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# recog

## A collection of group recognition methods

1.3.2

15 April 2018

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

Ákos Seress

Nurullah Ankaralioglu
Email: ankarali@atauni.edu.tr

Peter Brooksbank
Email: pbrooksb@bucknell.edu
Homepage: http://www.facstaff.bucknell.edu/pbrooksb/
Address:
Peter A. Brooksbank
Mathematics Department
Bucknell University
Lewisburg, PA 17837
USA

Frank Celler
Email: frank@celler.de
Homepage: http://www.celler.de/

Stephen Howe
Address:
Unknown

Maska Law
Email: maska@maths.uwa.edu.au
Address:
Maska Law
University of Western Australia
School of Mathematics and Statistics
35 Stirling Highway
Crawley 6009
Western Australia

Steve Linton
Email: sal@cs.st-andrews.ac.uk
Homepage: http://www-circa.mcs.st-and.ac.uk/~sal/
Address:
School of Computer Science
Jack Cole Building
North Haugh
St Andrews, Fife KY16 9SX
Scotland, UK

Gunter Malle
Email: malle@mathematik.uni-kl.de
Homepage: http://www.mathematik.uni-kl.de/~malle/

Alice Niemeyer
Email: alice@maths.uwa.edu.au
Homepage: http://www.maths.uwa.edu.au/~alice/
Address:
Alice C. Niemeyer
University of Western Australia
School of Mathematics and Statistics
35 Stirling Highway
Crawley 6009
Western Australia

Eamonn O'Brien
Email: obrien@math.auckland.ac.nz
Homepage: http://www.math.auckland.ac.nz/~obrien/

Colva M. Roney-Dougal
Email: colva@mcs.st-and.ac.uk
Homepage: http://www-groups.mcs.st-and.ac.uk/~colva
Address:
School of Mathematics and Statistics
Mathematical Institute
North Haugh
St Andrews, Fife KY16 9SS
Scotland, UK

Max Horn
Email: max.horn@uni-siegen.de
Homepage: https://www.quendi.de/math
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57072 Siegen
Germany

### Copyright

© 2005-2014 by Max Neunhöffer and Ákos Seress

This package may be distributed under the terms and conditions of the GNU Public License Version 3 or (at your option) any later version.

### Contents

1 Introduction
1.1 Philosophy
1.2 Overview over this manual
1.3 Feedback
2 Installation of the recog-Package
3 Group Recognition
3.1 The recursive procedure
3.1-1 RecogniseGeneric
3.1-2 RecognisePermGroup
3.1-3 RecogniseMatrixGroup
3.1-4 RecogniseProjectiveGroup
3.1-5 RecogniseGroup
3.1-6 FindHomDbPerm
3.1-7 FindHomMethodsPerm
3.1-8 SLPforElementFuncsPerm
3.1-9 FindHomDbMatrix
3.1-10 FindHomMethodsMatrix
3.1-11 SLPforElementFuncsMatrix
3.1-12 FindHomDbProjective
3.1-13 FindHomMethodsProjective
3.1-14 SLPforElementFuncsProjective
3.1-15 TryFindHomMethod
3.2 Recognition info records
3.2-1 RecognitionInfoFamily
3.2-2 IsRecognitionInfo
3.2-3 IsLeaf
3.2-4 IsReady
3.2-5 Grp
3.2-6 Homom
3.2-7 NiceGens
3.2-8 pregensfac
3.2-9 RIFac
3.2-10 RIKer
3.2-11 RIParent
3.2-12 fhmethsel
3.2-13 slpforelement
3.2-14 SLPforElement
3.2-15 StdPresentation
3.2-16 methodsforfactor
3.2-17 calcnicegens
3.2-18 CalcNiceGensGeneric
3.2-19 CalcNiceGensHomNode
3.2-20 CalcNiceGens
3.2-21 slptonice
3.2-22 gensN
3.2-23 findgensNmeth
3.2-24 FindKernelRandom
3.2-25 FindKernelDoNothing
3.2-26 FindKernelFastNormalClosure
3.2-27 gensNslp
3.2-28 immediateverification
3.2-29 forkernel
3.2-30 forfactor
3.2-31 isone
3.2-32 isequal
3.2-33 Other components of recognition info records
3.3 Methods to find homomorphisms
3.3-1 FindHomomorphism
3.3-2 SLPforElementGeneric
3.4 Conventions for the recognition of permutation groups
3.5 Conventions for the recognition of matrix groups
3.6 Conventions for the recognition of projective groups
4 Method Selection
4.1 What are methods?
4.1-1 AddMethod
4.2 How methods are called
4.2-1 CallMethods
5 After successful recognition
5.1 Functions and methods for recognition info records
5.1-1 SLPforNiceGens
`5.1-2 \in`
5.1-3 Size
5.1-4 DisplayCompositionFactors
6 Methods for recognition
6.1 Methods for permutation groups
6.1-1 `TrivialPermGroup`
6.1-2 `ThrowAwayFixedPoints`
6.1-3 `Pcgs`
6.1-4 `VeryFewPoints`
6.1-5 `Nontransitive`
6.1-6 `Giant`
6.1-7 `Imprimitive`
6.1-8 `SnkSetswrSr`
6.1-9 `StabChain`
6.2 Methods for matrix groups
6.2-1 `TrivialMatrixGroup`
6.2-2 `KnownStabilizerChain`
6.2-3 `DiagonalMatrices`
6.2-4 `FewGensAbelian`
6.2-5 `ReducibleIso`
6.2-6 `BlockLowerTriangular`
6.2-7 `BlockDiagonal`
6.2-8 `GoProjective`
6.2-9 `BlockScalar`
6.2-10 `LowerLeftPGroup`
6.3 Methods for projective groups
6.3-1 `TrivialProjectiveGroup`
6.3-2 `ProjDeterminant`
6.3-3 `ReducibleIso`
6.3-4 `BlocksModScalars`
6.3-5 `NotAbsolutelyIrred`
6.3-6 `Subfield`
6.3-7 `Derived`
6.3-8 `LowIndex`
6.3-9 `C6`
6.3-10 `Tensor`
6.3-11 `TwoLargeElOrders`
6.3-12 `StabilizerChain`
6.3-13 `BlockScalarProj`
7 Examples
References
Index
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