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orb

Methods to enumerate orbits

4.9.1

26 August 2024

Juergen Mueller
Email: juergen.mueller@math.rwth-aachen.de
Homepage: http://www.math.rwth-aachen.de/~Juergen.Mueller
Address:
Juergen Mueller
Lehrstuhl D fuer Mathematik, RWTH Aachen
Templergraben 64
52056 Aachen
Germany

Max Neunhöffer
Email: max@9hoeffer.de
Homepage: http://www-groups.mcs.st-and.ac.uk/~neunhoef
Address:
Gustav-Freytag-Straße 40
50354 Hürth
Germany

Felix Noeske
Email: felix.noeske@math.rwth-aachen.de
Homepage: http://www.math.rwth-aachen.de/~Felix.Noeske
Address:
Felix Noeske
Lehrstuhl D fuer Mathematik, RWTH Aachen
Templergraben 64
52056 Aachen
Germany

Copyright

© 2005-2014 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 https://www.gnu.org/licenses/.

Contents

1.1 Motivation for this package

1.2 Overview over this manual

2 Installation of the orb-Package

2.1 Recompiling the documentation

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
  6.2-4 AddGeneratorToProductReplacer

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

9 Orbit enumeration by suborbits

9.1 OrbitBySuborbits and its resulting objects

  9.1-1 OrbitBySuborbit
  9.1-2 OrbitBySuborbitKnownSize
  9.1-3 Size
  9.1-4 Seed
  9.1-5 SuborbitsDb
  9.1-6 WordsToSuborbits
  9.1-7 Memory
  9.1-8 Stabilizer
  9.1-9 StabWords
  9.1-10 SavingFactor
  9.1-11 TotalLength
  9.1-12 Representatives
  9.1-13 SavingFactor
  9.1-14 OrigSeed

9.2 Preparation functions for OrbitBySuborbit (9.1-1)

  9.2-1 OrbitBySuborbitBootstrapForVectors
  9.2-2 OrbitBySuborbitBootstrapForLines
  9.2-3 OrbitBySuborbitBootstrapForSpaces

9.3 Data structures for orbit-by-suborbits

9.3-1 IsOrbitBySuborbitSetup
9.3-2 The global record ORB

9.4 Lists of orbit-by-suborbit objects

  9.4-1 InitOrbitBySuborbitList
  9.4-2 IsVectorInOrbitBySuborbitList
  9.4-3 OrbitsFromSeedsToOrbitList
  9.4-4 VerifyDisjointness
  9.4-5 Memory
  9.4-6 TotalLength
  9.4-7 Size
  9.4-8 SavingFactor

10 Finding nice quotients

11.1 The Mathieu group \(M_{{11}}\) acting in dimension \(24\)

11.2 The Fischer group \(Fi_{{23}}\) acting in dimension \(1494\)

11.3 The Conway group \(Co_1\) acting in dimension \(24\)

11.4 The Baby Monster \(B\) acting on its 2A involutions

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