PAG

Prescribed Automorphism Groups

0.2.4

12 November 2024

Abstract

PAG is a GAP package for constructing combinatorial objects with prescribed automorphism groups.

The PAG package 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 2 of the License, or (at your option) any later version.

Acknowledgements

Development of the PAG package has been supported by the Croatian Science Foundation under the project IP-2020-02-9752.

1 The PAG Package

1.1 Getting Started

1.2 Installation

1.3 Examples: Designs

  1.3-1 6-(14,7,4) Designs
  1.3-2 6-(28,8,λ) Designs
  1.3-3 2-(81,6,2) Designs
  1.3-4 Quasi-symmetric 2-(56,16,18) Designs

1.4 Examples: Latin Squares

1.5 Examples: Cubes of Symmetric Designs

1.6 Examples: Projection Cubes of Symmetric Designs

1.7 Examples: Mosaics of Combinatorial Designs

2 The PAG Functions

2.1 Working With Permutation Groups

  2.1-1 CyclicPerm
  2.1-2 ToGroup
  2.1-3 MovePerm
  2.1-4 MoveGroup
  2.1-5 MultiPerm
  2.1-6 MultiGroup
  2.1-7 RestrictedGroup
  2.1-8 PrimitiveGroupsOfDegree
  2.1-9 TransitiveGroupsOfDegree
  2.1-10 Homogeneity
  2.1-11 AllSubgroupsConjugation
  2.1-12 PermRepresentationRight
  2.1-13 PermRepresentationLeft
  2.1-14 ExtendedPermRepresentation

2.2 Generating Orbits

  2.2-1 SubsetOrbitRep
  2.2-2 SubsetOrbitRepShort1
  2.2-3 SubsetOrbitRepIN
  2.2-4 IsGoodSubsetOrbit
  2.2-5 SmallLambdaFilter
  2.2-6 OrbitFilter1
  2.2-7 OrbitFilter2

2.3 Constructing Objects

  2.3-1 KramerMesnerSearch
  2.3-2 KramerMesnerMat
  2.3-3 CompatibilityMat
  2.3-4 SolveKramerMesner
  2.3-5 BaseBlocks
  2.3-6 ExpandMatRHS
  2.3-7 CameronSeidelSet
  2.3-8 OrthogonalNormalBasis
  2.3-9 KerdockSet
  2.3-10 SingerDifferenceSets
  2.3-11 NormalizedSingerDifferenceSets
  2.3-12 RightDevelopment
  2.3-13 LeftDevelopment
  2.3-14 EquivalentDifferenceSets

2.4 Inspecting Objects and Other Functions

  2.4-1 BlockDesignAut
  2.4-2 BlockDesignFilter
  2.4-3 Cliquer
  2.4-4 DisjointCliques
  2.4-5 IntersectionNumbers
  2.4-6 BlockScheme
  2.4-7 PointPairScheme
  2.4-8 TDesignB
  2.4-9 IversonBracket
  2.4-10 SymmetricDifference
  2.4-11 AddWeights
  2.4-12 AdjacencyMat

2.5 Latin Squares

  2.5-1 ReadMOLS
  2.5-2 WriteMOLS
  2.5-3 FieldToMOLS
  2.5-4 MOLSToOrthogonalArray
  2.5-5 OrthogonalArrayToMOLS
  2.5-6 MOLSToTransversalDesign
  2.5-7 TransversalDesignToMOLS
  2.5-8 MOLSAut
  2.5-9 MOLSFilter
  2.5-10 IsAutotopyGroup
  2.5-11 MOLSSubsetOrbitRep
  2.5-12 KramerMesnerMOLS
  2.5-13 KramerMesnerMOLSAutotopy
  2.5-14 KramerMesnerMOLSAutoparatopy

2.6 Cubes of Symmetric Designs

  2.6-1 DifferenceCube
  2.6-2 GroupCube
  2.6-3 CubeSlice
  2.6-4 CubeSlices
  2.6-5 CubeLayer
  2.6-6 CubeLayers
  2.6-7 CubeToOrthogonalArray
  2.6-8 OrthogonalArrayToCube
  2.6-9 OrthogonalArrayToTransversalDesign
  2.6-10 CubeToTransversalDesign
  2.6-11 TransversalDesignToCube
  2.6-12 LatinSquareToCube
  2.6-13 CubeTest
  2.6-14 SliceInvariant
  2.6-15 CubeAut
  2.6-16 CubeFilter
  2.6-17 SDPSeriesGroup
  2.6-18 SDPSeriesDesign

2.7 Projection Cubes of Symmetric Designs

  2.7-1 CubeProjection
  2.7-2 CubeProjections
  2.7-3 CubeProjectionTest
  2.7-4 OrthogonalArrayProjection
  2.7-5 OrthogonalArrayProjections
  2.7-6 OrthogonalArrayProjectionTest
  2.7-7 DifferenceSetToOrthogonalArray
  2.7-8 PaleyDifferenceSet
  2.7-9 PowerDifferenceSet
  2.7-10 TwinPrimePowerDifferenceSet
  2.7-11 OrthogonalArrayAut
  2.7-12 OrthogonalArrayFilter

2.8 Hadamard Matrices

  2.8-1 IsHadamardMat
  2.8-2 IsProperHadamardMat
  2.8-3 Paley1Mat
  2.8-4 Paley2Mat
  2.8-5 Paley3DMat
  2.8-6 SDPSeriesHadamardMat
  2.8-7 AllOnesMat
  2.8-8 ProductConstructionMat
  2.8-9 DigitConstructionMat
  2.8-10 CyclicDimensionIncrease
  2.8-11 HadamardMatAut
  2.8-12 HadamardMatFilter
  2.8-13 HadamardToIncidence
  2.8-14 IncidenceToHadamard

2.9 Mosaics of Combinatorial Designs

  2.9-1 MosaicParameters
  2.9-2 BlocksToIncidenceMat
  2.9-3 IncidenceMatToBlocks
  2.9-4 MosaicToBlockDesigns
  2.9-5 ReadMat
  2.9-6 WriteMat
  2.9-7 AffineMosaic
  2.9-8 DifferenceMosaic
  2.9-9 PowersMosaic
  2.9-10 MatAut
  2.9-11 MatFilter

2.10 Global Options

2.10-1 PAGGlobalOptions

References

Index

PAG

Prescribed Automorphism Groups

Abstract

Copyright

Acknowledgements

Contents