\contentsline {chapter}{\numberline {1}\textcolor {Chapter }{Introduction}}{4}{chapter.1}%
\contentsline {section}{\numberline {1.1}\textcolor {Chapter }{General aims}}{4}{section.1.1}%
\contentsline {section}{\numberline {1.2}\textcolor {Chapter }{General computations in group rings}}{4}{section.1.2}%
\contentsline {section}{\numberline {1.3}\textcolor {Chapter }{Computations in the normalized unit group}}{5}{section.1.3}%
\contentsline {section}{\numberline {1.4}\textcolor {Chapter }{Computing Lie properties of the group algebra }}{5}{section.1.4}%
\contentsline {section}{\numberline {1.5}\textcolor {Chapter }{Installation and system requirements}}{5}{section.1.5}%
\contentsline {chapter}{\numberline {2}\textcolor {Chapter }{A sample calculation with \textsf {LAGUNA}}}{6}{chapter.2}%
\contentsline {chapter}{\numberline {3}\textcolor {Chapter }{The basic theory behind \textsf {LAGUNA}}}{12}{chapter.3}%
\contentsline {section}{\numberline {3.1}\textcolor {Chapter }{Notation and definitions}}{12}{section.3.1}%
\contentsline {section}{\numberline {3.2}\textcolor {Chapter }{$p$-modular group algebras}}{13}{section.3.2}%
\contentsline {section}{\numberline {3.3}\textcolor {Chapter }{Polycyclic generating set for $V$}}{13}{section.3.3}%
\contentsline {section}{\numberline {3.4}\textcolor {Chapter }{Computing the canonical form}}{14}{section.3.4}%
\contentsline {section}{\numberline {3.5}\textcolor {Chapter }{Computing a power commutator presentation for $V$}}{15}{section.3.5}%
\contentsline {section}{\numberline {3.6}\textcolor {Chapter }{Verifying Lie properties of $FG$}}{15}{section.3.6}%
\contentsline {chapter}{\numberline {4}\textcolor {Chapter }{\textsf {LAGUNA} functions}}{16}{chapter.4}%
\contentsline {section}{\numberline {4.1}\textcolor {Chapter }{General functions for group algebras}}{16}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}\textcolor {Chapter }{IsGroupAlgebra}}{16}{subsection.4.1.1}%
\contentsline {subsection}{\numberline {4.1.2}\textcolor {Chapter }{IsFModularGroupAlgebra}}{16}{subsection.4.1.2}%
\contentsline {subsection}{\numberline {4.1.3}\textcolor {Chapter }{IsPModularGroupAlgebra}}{17}{subsection.4.1.3}%
\contentsline {subsection}{\numberline {4.1.4}\textcolor {Chapter }{UnderlyingGroup (of a group ring)}}{17}{subsection.4.1.4}%
\contentsline {subsection}{\numberline {4.1.5}\textcolor {Chapter }{UnderlyingRing}}{17}{subsection.4.1.5}%
\contentsline {subsection}{\numberline {4.1.6}\textcolor {Chapter }{UnderlyingField}}{18}{subsection.4.1.6}%
\contentsline {section}{\numberline {4.2}\textcolor {Chapter }{Operations with group algebra elements}}{18}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}\textcolor {Chapter }{Support}}{18}{subsection.4.2.1}%
\contentsline {subsection}{\numberline {4.2.2}\textcolor {Chapter }{CoefficientsBySupport}}{18}{subsection.4.2.2}%
\contentsline {subsection}{\numberline {4.2.3}\textcolor {Chapter }{TraceOfMagmaRingElement}}{19}{subsection.4.2.3}%
\contentsline {subsection}{\numberline {4.2.4}\textcolor {Chapter }{Length}}{19}{subsection.4.2.4}%
\contentsline {subsection}{\numberline {4.2.5}\textcolor {Chapter }{Augmentation}}{19}{subsection.4.2.5}%
\contentsline {subsection}{\numberline {4.2.6}\textcolor {Chapter }{PartialAugmentations}}{20}{subsection.4.2.6}%
\contentsline {subsection}{\numberline {4.2.7}\textcolor {Chapter }{Involution}}{20}{subsection.4.2.7}%
\contentsline {subsection}{\numberline {4.2.8}\textcolor {Chapter }{IsSymmetric}}{21}{subsection.4.2.8}%
\contentsline {subsection}{\numberline {4.2.9}\textcolor {Chapter }{IsUnitary}}{21}{subsection.4.2.9}%
\contentsline {subsection}{\numberline {4.2.10}\textcolor {Chapter }{IsUnit}}{21}{subsection.4.2.10}%
\contentsline {subsection}{\numberline {4.2.11}\textcolor {Chapter }{InverseOp}}{22}{subsection.4.2.11}%
\contentsline {subsection}{\numberline {4.2.12}\textcolor {Chapter }{BicyclicUnitOfType1}}{22}{subsection.4.2.12}%
\contentsline {subsection}{\numberline {4.2.13}\textcolor {Chapter }{BassCyclicUnit}}{23}{subsection.4.2.13}%
\contentsline {section}{\numberline {4.3}\textcolor {Chapter }{Important attributes of group algebras}}{23}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}\textcolor {Chapter }{AugmentationHomomorphism}}{23}{subsection.4.3.1}%
\contentsline {subsection}{\numberline {4.3.2}\textcolor {Chapter }{AugmentationIdeal}}{24}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}\textcolor {Chapter }{RadicalOfAlgebra}}{24}{subsection.4.3.3}%
\contentsline {subsection}{\numberline {4.3.4}\textcolor {Chapter }{WeightedBasis}}{25}{subsection.4.3.4}%
\contentsline {subsection}{\numberline {4.3.5}\textcolor {Chapter }{AugmentationIdealPowerSeries}}{25}{subsection.4.3.5}%
\contentsline {subsection}{\numberline {4.3.6}\textcolor {Chapter }{AugmentationIdealNilpotencyIndex}}{26}{subsection.4.3.6}%
\contentsline {subsection}{\numberline {4.3.7}\textcolor {Chapter }{AugmentationIdealOfDerivedSubgroupNilpotencyIndex}}{26}{subsection.4.3.7}%
\contentsline {subsection}{\numberline {4.3.8}\textcolor {Chapter }{LeftIdealBySubgroup}}{26}{subsection.4.3.8}%
\contentsline {section}{\numberline {4.4}\textcolor {Chapter }{Computations with the unit group}}{27}{section.4.4}%
\contentsline {subsection}{\numberline {4.4.1}\textcolor {Chapter }{NormalizedUnitGroup}}{27}{subsection.4.4.1}%
\contentsline {subsection}{\numberline {4.4.2}\textcolor {Chapter }{PcNormalizedUnitGroup}}{27}{subsection.4.4.2}%
\contentsline {subsection}{\numberline {4.4.3}\textcolor {Chapter }{NaturalBijectionToPcNormalizedUnitGroup}}{28}{subsection.4.4.3}%
\contentsline {subsection}{\numberline {4.4.4}\textcolor {Chapter }{NaturalBijectionToNormalizedUnitGroup}}{28}{subsection.4.4.4}%
\contentsline {subsection}{\numberline {4.4.5}\textcolor {Chapter }{Embedding (from group to unit group)}}{29}{subsection.4.4.5}%
\contentsline {subsection}{\numberline {4.4.6}\textcolor {Chapter }{Units}}{29}{subsection.4.4.6}%
\contentsline {subsection}{\numberline {4.4.7}\textcolor {Chapter }{PcUnits}}{30}{subsection.4.4.7}%
\contentsline {subsection}{\numberline {4.4.8}\textcolor {Chapter }{IsGroupOfUnitsOfMagmaRing}}{30}{subsection.4.4.8}%
\contentsline {subsection}{\numberline {4.4.9}\textcolor {Chapter }{IsUnitGroupOfGroupRing}}{31}{subsection.4.4.9}%
\contentsline {subsection}{\numberline {4.4.10}\textcolor {Chapter }{IsNormalizedUnitGroupOfGroupRing}}{31}{subsection.4.4.10}%
\contentsline {subsection}{\numberline {4.4.11}\textcolor {Chapter }{UnderlyingGroupRing}}{31}{subsection.4.4.11}%
\contentsline {subsection}{\numberline {4.4.12}\textcolor {Chapter }{UnitarySubgroup}}{32}{subsection.4.4.12}%
\contentsline {subsection}{\numberline {4.4.13}\textcolor {Chapter }{BicyclicUnitGroup}}{32}{subsection.4.4.13}%
\contentsline {subsection}{\numberline {4.4.14}\textcolor {Chapter }{GroupBases}}{33}{subsection.4.4.14}%
\contentsline {section}{\numberline {4.5}\textcolor {Chapter }{The Lie algebra of a group algebra}}{33}{section.4.5}%
\contentsline {subsection}{\numberline {4.5.1}\textcolor {Chapter }{LieAlgebraByDomain}}{33}{subsection.4.5.1}%
\contentsline {subsection}{\numberline {4.5.2}\textcolor {Chapter }{IsLieAlgebraByAssociativeAlgebra}}{34}{subsection.4.5.2}%
\contentsline {subsection}{\numberline {4.5.3}\textcolor {Chapter }{UnderlyingAssociativeAlgebra}}{34}{subsection.4.5.3}%
\contentsline {subsection}{\numberline {4.5.4}\textcolor {Chapter }{NaturalBijectionToLieAlgebra}}{34}{subsection.4.5.4}%
\contentsline {subsection}{\numberline {4.5.5}\textcolor {Chapter }{NaturalBijectionToAssociativeAlgebra}}{35}{subsection.4.5.5}%
\contentsline {subsection}{\numberline {4.5.6}\textcolor {Chapter }{IsLieAlgebraOfGroupRing}}{35}{subsection.4.5.6}%
\contentsline {subsection}{\numberline {4.5.7}\textcolor {Chapter }{UnderlyingGroup (of Lie algebra of a group ring)}}{36}{subsection.4.5.7}%
\contentsline {subsection}{\numberline {4.5.8}\textcolor {Chapter }{Embedding (from group to Lie algebra)}}{36}{subsection.4.5.8}%
\contentsline {subsection}{\numberline {4.5.9}\textcolor {Chapter }{LieCentre}}{37}{subsection.4.5.9}%
\contentsline {subsection}{\numberline {4.5.10}\textcolor {Chapter }{LieDerivedSubalgebra}}{37}{subsection.4.5.10}%
\contentsline {subsection}{\numberline {4.5.11}\textcolor {Chapter }{IsLieAbelian}}{38}{subsection.4.5.11}%
\contentsline {subsection}{\numberline {4.5.12}\textcolor {Chapter }{IsLieSolvable}}{38}{subsection.4.5.12}%
\contentsline {subsection}{\numberline {4.5.13}\textcolor {Chapter }{IsLieNilpotent}}{39}{subsection.4.5.13}%
\contentsline {subsection}{\numberline {4.5.14}\textcolor {Chapter }{IsLieMetabelian}}{39}{subsection.4.5.14}%
\contentsline {subsection}{\numberline {4.5.15}\textcolor {Chapter }{IsLieCentreByMetabelian}}{39}{subsection.4.5.15}%
\contentsline {subsection}{\numberline {4.5.16}\textcolor {Chapter }{CanonicalBasis}}{40}{subsection.4.5.16}%
\contentsline {subsection}{\numberline {4.5.17}\textcolor {Chapter }{IsBasisOfLieAlgebraOfGroupRing}}{40}{subsection.4.5.17}%
\contentsline {subsection}{\numberline {4.5.18}\textcolor {Chapter }{StructureConstantsTable}}{41}{subsection.4.5.18}%
\contentsline {subsection}{\numberline {4.5.19}\textcolor {Chapter }{LieUpperNilpotencyIndex}}{41}{subsection.4.5.19}%
\contentsline {subsection}{\numberline {4.5.20}\textcolor {Chapter }{LieLowerNilpotencyIndex}}{42}{subsection.4.5.20}%
\contentsline {subsection}{\numberline {4.5.21}\textcolor {Chapter }{LieDerivedLength}}{42}{subsection.4.5.21}%
\contentsline {section}{\numberline {4.6}\textcolor {Chapter }{Other commands}}{42}{section.4.6}%
\contentsline {subsection}{\numberline {4.6.1}\textcolor {Chapter }{SubgroupsOfIndexTwo}}{42}{subsection.4.6.1}%
\contentsline {subsection}{\numberline {4.6.2}\textcolor {Chapter }{DihedralDepth}}{43}{subsection.4.6.2}%
\contentsline {subsection}{\numberline {4.6.3}\textcolor {Chapter }{DimensionBasis}}{43}{subsection.4.6.3}%
\contentsline {subsection}{\numberline {4.6.4}\textcolor {Chapter }{LieDimensionSubgroups}}{43}{subsection.4.6.4}%
\contentsline {subsection}{\numberline {4.6.5}\textcolor {Chapter }{LieUpperCodimensionSeries (for group ring)}}{44}{subsection.4.6.5}%
\contentsline {subsection}{\numberline {4.6.6}\textcolor {Chapter }{LAGInfo}}{44}{subsection.4.6.6}%
\contentsline {chapter}{References}{46}{chapter*.5}%
\contentsline {chapter}{Index}{47}{section*.6}%
