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# HAPprime

## Datatypes reference manual

Version 0.5.1

10 May 2011

Paul Smith
Email: paul.smith@st-andrews.ac.uk
Homepage: http://www.cs.st-andrews.ac.uk/~pas
Address:
School of Computer Science,
University of St Andrews
St Andrews,
UK.

### Copyright

© 2006-2011 Paul Smith

HAPprime is released under the GNU General Public License (GPL). This file is part of HAPprime, though as documentation it is released under the GNU Free Documentation License (see http://www.gnu.org/licenses/licenses.html#FDL).

HAPprime 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.

HAPprime 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 HAPprime; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

For more details, see http://www.fsf.org/licenses/gpl.html.

### Acknowledgements

HAPprime 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.

### Contents

1 Introduction
1.1 Internal function reference
2 Resolutions
2.1 The `HAPResolution` datatype in HAPprime
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 `FpGModuleGF` 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 `FpGModuleGF`
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 `FpGModuleGF`
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 `FpGModuleGF`
3.6 Block echelon functions
3.6-1 EchelonModuleGenerators
3.6-2 ReverseEchelonModuleGenerators
3.6-3 Example: Converting a `FpGModuleGF` 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 `FpGModuleGF`s
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 `FpGModuleHomomorphismGF` 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 `FpGModuleHomomorphismGF`
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 `FpGModuleHomomorphismGF`
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 `FpGModuleHomomorphismGF`
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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