\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Introduction}}{7}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }Motivation for this package}}{7}{section.1.1}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }Overview over this manual}}{7}{section.1.2}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Installation of the \textsf {orb}-Package}}{8}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }Static linking}}{8}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }Recompiling the documentation}}{9}{section.2.2}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Basic orbit enumeration}}{10}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Enumerating orbits}}{10}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}\leavevmode {\color {Chapter }Orb}}{10}{subsection.3.1.1}
\contentsline {subsection}{\numberline {3.1.2}\leavevmode {\color {Chapter }Enumerate}}{11}{subsection.3.1.2}
\contentsline {subsection}{\numberline {3.1.3}\leavevmode {\color {Chapter }IsClosed}}{11}{subsection.3.1.3}
\contentsline {subsection}{\numberline {3.1.4}\leavevmode {\color {Chapter }Options for orbits}}{12}{subsection.3.1.4}
\contentsline {subsection}{\numberline {3.1.5}\leavevmode {\color {Chapter }Output components of orbits}}{15}{subsection.3.1.5}
\contentsline {subsection}{\numberline {3.1.6}\leavevmode {\color {Chapter }StabWords (basic)}}{16}{subsection.3.1.6}
\contentsline {subsection}{\numberline {3.1.7}\leavevmode {\color {Chapter }PositionOfFound}}{16}{subsection.3.1.7}
\contentsline {subsection}{\numberline {3.1.8}\leavevmode {\color {Chapter }DepthOfSchreierTree}}{16}{subsection.3.1.8}
\contentsline {subsection}{\numberline {3.1.9}\leavevmode {\color {Chapter }ActionOnOrbit}}{16}{subsection.3.1.9}
\contentsline {subsection}{\numberline {3.1.10}\leavevmode {\color {Chapter }OrbActionHomomorphism}}{16}{subsection.3.1.10}
\contentsline {subsection}{\numberline {3.1.11}\leavevmode {\color {Chapter }TraceSchreierTreeForward}}{16}{subsection.3.1.11}
\contentsline {subsection}{\numberline {3.1.12}\leavevmode {\color {Chapter }TraceSchreierTreeBack}}{17}{subsection.3.1.12}
\contentsline {subsection}{\numberline {3.1.13}\leavevmode {\color {Chapter }ActWithWord}}{17}{subsection.3.1.13}
\contentsline {subsection}{\numberline {3.1.14}\leavevmode {\color {Chapter }EvaluateWord}}{17}{subsection.3.1.14}
\contentsline {subsection}{\numberline {3.1.15}\leavevmode {\color {Chapter }AddGeneratorsToOrbit}}{17}{subsection.3.1.15}
\contentsline {subsection}{\numberline {3.1.16}\leavevmode {\color {Chapter }MakeSchreierTreeShallow}}{18}{subsection.3.1.16}
\contentsline {subsection}{\numberline {3.1.17}\leavevmode {\color {Chapter }FindSuborbits}}{18}{subsection.3.1.17}
\contentsline {subsection}{\numberline {3.1.18}\leavevmode {\color {Chapter }OrbitIntersectionMatrix}}{18}{subsection.3.1.18}
\contentsline {subsection}{\numberline {3.1.19}\leavevmode {\color {Chapter }ORB{\textunderscore }EstimateOrbitSize}}{18}{subsection.3.1.19}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Hashing techniques}}{20}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }The idea of hashing}}{20}{section.4.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Hash functions}}{20}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }ChooseHashFunction}}{20}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}\leavevmode {\color {Chapter }ChooseHashFunction (gf2vec)}}{21}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}\leavevmode {\color {Chapter }ChooseHashFunction (8bitvec)}}{21}{subsection.4.2.3}
\contentsline {subsection}{\numberline {4.2.4}\leavevmode {\color {Chapter }ChooseHashFunction (gf2mat)}}{21}{subsection.4.2.4}
\contentsline {subsection}{\numberline {4.2.5}\leavevmode {\color {Chapter }ChooseHashFunction (8bitmat)}}{21}{subsection.4.2.5}
\contentsline {subsection}{\numberline {4.2.6}\leavevmode {\color {Chapter }ChooseHashFunction (int)}}{22}{subsection.4.2.6}
\contentsline {subsection}{\numberline {4.2.7}\leavevmode {\color {Chapter }ChooseHashFunction (perm)}}{22}{subsection.4.2.7}
\contentsline {subsection}{\numberline {4.2.8}\leavevmode {\color {Chapter }ChooseHashFunction (intlist)}}{22}{subsection.4.2.8}
\contentsline {subsection}{\numberline {4.2.9}\leavevmode {\color {Chapter }ChooseHashFunction (NBitsPcWord)}}{22}{subsection.4.2.9}
\contentsline {subsection}{\numberline {4.2.10}\leavevmode {\color {Chapter }ChooseHashFunction (IntLists)}}{22}{subsection.4.2.10}
\contentsline {subsection}{\numberline {4.2.11}\leavevmode {\color {Chapter }ChooseHashFunction (MatLists)}}{22}{subsection.4.2.11}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Using hash tables}}{22}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}\leavevmode {\color {Chapter }HTCreate}}{22}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}\leavevmode {\color {Chapter }HTAdd}}{23}{subsection.4.3.2}
\contentsline {subsection}{\numberline {4.3.3}\leavevmode {\color {Chapter }HTValue}}{24}{subsection.4.3.3}
\contentsline {subsection}{\numberline {4.3.4}\leavevmode {\color {Chapter }HTUpdate}}{24}{subsection.4.3.4}
\contentsline {subsection}{\numberline {4.3.5}\leavevmode {\color {Chapter }HTDelete}}{24}{subsection.4.3.5}
\contentsline {subsection}{\numberline {4.3.6}\leavevmode {\color {Chapter }HTGrow}}{24}{subsection.4.3.6}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter }Using hash tables (legacy code)}}{25}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}\leavevmode {\color {Chapter }NewHT}}{25}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}\leavevmode {\color {Chapter }AddHT}}{25}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}\leavevmode {\color {Chapter }ValueHT}}{25}{subsection.4.4.3}
\contentsline {subsection}{\numberline {4.4.4}\leavevmode {\color {Chapter }InitHT}}{25}{subsection.4.4.4}
\contentsline {subsection}{\numberline {4.4.5}\leavevmode {\color {Chapter }GrowHT}}{26}{subsection.4.4.5}
\contentsline {section}{\numberline {4.5}\leavevmode {\color {Chapter } The data structures for hash tables }}{26}{section.4.5}
\contentsline {subsection}{\numberline {4.5.1}\leavevmode {\color {Chapter } Memory requirements }}{27}{subsection.4.5.1}
\contentsline {subsection}{\numberline {4.5.2}\leavevmode {\color {Chapter } Handling of collisions }}{27}{subsection.4.5.2}
\contentsline {subsection}{\numberline {4.5.3}\leavevmode {\color {Chapter } Efficiency }}{27}{subsection.4.5.3}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Caching techniques}}{28}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }The idea of caching}}{28}{section.5.1}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Using caches}}{28}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }LinkedListCache}}{28}{subsection.5.2.1}
\contentsline {subsection}{\numberline {5.2.2}\leavevmode {\color {Chapter }CacheObject}}{28}{subsection.5.2.2}
\contentsline {subsection}{\numberline {5.2.3}\leavevmode {\color {Chapter }ClearCache}}{29}{subsection.5.2.3}
\contentsline {subsection}{\numberline {5.2.4}\leavevmode {\color {Chapter }UseCacheObject}}{29}{subsection.5.2.4}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Random elements}}{30}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }Randomizing mutable objects}}{30}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}\leavevmode {\color {Chapter }Randomize}}{30}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}\leavevmode {\color {Chapter }MakeRandomVectors}}{30}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}\leavevmode {\color {Chapter }MakeRandomLines}}{30}{subsection.6.1.3}
\contentsline {section}{\numberline {6.2}\leavevmode {\color {Chapter }Product replacement}}{31}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}\leavevmode {\color {Chapter }ProductReplacer}}{31}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}\leavevmode {\color {Chapter }Next}}{32}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}\leavevmode {\color {Chapter }Reset}}{32}{subsection.6.2.3}
\contentsline {subsection}{\numberline {6.2.4}\leavevmode {\color {Chapter }AddGeneratorToProductReplacer}}{33}{subsection.6.2.4}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Searching in groups and orbits}}{34}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter }Searching using orbit enumeration}}{34}{section.7.1}
\contentsline {section}{\numberline {7.2}\leavevmode {\color {Chapter }Random searches in groups}}{35}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}\leavevmode {\color {Chapter }RandomSearcher}}{35}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}\leavevmode {\color {Chapter }Search}}{35}{subsection.7.2.2}
\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }The dihedral trick and applications}}{36}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}\leavevmode {\color {Chapter }FindInvolution}}{36}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}\leavevmode {\color {Chapter }FindCentralisingElementOfInvolution}}{36}{subsection.7.3.2}
\contentsline {subsection}{\numberline {7.3.3}\leavevmode {\color {Chapter }FindInvolutionCentralizer}}{36}{subsection.7.3.3}
\contentsline {section}{\numberline {7.4}\leavevmode {\color {Chapter }Orbit statistics on vector spaces}}{36}{section.7.4}
\contentsline {subsection}{\numberline {7.4.1}\leavevmode {\color {Chapter }OrbitStatisticOnVectorSpace}}{36}{subsection.7.4.1}
\contentsline {subsection}{\numberline {7.4.2}\leavevmode {\color {Chapter }OrbitStatisticOnVectorSpaceLines}}{37}{subsection.7.4.2}
\contentsline {section}{\numberline {7.5}\leavevmode {\color {Chapter }Finding generating sets of subgroups}}{37}{section.7.5}
\contentsline {subsection}{\numberline {7.5.1}\leavevmode {\color {Chapter }FindShortGeneratorsOfSubgroup}}{37}{subsection.7.5.1}
\contentsline {chapter}{\numberline {8}\leavevmode {\color {Chapter }AVL trees}}{38}{chapter.8}
\contentsline {section}{\numberline {8.1}\leavevmode {\color {Chapter }The idea of AVL trees}}{38}{section.8.1}
\contentsline {section}{\numberline {8.2}\leavevmode {\color {Chapter }Using AVL trees}}{39}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}\leavevmode {\color {Chapter }AVLTree}}{39}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2}\leavevmode {\color {Chapter }AVLCmp}}{39}{subsection.8.2.2}
\contentsline {subsection}{\numberline {8.2.3}\leavevmode {\color {Chapter }AVLAdd}}{39}{subsection.8.2.3}
\contentsline {subsection}{\numberline {8.2.4}\leavevmode {\color {Chapter }AVLLookup}}{39}{subsection.8.2.4}
\contentsline {subsection}{\numberline {8.2.5}\leavevmode {\color {Chapter }AVLDelete}}{40}{subsection.8.2.5}
\contentsline {subsection}{\numberline {8.2.6}\leavevmode {\color {Chapter }AVLFindIndex}}{40}{subsection.8.2.6}
\contentsline {subsection}{\numberline {8.2.7}\leavevmode {\color {Chapter }AVLIndex}}{40}{subsection.8.2.7}
\contentsline {subsection}{\numberline {8.2.8}\leavevmode {\color {Chapter }AVLIndexLookup}}{40}{subsection.8.2.8}
\contentsline {subsection}{\numberline {8.2.9}\leavevmode {\color {Chapter }AVLIndexAdd}}{40}{subsection.8.2.9}
\contentsline {subsection}{\numberline {8.2.10}\leavevmode {\color {Chapter }AVLIndexDelete}}{41}{subsection.8.2.10}
\contentsline {subsection}{\numberline {8.2.11}\leavevmode {\color {Chapter }AVLFind}}{41}{subsection.8.2.11}
\contentsline {subsection}{\numberline {8.2.12}\leavevmode {\color {Chapter }AVLIndexFind}}{41}{subsection.8.2.12}
\contentsline {subsection}{\numberline {8.2.13}\leavevmode {\color {Chapter }AVLData}}{41}{subsection.8.2.13}
\contentsline {subsection}{\numberline {8.2.14}\leavevmode {\color {Chapter }AVLValue}}{42}{subsection.8.2.14}
\contentsline {subsection}{\numberline {8.2.15}\leavevmode {\color {Chapter }Display}}{42}{subsection.8.2.15}
\contentsline {subsection}{\numberline {8.2.16}\leavevmode {\color {Chapter }ELM{\textunderscore }LIST}}{42}{subsection.8.2.16}
\contentsline {subsection}{\numberline {8.2.17}\leavevmode {\color {Chapter }Position}}{42}{subsection.8.2.17}
\contentsline {subsection}{\numberline {8.2.18}\leavevmode {\color {Chapter }Add}}{42}{subsection.8.2.18}
\contentsline {subsection}{\numberline {8.2.19}\leavevmode {\color {Chapter }Remove}}{42}{subsection.8.2.19}
\contentsline {subsection}{\numberline {8.2.20}\leavevmode {\color {Chapter }Length}}{43}{subsection.8.2.20}
\contentsline {subsection}{\numberline {8.2.21}\leavevmode {\color {Chapter }\texttt {\char 92\relax }in}}{43}{subsection.8.2.21}
\contentsline {section}{\numberline {8.3}\leavevmode {\color {Chapter }The internal data structures}}{43}{section.8.3}
\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Orbit enumeration by suborbits}}{45}{chapter.9}
\contentsline {section}{\numberline {9.1}\leavevmode {\color {Chapter }\texttt {OrbitBySuborbits} and its resulting objects}}{45}{section.9.1}
\contentsline {subsection}{\numberline {9.1.1}\leavevmode {\color {Chapter }OrbitBySuborbit}}{45}{subsection.9.1.1}
\contentsline {subsection}{\numberline {9.1.2}\leavevmode {\color {Chapter }OrbitBySuborbitKnownSize}}{46}{subsection.9.1.2}
\contentsline {subsection}{\numberline {9.1.3}\leavevmode {\color {Chapter }Size (fororb)}}{46}{subsection.9.1.3}
\contentsline {subsection}{\numberline {9.1.4}\leavevmode {\color {Chapter }Seed}}{46}{subsection.9.1.4}
\contentsline {subsection}{\numberline {9.1.5}\leavevmode {\color {Chapter }SuborbitsDb}}{46}{subsection.9.1.5}
\contentsline {subsection}{\numberline {9.1.6}\leavevmode {\color {Chapter }WordsToSuborbits}}{46}{subsection.9.1.6}
\contentsline {subsection}{\numberline {9.1.7}\leavevmode {\color {Chapter }Memory (forob)}}{47}{subsection.9.1.7}
\contentsline {subsection}{\numberline {9.1.8}\leavevmode {\color {Chapter }Stabilizer (obso)}}{47}{subsection.9.1.8}
\contentsline {subsection}{\numberline {9.1.9}\leavevmode {\color {Chapter }StabWords}}{47}{subsection.9.1.9}
\contentsline {subsection}{\numberline {9.1.10}\leavevmode {\color {Chapter }SavingFactor (fororb)}}{47}{subsection.9.1.10}
\contentsline {subsection}{\numberline {9.1.11}\leavevmode {\color {Chapter }TotalLength (fordb)}}{48}{subsection.9.1.11}
\contentsline {subsection}{\numberline {9.1.12}\leavevmode {\color {Chapter }Representatives}}{48}{subsection.9.1.12}
\contentsline {subsection}{\numberline {9.1.13}\leavevmode {\color {Chapter }SavingFactor (fordb)}}{48}{subsection.9.1.13}
\contentsline {subsection}{\numberline {9.1.14}\leavevmode {\color {Chapter }OrigSeed}}{48}{subsection.9.1.14}
\contentsline {section}{\numberline {9.2}\leavevmode {\color {Chapter }Preparation functions for \texttt {OrbitBySuborbit} (\ref {OrbitBySuborbit})}}{48}{section.9.2}
\contentsline {subsection}{\numberline {9.2.1}\leavevmode {\color {Chapter }OrbitBySuborbitBootstrapForVectors}}{48}{subsection.9.2.1}
\contentsline {subsection}{\numberline {9.2.2}\leavevmode {\color {Chapter }OrbitBySuborbitBootstrapForLines}}{49}{subsection.9.2.2}
\contentsline {subsection}{\numberline {9.2.3}\leavevmode {\color {Chapter }OrbitBySuborbitBootstrapForSpaces}}{49}{subsection.9.2.3}
\contentsline {section}{\numberline {9.3}\leavevmode {\color {Chapter }Data structures for orbit-by-suborbits}}{49}{section.9.3}
\contentsline {subsection}{\numberline {9.3.1}\leavevmode {\color {Chapter }IsOrbitBySuborbitSetup}}{50}{subsection.9.3.1}
\contentsline {subsection}{\numberline {9.3.2}\leavevmode {\color {Chapter }The global record \texttt {ORB}}}{51}{subsection.9.3.2}
\contentsline {section}{\numberline {9.4}\leavevmode {\color {Chapter }Lists of orbit-by-suborbit objects}}{52}{section.9.4}
\contentsline {subsection}{\numberline {9.4.1}\leavevmode {\color {Chapter }InitOrbitBySuborbitList}}{52}{subsection.9.4.1}
\contentsline {subsection}{\numberline {9.4.2}\leavevmode {\color {Chapter }IsVectorInOrbitBySuborbitList}}{53}{subsection.9.4.2}
\contentsline {subsection}{\numberline {9.4.3}\leavevmode {\color {Chapter }OrbitsFromSeedsToOrbitList}}{53}{subsection.9.4.3}
\contentsline {subsection}{\numberline {9.4.4}\leavevmode {\color {Chapter }VerifyDisjointness}}{53}{subsection.9.4.4}
\contentsline {subsection}{\numberline {9.4.5}\leavevmode {\color {Chapter }Memory (forobsol)}}{53}{subsection.9.4.5}
\contentsline {subsection}{\numberline {9.4.6}\leavevmode {\color {Chapter }TotalLength (forobsol)}}{53}{subsection.9.4.6}
\contentsline {subsection}{\numberline {9.4.7}\leavevmode {\color {Chapter }Size (forobsol)}}{54}{subsection.9.4.7}
\contentsline {subsection}{\numberline {9.4.8}\leavevmode {\color {Chapter }SavingFactor (forobsol)}}{54}{subsection.9.4.8}
\contentsline {chapter}{\numberline {10}\leavevmode {\color {Chapter }Finding nice quotients}}{55}{chapter.10}
\contentsline {chapter}{\numberline {11}\leavevmode {\color {Chapter }Examples}}{56}{chapter.11}
\contentsline {section}{\numberline {11.1}\leavevmode {\color {Chapter }The Mathieu group $M_{{11}}$ acting in dimension $24$}}{56}{section.11.1}
\contentsline {section}{\numberline {11.2}\leavevmode {\color {Chapter }The Fischer group $Fi_{{23}}$ acting in dimension $1494$ }}{58}{section.11.2}
\contentsline {section}{\numberline {11.3}\leavevmode {\color {Chapter }The Conway group $Co_1$ acting in dimension $24$ }}{58}{section.11.3}
\contentsline {section}{\numberline {11.4}\leavevmode {\color {Chapter }The Baby Monster $B$ acting on its 2A involutions}}{59}{section.11.4}
