\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Introduction}}{8}{chapter.1}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Resolutions}}{9}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }The \texttt {HAPResolution} datatype in \textsf {HAPprime}}}{9}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }Implementation: Constructing resolutions}}{9}{section.2.2}
\contentsline {section}{\numberline {2.3}\leavevmode {\color {Chapter }Resolution construction functions}}{10}{section.2.3}
\contentsline {subsection}{\numberline {2.3.1}\leavevmode {\color {Chapter }LengthOneResolutionPrimePowerGroup}}{10}{subsection.2.3.1}
\contentsline {subsection}{\numberline {2.3.2}\leavevmode {\color {Chapter }LengthZeroResolutionPrimePowerGroup}}{10}{subsection.2.3.2}
\contentsline {section}{\numberline {2.4}\leavevmode {\color {Chapter }Resolution data access functions}}{11}{section.2.4}
\contentsline {subsection}{\numberline {2.4.1}\leavevmode {\color {Chapter }ResolutionLength}}{11}{subsection.2.4.1}
\contentsline {subsection}{\numberline {2.4.2}\leavevmode {\color {Chapter }ResolutionGroup}}{11}{subsection.2.4.2}
\contentsline {subsection}{\numberline {2.4.3}\leavevmode {\color {Chapter }ResolutionFpGModuleGF}}{11}{subsection.2.4.3}
\contentsline {subsection}{\numberline {2.4.4}\leavevmode {\color {Chapter }ResolutionModuleRank}}{11}{subsection.2.4.4}
\contentsline {subsection}{\numberline {2.4.5}\leavevmode {\color {Chapter }ResolutionModuleRanks}}{11}{subsection.2.4.5}
\contentsline {subsection}{\numberline {2.4.6}\leavevmode {\color {Chapter }BoundaryFpGModuleHomomorphismGF}}{11}{subsection.2.4.6}
\contentsline {subsection}{\numberline {2.4.7}\leavevmode {\color {Chapter }ResolutionsAreEqual}}{11}{subsection.2.4.7}
\contentsline {section}{\numberline {2.5}\leavevmode {\color {Chapter }Example: Computing and working with resolutions}}{12}{section.2.5}
\contentsline {section}{\numberline {2.6}\leavevmode {\color {Chapter }Miscellaneous resolution functions}}{13}{section.2.6}
\contentsline {subsection}{\numberline {2.6.1}\leavevmode {\color {Chapter }BestCentralSubgroupForResolutionFiniteExtension}}{13}{subsection.2.6.1}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Graded algebras}}{14}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Graded algebras in \textsf {HAP} (and \textsf {HAPprime})}}{14}{section.3.1}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }Data access functions}}{14}{section.3.2}
\contentsline {subsection}{\numberline {3.2.1}\leavevmode {\color {Chapter }ModPRingGeneratorDegrees}}{14}{subsection.3.2.1}
\contentsline {subsection}{\numberline {3.2.2}\leavevmode {\color {Chapter }ModPRingNiceBasis}}{15}{subsection.3.2.2}
\contentsline {subsection}{\numberline {3.2.3}\leavevmode {\color {Chapter }ModPRingNiceBasisAsPolynomials}}{15}{subsection.3.2.3}
\contentsline {subsection}{\numberline {3.2.4}\leavevmode {\color {Chapter }ModPRingBasisAsPolynomials}}{15}{subsection.3.2.4}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter }Other functions}}{15}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}\leavevmode {\color {Chapter }PresentationOfGradedStructureConstantAlgebra}}{15}{subsection.3.3.1}
\contentsline {section}{\numberline {3.4}\leavevmode {\color {Chapter }Example: Graded algebras and mod-$p$ cohomology rings}}{16}{section.3.4}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Presentations of graded algebras}}{17}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }The \texttt {GradedAlgebraPresentation} datatype}}{17}{section.4.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Construction function}}{17}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }GradedAlgebraPresentation construction functions}}{17}{subsection.4.2.1}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Data access functions}}{18}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}\leavevmode {\color {Chapter }BaseRing}}{18}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}\leavevmode {\color {Chapter }CoefficientsRing}}{18}{subsection.4.3.2}
\contentsline {subsection}{\numberline {4.3.3}\leavevmode {\color {Chapter }IndeterminatesOfGradedAlgebraPresentation}}{18}{subsection.4.3.3}
\contentsline {subsection}{\numberline {4.3.4}\leavevmode {\color {Chapter }GeneratorsOfPresentationIdeal}}{18}{subsection.4.3.4}
\contentsline {subsection}{\numberline {4.3.5}\leavevmode {\color {Chapter }PresentationIdeal}}{18}{subsection.4.3.5}
\contentsline {subsection}{\numberline {4.3.6}\leavevmode {\color {Chapter }IndeterminateDegrees}}{18}{subsection.4.3.6}
\contentsline {subsection}{\numberline {4.3.7}\leavevmode {\color {Chapter }Example: Constructing and accessing data of a \texttt {GradedAlgebraPresentation}}}{18}{subsection.4.3.7}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter }Other functions}}{19}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}\leavevmode {\color {Chapter }TensorProduct}}{19}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}\leavevmode {\color {Chapter }IsIsomorphicGradedAlgebra}}{19}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}\leavevmode {\color {Chapter }IsAssociatedGradedRing}}{19}{subsection.4.4.3}
\contentsline {subsection}{\numberline {4.4.4}\leavevmode {\color {Chapter }DegreeOfRepresentative}}{19}{subsection.4.4.4}
\contentsline {subsection}{\numberline {4.4.5}\leavevmode {\color {Chapter }MaximumDegreeForPresentation}}{20}{subsection.4.4.5}
\contentsline {subsection}{\numberline {4.4.6}\leavevmode {\color {Chapter }SubspaceDimensionDegree}}{20}{subsection.4.4.6}
\contentsline {subsection}{\numberline {4.4.7}\leavevmode {\color {Chapter }SubspaceBasisRepsByDegree}}{20}{subsection.4.4.7}
\contentsline {subsection}{\numberline {4.4.8}\leavevmode {\color {Chapter }CoefficientsOfPoincareSeries}}{20}{subsection.4.4.8}
\contentsline {subsection}{\numberline {4.4.9}\leavevmode {\color {Chapter }HilbertPoincareSeries}}{20}{subsection.4.4.9}
\contentsline {subsection}{\numberline {4.4.10}\leavevmode {\color {Chapter }LHSSpectralSequence}}{20}{subsection.4.4.10}
\contentsline {section}{\numberline {4.5}\leavevmode {\color {Chapter }Example: Computing the Lyndon-Hoschild-Serre spectral sequence and mod-$p$ cohomology ring for a small $p$-group}}{21}{section.4.5}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }$\@mathbb {F}G$-modules}}{23}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }The \texttt {FpGModuleGF} datatype}}{23}{section.5.1}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Implementation details: Block echelon form}}{24}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }Generating vectors and their block structure}}{24}{subsection.5.2.1}
\contentsline {subsection}{\numberline {5.2.2}\leavevmode {\color {Chapter }Matrix echelon reduction and head elements}}{24}{subsection.5.2.2}
\contentsline {subsection}{\numberline {5.2.3}\leavevmode {\color {Chapter }Echelon block structure and minimal generators}}{25}{subsection.5.2.3}
\contentsline {subsection}{\numberline {5.2.4}\leavevmode {\color {Chapter }Intersection of two modules}}{26}{subsection.5.2.4}
\contentsline {section}{\numberline {5.3}\leavevmode {\color {Chapter }Construction functions}}{27}{section.5.3}
\contentsline {subsection}{\numberline {5.3.1}\leavevmode {\color {Chapter }FpGModuleGF construction functions}}{27}{subsection.5.3.1}
\contentsline {subsection}{\numberline {5.3.2}\leavevmode {\color {Chapter }FpGModuleFromFpGModuleGF}}{28}{subsection.5.3.2}
\contentsline {subsection}{\numberline {5.3.3}\leavevmode {\color {Chapter }MutableCopyModule}}{28}{subsection.5.3.3}
\contentsline {subsection}{\numberline {5.3.4}\leavevmode {\color {Chapter }CanonicalAction}}{28}{subsection.5.3.4}
\contentsline {subsection}{\numberline {5.3.5}\leavevmode {\color {Chapter }Example: Constructing a \texttt {FpGModuleGF}}}{29}{subsection.5.3.5}
\contentsline {section}{\numberline {5.4}\leavevmode {\color {Chapter }Data access functions}}{30}{section.5.4}
\contentsline {subsection}{\numberline {5.4.1}\leavevmode {\color {Chapter }ModuleGroup}}{30}{subsection.5.4.1}
\contentsline {subsection}{\numberline {5.4.2}\leavevmode {\color {Chapter }ModuleGroupOrder}}{30}{subsection.5.4.2}
\contentsline {subsection}{\numberline {5.4.3}\leavevmode {\color {Chapter }ModuleAction}}{30}{subsection.5.4.3}
\contentsline {subsection}{\numberline {5.4.4}\leavevmode {\color {Chapter }ModuleActionBlockSize}}{30}{subsection.5.4.4}
\contentsline {subsection}{\numberline {5.4.5}\leavevmode {\color {Chapter }ModuleGroupAndAction}}{30}{subsection.5.4.5}
\contentsline {subsection}{\numberline {5.4.6}\leavevmode {\color {Chapter }ModuleCharacteristic}}{31}{subsection.5.4.6}
\contentsline {subsection}{\numberline {5.4.7}\leavevmode {\color {Chapter }ModuleField}}{31}{subsection.5.4.7}
\contentsline {subsection}{\numberline {5.4.8}\leavevmode {\color {Chapter }ModuleAmbientDimension}}{31}{subsection.5.4.8}
\contentsline {subsection}{\numberline {5.4.9}\leavevmode {\color {Chapter }AmbientModuleDimension}}{31}{subsection.5.4.9}
\contentsline {subsection}{\numberline {5.4.10}\leavevmode {\color {Chapter }DisplayBlocks (for FpGModuleGF)}}{31}{subsection.5.4.10}
\contentsline {subsection}{\numberline {5.4.11}\leavevmode {\color {Chapter }Example: Accessing data about a \texttt {FpGModuleGF}}}{32}{subsection.5.4.11}
\contentsline {section}{\numberline {5.5}\leavevmode {\color {Chapter }Generator and vector space functions}}{33}{section.5.5}
\contentsline {subsection}{\numberline {5.5.1}\leavevmode {\color {Chapter }ModuleGenerators}}{33}{subsection.5.5.1}
\contentsline {subsection}{\numberline {5.5.2}\leavevmode {\color {Chapter }ModuleGeneratorsAreMinimal}}{33}{subsection.5.5.2}
\contentsline {subsection}{\numberline {5.5.3}\leavevmode {\color {Chapter }ModuleGeneratorsAreEchelonForm}}{33}{subsection.5.5.3}
\contentsline {subsection}{\numberline {5.5.4}\leavevmode {\color {Chapter }ModuleIsFullCanonical}}{33}{subsection.5.5.4}
\contentsline {subsection}{\numberline {5.5.5}\leavevmode {\color {Chapter }ModuleGeneratorsForm}}{33}{subsection.5.5.5}
\contentsline {subsection}{\numberline {5.5.6}\leavevmode {\color {Chapter }ModuleRank}}{34}{subsection.5.5.6}
\contentsline {subsection}{\numberline {5.5.7}\leavevmode {\color {Chapter }ModuleVectorSpaceBasis}}{34}{subsection.5.5.7}
\contentsline {subsection}{\numberline {5.5.8}\leavevmode {\color {Chapter }ModuleVectorSpaceDimension}}{34}{subsection.5.5.8}
\contentsline {subsection}{\numberline {5.5.9}\leavevmode {\color {Chapter }MinimalGeneratorsModule}}{34}{subsection.5.5.9}
\contentsline {subsection}{\numberline {5.5.10}\leavevmode {\color {Chapter }RadicalOfModule}}{35}{subsection.5.5.10}
\contentsline {subsection}{\numberline {5.5.11}\leavevmode {\color {Chapter }Example: Generators and basis vectors of a \texttt {FpGModuleGF}}}{35}{subsection.5.5.11}
\contentsline {section}{\numberline {5.6}\leavevmode {\color {Chapter }Block echelon functions}}{36}{section.5.6}
\contentsline {subsection}{\numberline {5.6.1}\leavevmode {\color {Chapter }EchelonModuleGenerators}}{36}{subsection.5.6.1}
\contentsline {subsection}{\numberline {5.6.2}\leavevmode {\color {Chapter }ReverseEchelonModuleGenerators}}{37}{subsection.5.6.2}
\contentsline {subsection}{\numberline {5.6.3}\leavevmode {\color {Chapter }Example: Converting a \texttt {FpGModuleGF} to block echelon form}}{38}{subsection.5.6.3}
\contentsline {section}{\numberline {5.7}\leavevmode {\color {Chapter }Sum and intersection functions}}{39}{section.5.7}
\contentsline {subsection}{\numberline {5.7.1}\leavevmode {\color {Chapter }DirectSumOfModules}}{39}{subsection.5.7.1}
\contentsline {subsection}{\numberline {5.7.2}\leavevmode {\color {Chapter }DirectDecompositionOfModule}}{39}{subsection.5.7.2}
\contentsline {subsection}{\numberline {5.7.3}\leavevmode {\color {Chapter }IntersectionModules}}{40}{subsection.5.7.3}
\contentsline {subsection}{\numberline {5.7.4}\leavevmode {\color {Chapter }SumModules}}{40}{subsection.5.7.4}
\contentsline {subsection}{\numberline {5.7.5}\leavevmode {\color {Chapter }Example: Sum and intersection of \texttt {FpGModuleGF}s}}{41}{subsection.5.7.5}
\contentsline {section}{\numberline {5.8}\leavevmode {\color {Chapter }Miscellaneous functions}}{42}{section.5.8}
\contentsline {subsection}{\numberline {5.8.1}\leavevmode {\color {Chapter }= (for FpGModuleGF)}}{42}{subsection.5.8.1}
\contentsline {subsection}{\numberline {5.8.2}\leavevmode {\color {Chapter }IsModuleElement}}{42}{subsection.5.8.2}
\contentsline {subsection}{\numberline {5.8.3}\leavevmode {\color {Chapter }IsSubModule}}{42}{subsection.5.8.3}
\contentsline {subsection}{\numberline {5.8.4}\leavevmode {\color {Chapter }RandomElement}}{42}{subsection.5.8.4}
\contentsline {subsection}{\numberline {5.8.5}\leavevmode {\color {Chapter }Random Submodule}}{43}{subsection.5.8.5}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }$\@mathbb {F}G$-module homomorphisms}}{44}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }The \texttt {FpGModuleHomomorphismGF} datatype}}{44}{section.6.1}
\contentsline {section}{\numberline {6.2}\leavevmode {\color {Chapter }Calculating the kernel of a $\@mathbb {F}G$-module homorphism by splitting into two homomorphisms}}{44}{section.6.2}
\contentsline {section}{\numberline {6.3}\leavevmode {\color {Chapter }Calculating the kernel of a $\@mathbb {F}G$-module homorphism by column reduction and partitioning}}{45}{section.6.3}
\contentsline {section}{\numberline {6.4}\leavevmode {\color {Chapter }Construction functions}}{46}{section.6.4}
\contentsline {subsection}{\numberline {6.4.1}\leavevmode {\color {Chapter }FpGModuleHomomorphismGF construction functions}}{46}{subsection.6.4.1}
\contentsline {subsection}{\numberline {6.4.2}\leavevmode {\color {Chapter }Example: Constructing a \texttt {FpGModuleHomomorphismGF}}}{46}{subsection.6.4.2}
\contentsline {section}{\numberline {6.5}\leavevmode {\color {Chapter }Data access functions}}{47}{section.6.5}
\contentsline {subsection}{\numberline {6.5.1}\leavevmode {\color {Chapter }SourceModule}}{47}{subsection.6.5.1}
\contentsline {subsection}{\numberline {6.5.2}\leavevmode {\color {Chapter }TargetModule}}{47}{subsection.6.5.2}
\contentsline {subsection}{\numberline {6.5.3}\leavevmode {\color {Chapter }ModuleHomomorphismGeneratorMatrix}}{47}{subsection.6.5.3}
\contentsline {subsection}{\numberline {6.5.4}\leavevmode {\color {Chapter }DisplayBlocks (for FpGModuleHomomorphismGF)}}{47}{subsection.6.5.4}
\contentsline {subsection}{\numberline {6.5.5}\leavevmode {\color {Chapter }DisplayModuleHomomorphismGeneratorMatrix}}{47}{subsection.6.5.5}
\contentsline {subsection}{\numberline {6.5.6}\leavevmode {\color {Chapter }DisplayModuleHomomorphismGeneratorMatrixBlocks}}{47}{subsection.6.5.6}
\contentsline {subsection}{\numberline {6.5.7}\leavevmode {\color {Chapter }Example: Accessing data about a \texttt {FpGModuleHomomorphismGF}}}{48}{subsection.6.5.7}
\contentsline {section}{\numberline {6.6}\leavevmode {\color {Chapter }Image and kernel functions}}{49}{section.6.6}
\contentsline {subsection}{\numberline {6.6.1}\leavevmode {\color {Chapter }ImageOfModuleHomomorphism}}{49}{subsection.6.6.1}
\contentsline {subsection}{\numberline {6.6.2}\leavevmode {\color {Chapter }PreImageRepresentativeOfModuleHomomorphism}}{49}{subsection.6.6.2}
\contentsline {subsection}{\numberline {6.6.3}\leavevmode {\color {Chapter }KernelOfModuleHomomorphism}}{49}{subsection.6.6.3}
\contentsline {subsection}{\numberline {6.6.4}\leavevmode {\color {Chapter }Example: Kernel and Image of a \texttt {FpGModuleHomomorphismGF}}}{50}{subsection.6.6.4}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Ring homomorphisms}}{52}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter }The \texttt {HAPRingHomomorphism} datatype}}{52}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}\leavevmode {\color {Chapter }Implementation details}}{52}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}\leavevmode {\color {Chapter }Elimination orderings}}{53}{subsection.7.1.2}
\contentsline {section}{\numberline {7.2}\leavevmode {\color {Chapter }Construction functions}}{54}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}\leavevmode {\color {Chapter }HAPRingToSubringHomomorphism}}{54}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}\leavevmode {\color {Chapter }HAPSubringToRingHomomorphism}}{54}{subsection.7.2.2}
\contentsline {subsection}{\numberline {7.2.3}\leavevmode {\color {Chapter }HAPRingHomomorphismByIndeterminateMap}}{54}{subsection.7.2.3}
\contentsline {subsection}{\numberline {7.2.4}\leavevmode {\color {Chapter }HAPRingReductionHomomorphism (for ring presentation)}}{54}{subsection.7.2.4}
\contentsline {subsection}{\numberline {7.2.5}\leavevmode {\color {Chapter }HAPZeroRingHomomorphism}}{55}{subsection.7.2.5}
\contentsline {subsection}{\numberline {7.2.6}\leavevmode {\color {Chapter }InverseRingHomomorphism}}{55}{subsection.7.2.6}
\contentsline {subsection}{\numberline {7.2.7}\leavevmode {\color {Chapter }CompositionRingHomomorphism}}{55}{subsection.7.2.7}
\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }Data access functions}}{55}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}\leavevmode {\color {Chapter }SourceGenerators}}{55}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}\leavevmode {\color {Chapter }SourceRelations}}{55}{subsection.7.3.2}
\contentsline {subsection}{\numberline {7.3.3}\leavevmode {\color {Chapter }SourcePolynomialRing}}{55}{subsection.7.3.3}
\contentsline {subsection}{\numberline {7.3.4}\leavevmode {\color {Chapter }ImageGenerators}}{56}{subsection.7.3.4}
\contentsline {subsection}{\numberline {7.3.5}\leavevmode {\color {Chapter }ImageRelations}}{56}{subsection.7.3.5}
\contentsline {subsection}{\numberline {7.3.6}\leavevmode {\color {Chapter }ImagePolynomialRing}}{56}{subsection.7.3.6}
\contentsline {section}{\numberline {7.4}\leavevmode {\color {Chapter }General functions}}{56}{section.7.4}
\contentsline {subsection}{\numberline {7.4.1}\leavevmode {\color {Chapter }ImageOfRingHomomorphism}}{56}{subsection.7.4.1}
\contentsline {subsection}{\numberline {7.4.2}\leavevmode {\color {Chapter }PreimageOfRingHomomorphism}}{56}{subsection.7.4.2}
\contentsline {section}{\numberline {7.5}\leavevmode {\color {Chapter }Example: Constructing and using a \texttt {HAPRingHomomorphism}}}{56}{section.7.5}
\contentsline {chapter}{\numberline {8}\leavevmode {\color {Chapter }Derivations}}{59}{chapter.8}
\contentsline {section}{\numberline {8.1}\leavevmode {\color {Chapter }The \texttt {HAPDerivation} datatype}}{59}{section.8.1}
\contentsline {section}{\numberline {8.2}\leavevmode {\color {Chapter }Computing the kernel and homology of a derivation}}{59}{section.8.2}
\contentsline {section}{\numberline {8.3}\leavevmode {\color {Chapter }Construction function}}{60}{section.8.3}
\contentsline {subsection}{\numberline {8.3.1}\leavevmode {\color {Chapter }HAPDerivation construction functions}}{60}{subsection.8.3.1}
\contentsline {section}{\numberline {8.4}\leavevmode {\color {Chapter }Data access function}}{60}{section.8.4}
\contentsline {subsection}{\numberline {8.4.1}\leavevmode {\color {Chapter }DerivationRing}}{60}{subsection.8.4.1}
\contentsline {subsection}{\numberline {8.4.2}\leavevmode {\color {Chapter }DerivationImages}}{61}{subsection.8.4.2}
\contentsline {subsection}{\numberline {8.4.3}\leavevmode {\color {Chapter }DerivationRelations}}{61}{subsection.8.4.3}
\contentsline {subsection}{\numberline {8.4.4}\leavevmode {\color {Chapter }Example: Constructing and accessing data of a \texttt {HAPDerivation}}}{61}{subsection.8.4.4}
\contentsline {section}{\numberline {8.5}\leavevmode {\color {Chapter }Image, kernel and homology functions}}{61}{section.8.5}
\contentsline {subsection}{\numberline {8.5.1}\leavevmode {\color {Chapter }ImageOfDerivation}}{61}{subsection.8.5.1}
\contentsline {subsection}{\numberline {8.5.2}\leavevmode {\color {Chapter }KernelOfDerivation}}{61}{subsection.8.5.2}
\contentsline {subsection}{\numberline {8.5.3}\leavevmode {\color {Chapter }HomologyOfDerivation}}{62}{subsection.8.5.3}
\contentsline {subsection}{\numberline {8.5.4}\leavevmode {\color {Chapter }Example: Homology of a \texttt {HAPDerivation}}}{62}{subsection.8.5.4}
\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Poincar{\'e} series}}{64}{chapter.9}
\contentsline {section}{\numberline {9.1}\leavevmode {\color {Chapter }Computing the Poincar{\'e} series using spectral sequences}}{64}{section.9.1}
\contentsline {section}{\numberline {9.2}\leavevmode {\color {Chapter }Computing the Poincar{\'e} series using a minimal resolution}}{64}{section.9.2}
\contentsline {subsection}{\numberline {9.2.1}\leavevmode {\color {Chapter }PoincareSeriesAutoMem}}{65}{subsection.9.2.1}
\contentsline {section}{\numberline {9.3}\leavevmode {\color {Chapter }Example Poincar{\'e} series computations}}{65}{section.9.3}
\contentsline {section}{\numberline {9.4}\leavevmode {\color {Chapter }The Poincar{\'e} series of groups of order 64 and 128}}{66}{section.9.4}
\contentsline {chapter}{\numberline {10}\leavevmode {\color {Chapter }General Functions}}{67}{chapter.10}
\contentsline {section}{\numberline {10.1}\leavevmode {\color {Chapter }Matrices}}{67}{section.10.1}
\contentsline {subsection}{\numberline {10.1.1}\leavevmode {\color {Chapter }SumIntersectionMatDestructive}}{67}{subsection.10.1.1}
\contentsline {subsection}{\numberline {10.1.2}\leavevmode {\color {Chapter }SolutionMat (for multiple vectors)}}{68}{subsection.10.1.2}
\contentsline {subsection}{\numberline {10.1.3}\leavevmode {\color {Chapter }IsSameSubspace}}{68}{subsection.10.1.3}
\contentsline {subsection}{\numberline {10.1.4}\leavevmode {\color {Chapter }PrintDimensionsMat}}{68}{subsection.10.1.4}
\contentsline {subsection}{\numberline {10.1.5}\leavevmode {\color {Chapter }Example: matrices and vector spaces}}{68}{subsection.10.1.5}
\contentsline {section}{\numberline {10.2}\leavevmode {\color {Chapter }Polynomials}}{69}{section.10.2}
\contentsline {subsection}{\numberline {10.2.1}\leavevmode {\color {Chapter }TermsOfPolynomial}}{69}{subsection.10.2.1}
\contentsline {subsection}{\numberline {10.2.2}\leavevmode {\color {Chapter }IsMonomial (for polynomial)}}{69}{subsection.10.2.2}
\contentsline {subsection}{\numberline {10.2.3}\leavevmode {\color {Chapter }UnivariateMonomialsOfMonomial}}{69}{subsection.10.2.3}
\contentsline {subsection}{\numberline {10.2.4}\leavevmode {\color {Chapter }IndeterminateAndExponentOfUnivariateMonomial}}{69}{subsection.10.2.4}
\contentsline {subsection}{\numberline {10.2.5}\leavevmode {\color {Chapter }IndeterminatesOfPolynomial}}{69}{subsection.10.2.5}
\contentsline {subsection}{\numberline {10.2.6}\leavevmode {\color {Chapter }ReduceIdeal}}{69}{subsection.10.2.6}
\contentsline {subsection}{\numberline {10.2.7}\leavevmode {\color {Chapter }ReducedPolynomialRingPresentation}}{70}{subsection.10.2.7}
\contentsline {subsection}{\numberline {10.2.8}\leavevmode {\color {Chapter }Example: monomials, polynomials and ring presentations}}{70}{subsection.10.2.8}
\contentsline {section}{\numberline {10.3}\leavevmode {\color {Chapter }Singular}}{71}{section.10.3}
\contentsline {subsection}{\numberline {10.3.1}\leavevmode {\color {Chapter }SingularSetNormalFormIdeal}}{71}{subsection.10.3.1}
\contentsline {subsection}{\numberline {10.3.2}\leavevmode {\color {Chapter }SingularPolynomialNormalForm}}{71}{subsection.10.3.2}
\contentsline {subsection}{\numberline {10.3.3}\leavevmode {\color {Chapter }SingularGroebnerBasis}}{71}{subsection.10.3.3}
\contentsline {subsection}{\numberline {10.3.4}\leavevmode {\color {Chapter }SingularReducedGroebnerBasis}}{72}{subsection.10.3.4}
\contentsline {section}{\numberline {10.4}\leavevmode {\color {Chapter }Groups}}{72}{section.10.4}
\contentsline {subsection}{\numberline {10.4.1}\leavevmode {\color {Chapter }HallSeniorNumber}}{72}{subsection.10.4.1}
