  
  
                                   [1X [5XHAPprime[0m[1X [0m
  
  
                          [1X Datatypes reference manual [0m
  
  
                                 Version 0.5.1
  
  
                                  10 May 2011
  
  
                                   Paul Smith
  
  
  
  Paul Smith
      Email:    [7Xmailto:paul.smith@st-andrews.ac.uk[0m
      Homepage: [7Xhttp://www.cs.st-andrews.ac.uk/~pas[0m
      Address:  School of Computer Science,
                University of St Andrews
                St Andrews,
                UK.
  
  
  
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  [1XCopyright[0m
  © 2006-2011 Paul Smith
  
  [5XHAPprime[0m  is  released under the GNU General Public License (GPL). This file
  is  part  of  [5XHAPprime[0m, though as documentation it is released under the GNU
  Free                Documentation                License                (see
  [7Xhttp://www.gnu.org/licenses/licenses.html#FDL[0m).
  
  [5XHAPprime[0m  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.
  
  [5XHAPprime[0m  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
  [5XHAPprime[0m;  if  not,  write  to the Free Software Foundation, Inc., 59 Temple
  Place, Suite 330, Boston, MA 02111-1307 USA
  
  For more details, see [7Xhttp://www.fsf.org/licenses/gpl.html[0m.
  
  
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  [1XAcknowledgements[0m
  [5XHAPprime[0m  was  developed  with  the  support  of  a  Marie Curie Transfer of
  Knowledge   grant  based  at  the  Department  of  Mathematics,  NUI  Galway
  (MTKD-CT-2006-042685)  and  is  maintained  with  the support of the HPC-GAP
  grant at the School of Computer Science, University of St Andrews.
  
  
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  [1XContents (HAPprime Datatypes)[0X
  
  1 Introduction
    1.1 Internal function reference
  2 Resolutions
    2.1 The [9XHAPResolution[0m datatype in [5XHAPprime[0m
    2.2 Implementation: Constructing resolutions
    2.3 Resolution construction functions
      2.3-1 LengthOneResolutionPrimePowerGroup
      2.3-2 LengthZeroResolutionPrimePowerGroup
    2.4 Resolution data access functions
      2.4-1 ResolutionLength
      2.4-2 ResolutionGroup
      2.4-3 ResolutionFpGModuleGF
      2.4-4 ResolutionModuleRank
      2.4-5 ResolutionModuleRanks
      2.4-6 BoundaryFpGModuleHomomorphismGF
      2.4-7 ResolutionsAreEqual
    2.5 Example: Computing and working with resolutions
    2.6 Miscellaneous resolution functions
      2.6-1 BestCentralSubgroupForResolutionFiniteExtension
  3 FG-modules
    3.1 The [9XFpGModuleGF[0m datatype
    3.2 Implementation details: Block echelon form
      3.2-1 Generating vectors and their block structure
      3.2-2 Matrix echelon reduction and head elements
      3.2-3 Echelon block structure and minimal generators
      3.2-4 Intersection of two modules
    3.3 Construction functions
      3.3-1 FpGModuleGF construction functions
      3.3-2 FpGModuleFromFpGModuleGF
      3.3-3 MutableCopyModule
      3.3-4 CanonicalAction
      3.3-5 Example: Constructing a [9XFpGModuleGF[0m
    3.4 Data access functions
      3.4-1 ModuleGroup
      3.4-2 ModuleGroupOrder
      3.4-3 ModuleAction
      3.4-4 ModuleActionBlockSize
      3.4-5 ModuleGroupAndAction
      3.4-6 ModuleCharacteristic
      3.4-7 ModuleField
      3.4-8 ModuleAmbientDimension
      3.4-9 AmbientModuleDimension
      3.4-10 DisplayBlocks
      3.4-11 Example: Accessing data about a [9XFpGModuleGF[0m
    3.5 Generator and vector space functions
      3.5-1 ModuleGenerators
      3.5-2 ModuleGeneratorsAreMinimal
      3.5-3 ModuleGeneratorsAreEchelonForm
      3.5-4 ModuleIsFullCanonical
      3.5-5 ModuleGeneratorsForm
      3.5-6 ModuleRank
      3.5-7 ModuleVectorSpaceBasis
      3.5-8 ModuleVectorSpaceDimension
      3.5-9 MinimalGeneratorsModule
      3.5-10 RadicalOfModule
      3.5-11 Example: Generators and basis vectors of a [9XFpGModuleGF[0m
    3.6 Block echelon functions
      3.6-1 EchelonModuleGenerators
      3.6-2 ReverseEchelonModuleGenerators
      3.6-3 Example: Converting a [9XFpGModuleGF[0m to block echelon form
    3.7 Sum and intersection functions
      3.7-1 DirectSumOfModules
      3.7-2 DirectDecompositionOfModule
      3.7-3 IntersectionModules
      3.7-4 SumModules
      3.7-5 Example: Sum and intersection of [9XFpGModuleGF[0ms
    3.8 Miscellaneous functions
      3.8-1 =
      3.8-2 IsModuleElement
      3.8-3 IsSubModule
      3.8-4 RandomElement
      3.8-5 Random Submodule
  4 FG-module homomorphisms
    4.1 The [9XFpGModuleHomomorphismGF[0m datatype
    4.2 Calculating the kernel of a FG-module homorphism by splitting into two
    homomorphisms
    4.3 Calculating the kernel of a FG-module homorphism by column reduction
    and partitioning
    4.4 Construction functions
      4.4-1 FpGModuleHomomorphismGF construction functions
      4.4-2 Example: Constructing a [9XFpGModuleHomomorphismGF[0m
    4.5 Data access functions
      4.5-1 SourceModule
      4.5-2 TargetModule
      4.5-3 ModuleHomomorphismGeneratorMatrix
      4.5-4 DisplayBlocks
      4.5-5 DisplayModuleHomomorphismGeneratorMatrix
      4.5-6 DisplayModuleHomomorphismGeneratorMatrixBlocks
      4.5-7 Example: Accessing data about a [9XFpGModuleHomomorphismGF[0m
    4.6 Image and kernel functions
      4.6-1 ImageOfModuleHomomorphism
      4.6-2 PreImageRepresentativeOfModuleHomomorphism
      4.6-3 KernelOfModuleHomomorphism
      4.6-4 Example: Kernel and Image of a [9XFpGModuleHomomorphismGF[0m
  5 General Functions
    5.1 Matrices
      5.1-1 SumIntersectionMatDestructive
      5.1-2 SolutionMat
      5.1-3 IsSameSubspace
      5.1-4 PrintDimensionsMat
      5.1-5 Example: matrices and vector spaces
    5.2 Groups
      5.2-1 HallSeniorNumber
  6 Internal functions
    6.1 Matrices as G-generators of a FG-module vector space
      6.1-1 HAPPRIME_ValueOptionMaxFGExpansionSize
      6.1-2 HAPPRIME_KernelOfGeneratingRowsDestructive
      6.1-3 HAPPRIME_GActMatrixColumns
      6.1-4 HAPPRIME_ExpandGeneratingRow
      6.1-5 HAPPRIME_AddGeneratingRowToSemiEchelonBasisDestructive
      6.1-6 HAPPRIME_ReduceVectorDestructive
      6.1-7 HAPPRIME_ReduceGeneratorsOfModuleByXX
      6.1-8 HAPPRIME_DisplayGeneratingRows
      6.1-9 HAPPRIME_GeneratingRowsBlockStructure
      6.1-10 HAPPRIME_DisplayGeneratingRowsBlocks
      6.1-11 HAPPRIME_IndependentGeneratingRows
      6.1-12 HAPPRIME_GactFGvector
      6.1-13 HAPPRIME_CoefficientsOfGeneratingRowsXX
      6.1-14 HAPPRIME_GenerateFromGeneratingRowsCoefficientsXX
      6.1-15 HAPPRIME_RemoveZeroBlocks
      6.1-16 HAPPRIME_AddZeroBlocks
    6.2 FG-modules
      6.2-1 HAPPRIME_DirectSumForm
      6.2-2 HAPPRIME_PrintModuleDescription
      6.2-3 HAPPRIME_ModuleGeneratorCoefficients
      6.2-4 HAPPRIME_ModuleElementFromGeneratorCoefficients
      6.2-5 HAPPRIME_MinimalGeneratorsVectorSpaceGeneratingRowsDestructive
      6.2-6 HAPPRIME_IsGroupAndAction
    6.3 Resolutions
      6.3-1 HAPPRIME_WordToVector
      6.3-2 HAPPRIME_VectorToWord
      6.3-3 HAPPRIME_BoundaryMatrices
      6.3-4 HAPPRIME_AddNextResolutionBoundaryMapMatNC
      6.3-5 HAPPRIME_CreateResolutionWithBoundaryMapMatsNC
    6.4 Test functions
      6.4-1 HAPPRIME_SingularIsAvailable
      6.4-2 HAPPRIME_Random2Group
      6.4-3 HAPPRIME_TestResolutionPrimePowerGroup
  
  
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