\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Introduction}}{6}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }Motivation for this package}}{6}{section.1.1}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }Overview over this manual}}{6}{section.1.2}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Installation of the \textsf {orb}-Package}}{7}{chapter.2}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Basic orbit enumeration}}{8}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Enumerating orbits}}{8}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}\leavevmode {\color {Chapter }Orb}}{8}{subsection.3.1.1}
\contentsline {subsection}{\numberline {3.1.2}\leavevmode {\color {Chapter }Enumerate}}{9}{subsection.3.1.2}
\contentsline {subsection}{\numberline {3.1.3}\leavevmode {\color {Chapter }IsClosed}}{9}{subsection.3.1.3}
\contentsline {subsection}{\numberline {3.1.4}\leavevmode {\color {Chapter }Options for orbits}}{10}{subsection.3.1.4}
\contentsline {subsection}{\numberline {3.1.5}\leavevmode {\color {Chapter }Output components of orbits}}{13}{subsection.3.1.5}
\contentsline {subsection}{\numberline {3.1.6}\leavevmode {\color {Chapter }StabWords (basic)}}{13}{subsection.3.1.6}
\contentsline {subsection}{\numberline {3.1.7}\leavevmode {\color {Chapter }PositionOfFound}}{13}{subsection.3.1.7}
\contentsline {subsection}{\numberline {3.1.8}\leavevmode {\color {Chapter }DepthOfSchreierTree}}{14}{subsection.3.1.8}
\contentsline {subsection}{\numberline {3.1.9}\leavevmode {\color {Chapter }ActionOnOrbit}}{14}{subsection.3.1.9}
\contentsline {subsection}{\numberline {3.1.10}\leavevmode {\color {Chapter }OrbActionHomomorphism}}{14}{subsection.3.1.10}
\contentsline {subsection}{\numberline {3.1.11}\leavevmode {\color {Chapter }TraceSchreierTreeForward}}{14}{subsection.3.1.11}
\contentsline {subsection}{\numberline {3.1.12}\leavevmode {\color {Chapter }TraceSchreierTreeBack}}{14}{subsection.3.1.12}
\contentsline {subsection}{\numberline {3.1.13}\leavevmode {\color {Chapter }ActWithWord}}{15}{subsection.3.1.13}
\contentsline {subsection}{\numberline {3.1.14}\leavevmode {\color {Chapter }EvaluateWord}}{15}{subsection.3.1.14}
\contentsline {subsection}{\numberline {3.1.15}\leavevmode {\color {Chapter }AddGeneratorsToOrbit}}{15}{subsection.3.1.15}
\contentsline {subsection}{\numberline {3.1.16}\leavevmode {\color {Chapter }MakeSchreierTreeShallow}}{15}{subsection.3.1.16}
\contentsline {subsection}{\numberline {3.1.17}\leavevmode {\color {Chapter }FindSuborbits}}{15}{subsection.3.1.17}
\contentsline {subsection}{\numberline {3.1.18}\leavevmode {\color {Chapter }OrbitIntersectionMatrix}}{16}{subsection.3.1.18}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Hashing techniques}}{17}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }The idea of hashing}}{17}{section.4.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Hash functions}}{17}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }ChooseHashFunction}}{17}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}\leavevmode {\color {Chapter }ChooseHashFunction (gf2vec)}}{18}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}\leavevmode {\color {Chapter }ChooseHashFunction (8bitvec)}}{18}{subsection.4.2.3}
\contentsline {subsection}{\numberline {4.2.4}\leavevmode {\color {Chapter }ChooseHashFunction (gf2mat)}}{18}{subsection.4.2.4}
\contentsline {subsection}{\numberline {4.2.5}\leavevmode {\color {Chapter }ChooseHashFunction (8bitmat)}}{18}{subsection.4.2.5}
\contentsline {subsection}{\numberline {4.2.6}\leavevmode {\color {Chapter }ChooseHashFunction (int)}}{18}{subsection.4.2.6}
\contentsline {subsection}{\numberline {4.2.7}\leavevmode {\color {Chapter }ChooseHashFunction (perm)}}{19}{subsection.4.2.7}
\contentsline {subsection}{\numberline {4.2.8}\leavevmode {\color {Chapter }ChooseHashFunction (intlist)}}{19}{subsection.4.2.8}
\contentsline {subsection}{\numberline {4.2.9}\leavevmode {\color {Chapter }ChooseHashFunction (NBitsPcWord)}}{19}{subsection.4.2.9}
\contentsline {subsection}{\numberline {4.2.10}\leavevmode {\color {Chapter }ChooseHashFunction (IntLists)}}{19}{subsection.4.2.10}
\contentsline {subsection}{\numberline {4.2.11}\leavevmode {\color {Chapter }ChooseHashFunction (MatLists)}}{19}{subsection.4.2.11}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Using hash tables}}{19}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}\leavevmode {\color {Chapter }NewHT}}{19}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}\leavevmode {\color {Chapter }AddHT}}{20}{subsection.4.3.2}
\contentsline {subsection}{\numberline {4.3.3}\leavevmode {\color {Chapter }ValueHT}}{20}{subsection.4.3.3}
\contentsline {subsection}{\numberline {4.3.4}\leavevmode {\color {Chapter }InitHT}}{20}{subsection.4.3.4}
\contentsline {subsection}{\numberline {4.3.5}\leavevmode {\color {Chapter }GrowHT}}{20}{subsection.4.3.5}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter } The data structures for hash tables }}{21}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}\leavevmode {\color {Chapter } Memory requirements }}{21}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}\leavevmode {\color {Chapter } Handling of collisions }}{21}{subsection.4.4.2}
\contentsline {subsection}{\numberline {4.4.3}\leavevmode {\color {Chapter } Efficiency }}{22}{subsection.4.4.3}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Caching techniques}}{23}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }The idea of caching}}{23}{section.5.1}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Using caches}}{23}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }LinkedListCache}}{23}{subsection.5.2.1}
\contentsline {subsection}{\numberline {5.2.2}\leavevmode {\color {Chapter }CacheObject}}{23}{subsection.5.2.2}
\contentsline {subsection}{\numberline {5.2.3}\leavevmode {\color {Chapter }ClearCache}}{24}{subsection.5.2.3}
\contentsline {subsection}{\numberline {5.2.4}\leavevmode {\color {Chapter }UseCacheObject}}{24}{subsection.5.2.4}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Random elements}}{25}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }Randomizing mutable objects}}{25}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}\leavevmode {\color {Chapter }Randomize}}{25}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}\leavevmode {\color {Chapter }MakeRandomVectors}}{25}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}\leavevmode {\color {Chapter }MakeRandomLines}}{25}{subsection.6.1.3}
\contentsline {section}{\numberline {6.2}\leavevmode {\color {Chapter }Product replacement}}{26}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}\leavevmode {\color {Chapter }ProductReplacer}}{26}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}\leavevmode {\color {Chapter }Next}}{27}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}\leavevmode {\color {Chapter }Reset}}{27}{subsection.6.2.3}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Searching in groups and orbits}}{28}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter }Searching using orbit enumeration}}{28}{section.7.1}
\contentsline {section}{\numberline {7.2}\leavevmode {\color {Chapter }Random searches in groups}}{29}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}\leavevmode {\color {Chapter }RandomSearcher}}{29}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}\leavevmode {\color {Chapter }Search}}{29}{subsection.7.2.2}
\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }The dihedral trick and applications}}{30}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}\leavevmode {\color {Chapter }FindInvolution}}{30}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}\leavevmode {\color {Chapter }FindCentralisingElementOfInvolution}}{30}{subsection.7.3.2}
\contentsline {subsection}{\numberline {7.3.3}\leavevmode {\color {Chapter }FindInvolutionCentralizer}}{30}{subsection.7.3.3}
\contentsline {section}{\numberline {7.4}\leavevmode {\color {Chapter }Orbit statistics on vector spaces}}{30}{section.7.4}
\contentsline {subsection}{\numberline {7.4.1}\leavevmode {\color {Chapter }OrbitStatisticOnVectorSpace}}{30}{subsection.7.4.1}
\contentsline {subsection}{\numberline {7.4.2}\leavevmode {\color {Chapter }OrbitStatisticOnVectorSpaceLines}}{31}{subsection.7.4.2}
\contentsline {section}{\numberline {7.5}\leavevmode {\color {Chapter }Finding generating sets of subgroups}}{31}{section.7.5}
\contentsline {subsection}{\numberline {7.5.1}\leavevmode {\color {Chapter }FindShortGeneratorsOfSubgroup}}{31}{subsection.7.5.1}
\contentsline {chapter}{\numberline {8}\leavevmode {\color {Chapter }Orbit enumeration by suborbits}}{32}{chapter.8}
\contentsline {section}{\numberline {8.1}\leavevmode {\color {Chapter }\texttt {OrbitBySuborbits} and its resulting objects}}{32}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\leavevmode {\color {Chapter }OrbitBySuborbit}}{32}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}\leavevmode {\color {Chapter }OrbitBySuborbitKnownSize}}{33}{subsection.8.1.2}
\contentsline {subsection}{\numberline {8.1.3}\leavevmode {\color {Chapter }Size (fororb)}}{33}{subsection.8.1.3}
\contentsline {subsection}{\numberline {8.1.4}\leavevmode {\color {Chapter }Seed}}{33}{subsection.8.1.4}
\contentsline {subsection}{\numberline {8.1.5}\leavevmode {\color {Chapter }SuborbitsDb}}{33}{subsection.8.1.5}
\contentsline {subsection}{\numberline {8.1.6}\leavevmode {\color {Chapter }WordsToSuborbits}}{33}{subsection.8.1.6}
\contentsline {subsection}{\numberline {8.1.7}\leavevmode {\color {Chapter }Memory (forob)}}{34}{subsection.8.1.7}
\contentsline {subsection}{\numberline {8.1.8}\leavevmode {\color {Chapter }Stabilizer (obso)}}{34}{subsection.8.1.8}
\contentsline {subsection}{\numberline {8.1.9}\leavevmode {\color {Chapter }StabWords}}{34}{subsection.8.1.9}
\contentsline {subsection}{\numberline {8.1.10}\leavevmode {\color {Chapter }SavingFactor (fororb)}}{34}{subsection.8.1.10}
\contentsline {subsection}{\numberline {8.1.11}\leavevmode {\color {Chapter }TotalLength (fordb)}}{35}{subsection.8.1.11}
\contentsline {subsection}{\numberline {8.1.12}\leavevmode {\color {Chapter }Representatives}}{35}{subsection.8.1.12}
\contentsline {subsection}{\numberline {8.1.13}\leavevmode {\color {Chapter }SavingFactor (fordb)}}{35}{subsection.8.1.13}
\contentsline {subsection}{\numberline {8.1.14}\leavevmode {\color {Chapter }OrigSeed}}{35}{subsection.8.1.14}
\contentsline {section}{\numberline {8.2}\leavevmode {\color {Chapter }Preparation functions for \texttt {OrbitBySuborbit} (\ref {OrbitBySuborbit})}}{35}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}\leavevmode {\color {Chapter }OrbitBySuborbitBootstrapForVectors}}{35}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2}\leavevmode {\color {Chapter }OrbitBySuborbitBootstrapForLines}}{36}{subsection.8.2.2}
\contentsline {subsection}{\numberline {8.2.3}\leavevmode {\color {Chapter }OrbitBySuborbitBootstrapForSpaces}}{36}{subsection.8.2.3}
\contentsline {section}{\numberline {8.3}\leavevmode {\color {Chapter }Data structures for orbit-by-suborbits}}{36}{section.8.3}
\contentsline {subsection}{\numberline {8.3.1}\leavevmode {\color {Chapter }IsOrbitBySuborbitSetup}}{36}{subsection.8.3.1}
\contentsline {subsection}{\numberline {8.3.2}\leavevmode {\color {Chapter }The global record \texttt {ORB}}}{38}{subsection.8.3.2}
\contentsline {section}{\numberline {8.4}\leavevmode {\color {Chapter }Lists of orbit-by-suborbit objects}}{39}{section.8.4}
\contentsline {subsection}{\numberline {8.4.1}\leavevmode {\color {Chapter }InitOrbitBySuborbitList}}{39}{subsection.8.4.1}
\contentsline {subsection}{\numberline {8.4.2}\leavevmode {\color {Chapter }IsVectorInOrbitBySuborbitList}}{39}{subsection.8.4.2}
\contentsline {subsection}{\numberline {8.4.3}\leavevmode {\color {Chapter }OrbitsFromSeedsToOrbitList}}{40}{subsection.8.4.3}
\contentsline {subsection}{\numberline {8.4.4}\leavevmode {\color {Chapter }VerifyDisjointness}}{40}{subsection.8.4.4}
\contentsline {subsection}{\numberline {8.4.5}\leavevmode {\color {Chapter }Memory (forobsol)}}{40}{subsection.8.4.5}
\contentsline {subsection}{\numberline {8.4.6}\leavevmode {\color {Chapter }TotalLength (forobsol)}}{40}{subsection.8.4.6}
\contentsline {subsection}{\numberline {8.4.7}\leavevmode {\color {Chapter }Size (forobsol)}}{40}{subsection.8.4.7}
\contentsline {subsection}{\numberline {8.4.8}\leavevmode {\color {Chapter }SavingFactor (forobsol)}}{40}{subsection.8.4.8}
\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Finding nice quotients}}{42}{chapter.9}
\contentsline {chapter}{\numberline {10}\leavevmode {\color {Chapter }Examples}}{43}{chapter.10}
\contentsline {section}{\numberline {10.1}\leavevmode {\color {Chapter }The Mathieu group $M_{{11}}$ acting in dimension $24$}}{43}{section.10.1}
\contentsline {section}{\numberline {10.2}\leavevmode {\color {Chapter }The Fischer group $Fi_{{23}}$ acting in dimension $1494$ }}{45}{section.10.2}
\contentsline {section}{\numberline {10.3}\leavevmode {\color {Chapter }The Conway group $Co_1$ acting in dimension $24$ }}{45}{section.10.3}
\contentsline {section}{\numberline {10.4}\leavevmode {\color {Chapter }The Baby Monster $B$ acting on its 2A involutions}}{46}{section.10.4}