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# GAP 4 Package orb

## orb --- Methods to enumerate Orbits

Version 2.0

June 2008

Jürgen Müller
Email: juergen.mueller@math.rwth-aachen.de
Homepage: http://www.math.rwth-aachen.de/~Juergen.Mueller
Address:
Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany

Max Neunhöffer
Email: neunhoef@mcs.st-and.ac.uk
Homepage: http://www-groups.mcs.st-and.ac.uk/~neunhoef
Address:
School of Mathematics and Statistics Mathematical Institute University of St Andrews North Haugh St Andrews, Fife KY16 9SS Scotland, UK

Felix Noeske
Email: felix.noeske@math.rwth-aachen.de
Homepage: http://www.math.rwth-aachen.de/~Felix.Noeske
Address:
Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany

### Copyright

© 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 http://www.gnu.org/licenses/.

### Contents

1 Introduction
1.1 Motivation for this package
1.2 Overview over this manual
2 Installation of the orb-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 `OrbitBySuborbits` 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 `OrbitBySuborbit` (8.1-1)
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 `ORB`
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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