  
  
                               [1XGAP 4 Package [5Xorb[0m[1X[0m
  
  
                      [1Xorb --- Methods to enumerate Orbits[0m
  
  
                                  Version 2.0
  
  
                                   June 2008
  
  
                                 Jürgen Müller
  
                                 Max Neunhöffer
  
                                  Felix Noeske
  
  
  
  Jürgen Müller
      Email:    [7Xmailto:juergen.mueller@math.rwth-aachen.de[0m
      Homepage: [7Xhttp://www.math.rwth-aachen.de/~Juergen.Mueller[0m
      Address:  Lehrstuhl  D  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52062 Aachen, Germany
  
  
  Max Neunhöffer
      Email:    [7Xmailto:neunhoef@mcs.st-and.ac.uk[0m
      Homepage: [7Xhttp://www-groups.mcs.st-and.ac.uk/~neunhoef[0m
      Address:  School  of  Mathematics  and Statistics Mathematical Institute
                University of St Andrews North Haugh St Andrews, Fife KY16 9SS
                Scotland, UK
  
  
  Felix Noeske
      Email:    [7Xmailto:felix.noeske@math.rwth-aachen.de[0m
      Homepage: [7Xhttp://www.math.rwth-aachen.de/~Felix.Noeske[0m
      Address:  Lehrstuhl  D  für  Mathematik,  RWTH Aachen, Templergraben 64,
                52062 Aachen, Germany
  
  
  
  -------------------------------------------------------
  [1XCopyright[0m
  © 2005-2008 by Jürgen Müller, Max Neunhöffer and Felix Noeske
  
  This  program  is  free  software:  you can redistribute it and/or modify it
  under  the  terms of the GNU General Public License as published by the Free
  Software  Foundation,  either  version 3 of the License, or (at your option)
  any  later  version. This program is distributed in the hope that it will be
  useful,  but  WITHOUT  ANY  WARRANTY;  without  even the implied warranty of
  MERCHANTABILITY  or  FITNESS  FOR  A PARTICULAR PURPOSE. See the GNU General
  Public  License for more details. You should have received a copy of the GNU
  General   Public   License   along   with   this   program.   If   not,  see
  [7Xhttp://www.gnu.org/licenses/[0m.
  
  
  -------------------------------------------------------
  
  
  [1XContents (orb)[0X
  
  1 Introduction
    1.1 Motivation for this package
    1.2 Overview over this manual
  2 Installation of the [5Xorb[0m-Package
  3 Basic orbit enumeration
    3.1 Enumerating orbits
      3.1-1 Orb
      3.1-2 Enumerate
      3.1-3 IsClosed
      3.1-4 Options for orbits
      3.1-5 Output components of orbits
      3.1-6 StabWords
      3.1-7 PositionOfFound
      3.1-8 DepthOfSchreierTree
      3.1-9 ActionOnOrbit
      3.1-10 OrbActionHomomorphism
      3.1-11 TraceSchreierTreeForward
      3.1-12 TraceSchreierTreeBack
      3.1-13 ActWithWord
      3.1-14 EvaluateWord
      3.1-15 AddGeneratorsToOrbit
      3.1-16 MakeSchreierTreeShallow
      3.1-17 FindSuborbits
      3.1-18 OrbitIntersectionMatrix
  4 Hashing techniques
    4.1 The idea of hashing
    4.2 Hash functions
      4.2-1 ChooseHashFunction
      4.2-2 ChooseHashFunction
      4.2-3 ChooseHashFunction
      4.2-4 ChooseHashFunction
      4.2-5 ChooseHashFunction
      4.2-6 ChooseHashFunction
      4.2-7 ChooseHashFunction
      4.2-8 ChooseHashFunction
      4.2-9 ChooseHashFunction
      4.2-10 ChooseHashFunction
      4.2-11 ChooseHashFunction
    4.3 Using hash tables
      4.3-1 NewHT
      4.3-2 AddHT
      4.3-3 ValueHT
      4.3-4 InitHT
      4.3-5 GrowHT
    4.4 The data structures for hash tables
      4.4-1 Memory requirements
      4.4-2 Handling of collisions
      4.4-3 Efficiency
  5 Caching techniques
    5.1 The idea of caching
    5.2 Using caches
      5.2-1 LinkedListCache
      5.2-2 CacheObject
      5.2-3 ClearCache
      5.2-4 UseCacheObject
  6 Random elements
    6.1 Randomizing mutable objects
      6.1-1 Randomize
      6.1-2 MakeRandomVectors
      6.1-3 MakeRandomLines
    6.2 Product replacement
      6.2-1 ProductReplacer
      6.2-2 Next
      6.2-3 Reset
  7 Searching in groups and orbits
    7.1 Searching using orbit enumeration
    7.2 Random searches in groups
      7.2-1 RandomSearcher
      7.2-2 Search
    7.3 The dihedral trick and applications
      7.3-1 FindInvolution
      7.3-2 FindCentralisingElementOfInvolution
      7.3-3 FindInvolutionCentralizer
    7.4 Orbit statistics on vector spaces
      7.4-1 OrbitStatisticOnVectorSpace
      7.4-2 OrbitStatisticOnVectorSpaceLines
    7.5 Finding generating sets of subgroups
      7.5-1 FindShortGeneratorsOfSubgroup
  8 Orbit enumeration by suborbits
    8.1 [10XOrbitBySuborbits[0m and its resulting objects
      8.1-1 OrbitBySuborbit
      8.1-2 OrbitBySuborbitKnownSize
      8.1-3 Size
      8.1-4 Seed
      8.1-5 SuborbitsDb
      8.1-6 WordsToSuborbits
      8.1-7 Memory
      8.1-8 Stabilizer
      8.1-9 StabWords
      8.1-10 SavingFactor
      8.1-11 TotalLength
      8.1-12 Representatives
      8.1-13 SavingFactor
      8.1-14 OrigSeed
    8.2 Preparation functions for [2XOrbitBySuborbit[0m ([14X8.1-1[0m)
      8.2-1 OrbitBySuborbitBootstrapForVectors
      8.2-2 OrbitBySuborbitBootstrapForLines
      8.2-3 OrbitBySuborbitBootstrapForSpaces
    8.3 Data structures for orbit-by-suborbits
      8.3-1 IsOrbitBySuborbitSetup
      8.3-2 The global record [10XORB[0m
    8.4 Lists of orbit-by-suborbit objects
      8.4-1 InitOrbitBySuborbitList
      8.4-2 IsVectorInOrbitBySuborbitList
      8.4-3 OrbitsFromSeedsToOrbitList
      8.4-4 VerifyDisjointness
      8.4-5 Memory
      8.4-6 TotalLength
      8.4-7 Size
      8.4-8 SavingFactor
  9 Finding nice quotients
  10 Examples
    10.1 The Mathieu group M_{11} acting in dimension 24
    10.2 The Fischer group Fi_{23} acting in dimension 1494
    10.3 The Conway group Co_1 acting in dimension 24
    10.4 The Baby Monster B acting on its 2A involutions
  
  
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