiplication coefficient S 69.11. Printing Character Tables I 69.11. ViewObj!for character tables I 69.11. PrintObj!for character tables I 69.11. Display!for character tables F 69.11. DisplayOptions F 69.11. PrintCharacterTable S 69.12. Computing the Irreducible Characters of a Group F 69.12. IrrDixonSchneider F 69.12. IrrConlon F 69.12. IrrBaumClausen F 69.12. IrreducibleRepresentations F 69.12. IrreducibleRepresentations F 69.12. IrreducibleRepresentationsDixon F 69.12. IrreducibleRepresentationsDixon F 69.12. IrreducibleRepresentationsDixon S 69.13. Representations given by modules F 69.13. IrreducibleModules F 69.13. AbsoluteIrreducibleModules F 69.13. AbsolutIrreducibleModules F 69.13. RegularModule S 69.14. The Dixon-Schneider Algorithm I 69.14. Dixon-Schneider algorithm S 69.15. Advanced Methods for Dixon-Schneider Calculations I 69.15. irreducible characters!computation F 69.15. DixonRecord F 69.15. DixonInit F 69.15. DixontinI F 69.15. DixonSplit F 69.15. BestSplittingMatrix F 69.15. DxIncludeIrreducibles F 69.15. SplitCharacters F 69.15. IsDxLargeGroup S 69.16. Components of a Dixon Record S 69.17. An Example of Advanced Dixon-Schneider Calculations S 69.18. Constructing Character Tables from Others F 69.18. CharacterTableDirectProduct F 69.18. FactorsOfDirectProduct F 69.18. CharacterTableFactorGroup F 69.18. CharacterTableIsoclinic F 69.18. CharacterTableIsoclinic F 69.18. CharacterTableIsoclinic F 69.18. SourceOfIsoclinicTable F 69.18. CharacterTableWreathSymmetric S 69.19. Sorted Character Tables F 69.19. CharacterTableWithSortedCharacters F 69.19. CharacterTableWithSortedCharacters F 69.19. SortedCharacters F 69.19. SortedCharacters F 69.19. SortedCharacters F 69.19. CharacterTableWithSortedClasses F 69.19. CharacterTableWithSortedClasses F 69.19. CharacterTableWithSortedClasses F 69.19. CharacterTableWithSortedClasses F 69.19. SortedCharacterTable F 69.19. SortedCharacterTable F 69.19. SortedCharacterTable F 69.19. ClassPermutation S 69.20. Automorphisms and Equivalence of Character Tables F 69.20. MatrixAutomorphisms F 69.20. TableAutomorphisms F 69.20. TableAutomorphisms F 69.20. TableAutomorphisms F 69.20. TransformingPermutations F 69.20. TransformingPermutationsCharacterTables F 69.20. FamiliesOfRows S 69.21. Storing Normal Subgroup Information F 69.21. NormalSubgroupClassesInfo F 69.21. ClassPositionsOfNormalSubgroup F 69.21. NormalSubgroupClasses F 69.21. FactorGroupNormalSubgroupClasses C ctblfuns.tex 70. Class Functions I 70.0. characters I 70.0. group characters I 70.0. virtual characters I 70.0. generalized characters F 70.0. IsClassFunction I 70.0. class function I 70.0. class function objects S 70.1. Why Class Functions? S 70.2. Basic Operations for Class Functions F 70.2. UnderlyingCharacterTable F 70.2. ValuesOfClassFunction S 70.3. Comparison of Class Functions S 70.4. Arithmetic Operations for Class Functions I 70.4. class functions!as ring elements I 70.4. inverse!of class function I 70.4. character value!of group element using powering operator I 70.4. power!meaning for class functions I 70.4. {\accent 94 }!for class functions I 70.4. characteristic!for class functions I 70.4. ComplexConjugate!for class functions I 70.4. GaloisCyc!for class functions I 70.4. Permuted!for class functions I 70.4. Order!of a class function S 70.5. Printing Class Functions I 70.5. ViewObj!for class functions I 70.5. PrintObj!for character tables I 70.5. Display!for character tables S 70.6. Creating Class Functions from Values Lists F 70.6. ClassFunction F 70.6. ClassFunction F 70.6. VirtualCharacter F 70.6. VirtualCharacter F 70.6. Character F 70.6. ClassFunctionSameType S 70.7. Creating Class Functions using Groups F 70.7. TrivialCharacter F 70.7. TrivialCharacter F 70.7. NaturalCharacter F 70.7. NaturalCharacter F 70.7. PermutationCharacter F 70.7. PermutationCharacter S 70.8. Operations for Class Functions F 70.8. IsCharacter I 70.8. ordinary character I 70.8. Brauer character F 70.8. IsVirtualCharacter I 70.8. virtual character F 70.8. IsIrreducibleCharacter I 70.8. irreducible character F 70.8. DegreeOfCharacter I 70.8. constituent!of a group character I 70.8. decompose!a group character I 70.8. multiplicity!of constituents of a group character I 70.8. inner product!of group characters F 70.8. ScalarProduct!for characters F 70.8. MatScalarProducts F 70.8. MatScalarProducts F 70.8. Norm!of character F 70.8. ConstituentsOfCharacter F 70.8. KernelOfCharacter F 70.8. ClassPositionsOfKernel F 70.8. CentreOfCharacter I 70.8. centre!of a character F 70.8. ClassPositionsOfCentre!for characters F 70.8. InertiaSubgroup F 70.8. CycleStructureClass F 70.8. IsTransitive!for characters I 70.8. IsTransitive!for class functions F 70.8. Transitivity!for characters I 70.8. Transitivity!for class functions F 70.8. CentralCharacter I 70.8. central character F 70.8. DeterminantOfCharacter I 70.8. determinant character F 70.8. EigenvaluesChar F 70.8. Tensored S 70.9. Restricted and Induced Class Functions I 70.9. inflated class functions F 70.9. RestrictedClassFunction F 70.9. RestrictedClassFunction F 70.9. RestrictedClassFunction F 70.9. RestrictedClassFunctions F 70.9. RestrictedClassFunctions F 70.9. RestrictedClassFunctions F 70.9. InducedClassFunction F 70.9. InducedClassFunction F 70.9. InducedClassFunction F 70.9. InducedClassFunctions F 70.9. InducedClassFunctions F 70.9. InducedClassFunctions F 70.9. InducedClassFunctionsByFusionMap F 70.9. InducedCyclic F 70.9. InducedCyclic F 70.9. InducedCyclic F 70.9. InducedCyclic S 70.10. Reducing Virtual Characters F 70.10. ReducedClassFunctions F 70.10. ReducedClassFunctions F 70.10. ReducedCharacters F 70.10. IrreducibleDifferences F 70.10. IrreducibleDifferences F 70.10. IrreducibleDifferences F 70.10. IrreducibleDifferences F 70.10. LLL I 70.10. LLL algorithm!for virtual characters I 70.10. short vectors spanning a lattice I 70.10. lattice basis reduction!for virtual characters F 70.10. Extract F 70.10. OrthogonalEmbeddingsSpecialDimension F 70.10. Decreased F 70.10. DnLattice F 70.10. DnLatticeIterative S 70.11. Symmetrizations of Class Functions I 70.11. characters!symmetrizations of F 70.11. Symmetrizations F 70.11. Symmetrizations F 70.11. SymmetricParts F 70.11. AntiSymmetricParts F 70.11. OrthogonalComponents I 70.11. symmetrizations!orthogonal I 70.11. Frame I 70.11. Murnaghan components I 70.11. symmetrizations!orthogonal I 70.11. Frame I 70.11. Murnaghan components F 70.11. SymplecticComponents I 70.11. symmetrizations!symplectic I 70.11. Murnaghan components S 70.12. Molien Series F 70.12. MolienSeries F 70.12. MolienSeries F 70.12. MolienSeries F 70.12. MolienSeries F 70.12. MolienSeriesInfo F 70.12. ValueMolienSeries F 70.12. MolienSeriesWithGivenDenominator S 70.13. Possible Permutation Characters I 70.13. characters!permutation I 70.13. candidates!for permutation characters I 70.13. possible permutation characters I 70.13. permutation characters!possible F 70.13. PermCharInfo F 70.13. PermCharInfo F 70.13. PermCharInfoRelative S 70.14. Computing Possible Permutation Characters I 70.14. characters!permutation I 70.14. candidates!for permutation characters I 70.14. possible permutation characters I 70.14. permutation characters!possible F 70.14. PermChars F 70.14. PermChars F 70.14. PermChars F 70.14. TestPerm1 F 70.14. TestPerm2 F 70.14. TestPerm3 F 70.14. TestPerm4 F 70.14. TestPerm5 F 70.14. PermBounds F 70.14. PermComb F 70.14. Inequalities S 70.15. Operations for Brauer Characters F 70.15. FrobeniusCharacterValue F 70.15. BrauerCharacterValue F 70.15. SizeOfFieldOfDefinition F 70.15. RealizableBrauerCharacters S 70.16. Domains Generated by Class Functions C ctblmaps.tex 71. Maps Concerning Character Tables I 71.0. maps I 71.0. parametrized maps S 71.1. Power Maps F 71.1. PowerMap F 71.1. PowerMapOp F 71.1. ComputedPowerMaps F 71.1. PossiblePowerMaps F 71.1. ElementOrdersPowerMap F 71.1. PowerMapByComposition F 71.1. OrbitPowerMaps I 71.1. matrix automorphisms F 71.1. RepresentativesPowerMaps S 71.2. Class Fusions between Character Tables I 71.2. fusions I 71.2. subgroup fusions F 71.2. FusionConjugacyClasses F 71.2. FusionConjugacyClasses F 71.2. FusionConjugacyClasses F 71.2. FusionConjugacyClassesOp F 71.2. FusionConjugacyClassesOp F 71.2. ComputedClassFusions F 71.2. GetFusionMap F 71.2. GetFusionMap F 71.2. StoreFusion F 71.2. NamesOfFusionSources F 71.2. PossibleClassFusions F 71.2. OrbitFusions I 71.2. table automorphisms F 71.2. RepresentativesFusions F 71.2. RepresentativesFusions F 71.2. ConsiderStructureConstants S 71.3. Parametrized Maps I 71.3. map!parametrized I 71.3. class functions F 71.3. CompositionMaps F 71.3. InverseMap F 71.3. ProjectionMap F 71.3. Indirected F 71.3. Parametrized F 71.3. ContainedMaps F 71.3. UpdateMap F 71.3. MeetMaps F 71.3. CommutativeDiagram F 71.3. CheckFixedPoints F 71.3. TransferDiagram F 71.3. TestConsistencyMaps F 71.3. Indeterminateness F 71.3. PrintAmbiguity F 71.3. ContainedSpecialVectors F 71.3. IntScalarProducts F 71.3. NonnegIntScalarProducts F 71.3. ContainedPossibleVirtualCharacters F 71.3. ContainedPossibleCharacters I 71.3. IntScalarProducts I 71.3. NonnegIntScalarProducts I 71.3. ContainedPossibleVirtualCharacters I 71.3. ContainedPossibleCharacters I 71.3. ContainedSpecialVectors F 71.3. CollapsedMat F 71.3. ContainedDecomposables F 71.3. ContainedCharacters S 71.4. Subroutines for the Construction of Power Maps F 71.4. InitPowerMap F 71.4. Congruences!for character tables F 71.4. ConsiderKernels F 71.4. ConsiderSmallerPowerMaps F 71.4. MinusCharacter F 71.4. PowerMapsAllowedBySymmetrizations S 71.5. Subroutines for the Construction of Class Fusions F 71.5. InitFusion F 71.5. CheckPermChar I 71.5. permutation character F 71.5. ConsiderTableAutomorphisms I 71.5. table automorphisms F 71.5. FusionsAllowedByRestrictions C ctblmono.tex 72. Monomiality Questions F 72.0. InfoMonomial S 72.1. Character Degrees and Derived Length F 72.1. Alpha F 72.1. Delta F 72.1. IsBergerCondition F 72.1. IsBergerCondition S 72.2. Primitivity of Characters F 72.2. TestHomogeneous F 72.2. IsPrimitiveCharacter F 72.2. TestQuasiPrimitive F 72.2. IsQuasiPrimitive F 72.2. TestInducedFromNormalSubgroup F 72.2. IsInducedFromNormalSubgroup S 72.3. Testing Monomiality I 72.3. IsMonomial!for groups I 72.3. IsMonomial!for characters F 72.3. TestMonomial F 72.3. TestMonomial F 72.3. TestMonomial F 72.3. TestMonomial F 72.3. TestMonomialUseLattice F 72.3. IsMonomialNumber I 72.3. IsMonomial!for positive integers F 72.3. TestMonomialQuick F 72.3. TestMonomialQuick F 72.3. TestSubnormallyMonomial F 72.3. TestSubnormallyMonomial F 72.3. IsSubnormallyMonomial F 72.3. IsSubnormallyMonomial F 72.3. TestRelativelySM F 72.3. TestRelativelySM F 72.3. TestRelativelySM F 72.3. TestRelativelySM F 72.3. IsRelativelySM F 72.3. IsRelativelySM S 72.4. Minimal Nonmonomial Groups F 72.4. IsMinimalNonmonomial F 72.4. MinimalNonmonomialGroup C install.tex 73. Installing GAP I 73.0. installation I 73.0. options S 73.1. Installation Overview S 73.2. Get the Archives S 73.3. Unpacking S 73.4. Compilation S 73.5. Test of the installation S 73.6. Packages S 73.7. Finish Installation and Cleanup S 73.8. The Documentation S 73.9. If Things Go Wrong I 73.9. problems I 73.9. FAQ I 73.9. support!email address I 73.9. bug reports!see If Things Go Wrong S 73.10. Known Problems of the Configure Process S 73.11. Problems on Particular Systems S 73.12. Optimization and Compiler Options S 73.13. Porting GAP S 73.14. GAP for Macintosh OS X I 73.14. OSX I 73.14. Apple I 73.14. Macintosh S 73.15. GAP for MacOS I 73.15. MacOS I 73.15. Apple I 73.15. Macintosh S 73.16. Installation of GAP for MacOS S 73.17. Expert Windows installation S 73.18. Copyrights C gappkg.tex 74. GAP Packages I 74.0. package S 74.1. Installing a GAP Package S 74.2. Loading a GAP Package I 74.2. automatic loading of GAP packages I 74.2. disable automatic loading F 74.2. LoadPackage F 74.2. LoadPackage I 74.2. NOAUTO S 74.3. Functions for GAP Packages F 74.3. ReadPackage F 74.3. ReadPackage F 74.3. RereadPackage F 74.3. RereadPackage F 74.3. TestPackageAvailability F 74.3. TestPackageAvailability F 74.3. InstalledPackageVersion F 74.3. DirectoriesPackageLibrary F 74.3. DirectoriesPackagePrograms F 74.3. CompareVersionNumbers F 74.3. CompareVersionNumbers C obsolete.tex 75. Replaced and Removed Command Names I 75.0. obsolete I 75.0. deprecated I 75.0. legacy S 75.1. Group Actions - Name Changes I 75.1. group operations I 75.1. Operation I 75.1. RepresentativeOperation I 75.1. OperationHomomorphism I 75.1. FunctionOperation S 75.2. Package Interface - Obsolete Functions and Name Changes I 75.2. DeclarePackage I 75.2. DeclareAutoPackage I 75.2. DeclarePackageDocumentation I 75.2. DeclarePackageAutoDocumentation I 75.2. RequirePackage I 75.2. ReadPkg I 75.2. RereadPkg I 75.2. CreateCompletionFilesPkg S 75.3. Normal Forms of Integer Matrices - Name Changes I 75.3. Smith normal form I 75.3. Hermite normal form S 75.4. Miscellaneous Name Changes or Removed Names I 75.4. QUIET I 75.4. BANNER I 75.4. GAPInfo I 75.4. MonomialTotalDegreeLess I 75.4. NormedVectors C copyrigh.tex 76. Copyright