LAGUNA
  
  
                    Lie AlGebras and UNits of group Algebras
  
  
                                  Version 3.4
  
  
                                 February 2007
  
  
                                  Victor Bovdi
  
                              Alexander Konovalov
  
                               Richard Rossmanith
  
                                Csaba Schneider
  
  
  
  Victor Bovdi
      Email:    mailto:vbovdi@math.klte.hu
      Address:  Institute of Mathematics and Informatics
                University of Debrecen
                P.O.Box 12, Debrecen, H-4010 Hungary
  
  
  Alexander Konovalov
      Email:    mailto:konovalov@member.ams.org
      Homepage: http://www.cs.st-andrews.ac.uk/~alexk/
      Address:  School of Computer Science
                University of St Andrews
                Jack Cole Building, North Haugh,
                St Andrews, Fife, KY16 9SX, Scotland
  
  
  Richard Rossmanith
      Email:    mailto:richard.rossmanith@d-fine.de
      Address:  d-fine GmbH
                Mergenthalerallee 55 65760 Eschborn/Frankfurt
                Germany
  
  
  Csaba Schneider
      Email:    mailto:csaba.schneider@sztaki.hu
      Homepage: http://www.sztaki.hu/~schneider
      Address:  Informatics Laboratory
                Computer and Automation Research Institute
                The Hungarian Academy of Sciences
                1111 Budapest, Lagymanyosi u. 11, Hungary
  
  
  
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  Abstract
  The   title  ``LAGUNA''  stands  for  ``Lie  AlGebras  and  UNits  of  group
  Algebras''.  This  is  the  new  name of the GAP4 package LAG, which is thus
  replaced by LAGUNA.
  
  LAGUNA  extends  the  GAP  functionality  for  computations  in group rings.
  Besides  computing some general properties and attributes of group rings and
  their  elements,  LAGUNA  is able to perform two main kinds of computations.
  Namely,  it  can  verify whether a group algebra of a finite group satisfies
  certain Lie properties; and it can calculate the structure of the normalized
  unit  group  of  a  group  algebra  of  a finite p-group over the field of p
  elements.
  
  
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  Copyright
  (C)  2003-2007 by Victor Bovdi, Alexander Konovalov, Richard Rossmanith, and
  Csaba Schneider
  
  LAGUNA  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.      For      details,      see      the     FSF's     own     site
  http://www.gnu.org/licenses/gpl.html.
  
  If  you  obtained LAGUNA, we would be grateful for a short notification sent
  to one of the authors.
  
  If  you  publish  a  result  which  was partially obtained with the usage of
  LAGUNA, please cite it in the following form:
  
  V.  Bovdi,  A.  Konovalov,  R.  Rossmanith  and C. Schneider. LAGUNA --- Lie
  AlGebras    and    UNits    of    group    Algebras,   Version   3.4;   2007
  (http://www.cs.st-andrews.ac.uk/~alexk/laguna.htm).
  
  
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  Acknowledgements
  Some of the features of LAGUNA were already included in the GAP4 package LAG
  written  by  the  third  author, Richard Rossmanith. The three other authors
  first  would like to thank Greg Gamble for maintaining LAG and for upgrading
  it from version 2.0 to version 2.1, and Richard Rossmanith for allowing them
  to  update  and  extend  the  LAG  package. We are also grateful to Wolfgang
  Kimmerle  for  organizing  the workshop ``Computational Group and Group Ring
  Theory''  (University of Stuttgart, 28--29 November, 2002), which allowed us
  to  meet  and  have  fruitful  discussions that led towards the final LAGUNA
  release.
  
  We  are  all  very  grateful  to the members of the GAP team: Thomas Breuer,
  Willem  de Graaf, Alexander Hulpke, Stefan Kohl, Steve Linton, Frank Lübeck,
  Max Neunhöffer and many other colleagues for helpful comments and advise. We
  acknowledge very much Herbert Pahlings for communicating the package and the
  referee for careful testing LAGUNA and useful suggestions.
  
  A  part  of  the  work  on  upgrading  LAG to LAGUNA was done in 2002 during
  Alexander   Konovalov's   visits  to  Debrecen,  St  Andrews  and  Stuttgart
  Universities.  He  would like to express his gratitude to Adalbert Bovdi and
  Victor  Bovdi,  Colin  Campbell, Edmund Robertson and Steve Linton, Wolfgang
  Kimmerle, Martin Hertweck and Stefan Kohl for their warm hospitality, and to
  the  NATO Science Fellowship Program, to the London Mathematical Society and
  to the DAAD for the support of these visits.
  
  
  -------------------------------------------------------
  
  
  Content (LAGUNA)
  
  1. Introduction
    1.1 General aims
    1.2 General computations in group rings
    1.3 Computations in the normalized unit group
    1.4 Computing Lie properties of the group algebra
    1.5 Installation and system requirements
  2. A sample calculation with LAGUNA
  3. The basic theory behind LAGUNA
    3.1 Notation and definitions
    3.2 p-modular group algebras
    3.3 Polycyclic generating set for V
    3.4 Computing the canonical form
    3.5 Computing a power commutator presentation for V
    3.6 Verifying Lie properties of FG
  4. LAGUNA functions
    4.1 General functions for group algebras
      4.1-1 IsGroupAlgebra
      4.1-2 IsFModularGroupAlgebra
      4.1-3 IsPModularGroupAlgebra
      4.1-4 UnderlyingGroup
      4.1-5 UnderlyingRing
      4.1-6 UnderlyingField
    4.2 Operations with group algebra elements
      4.2-1 Support
      4.2-2 CoefficientsBySupport
      4.2-3 TraceOfMagmaRingElement
      4.2-4 Length
      4.2-5 Augmentation
      4.2-6 PartialAugmentations
      4.2-7 Involution
      4.2-8 IsSymmetric
      4.2-9 IsUnitary
      4.2-10 IsUnit
      4.2-11 InverseOp
      4.2-12 BicyclicUnitOfType1
    4.3 Important attributes of group algebras
      4.3-1 AugmentationHomomorphism
      4.3-2 AugmentationIdeal
      4.3-3 RadicalOfAlgebra
      4.3-4 WeightedBasis
      4.3-5 AugmentationIdealPowerSeries
      4.3-6 AugmentationIdealNilpotencyIndex
      4.3-7 AugmentationIdealOfDerivedSubgroupNilpotencyIndex
      4.3-8 LeftIdealBySubgroup
    4.4 Computations with the unit group
      4.4-1 NormalizedUnitGroup
      4.4-2 PcNormalizedUnitGroup
      4.4-3 NaturalBijectionToPcNormalizedUnitGroup
      4.4-4 NaturalBijectionToNormalizedUnitGroup
      4.4-5 Embedding
      4.4-6 Units
      4.4-7 PcUnits
      4.4-8 IsGroupOfUnitsOfMagmaRing
      4.4-9 IsUnitGroupOfGroupRing
      4.4-10 IsNormalizedUnitGroupOfGroupRing
      4.4-11 UnderlyingGroupRing
      4.4-12 UnitarySubgroup
      4.4-13 BicyclicUnitGroup
      4.4-14 AugmentationIdealPowerFactorGroup
      4.4-15 GroupBases
    4.5 The Lie algebra of a group algebra
      4.5-1 LieAlgebraByDomain
      4.5-2 IsLieAlgebraByAssociativeAlgebra
      4.5-3 UnderlyingAssociativeAlgebra
      4.5-4 NaturalBijectionToLieAlgebra
      4.5-5 NaturalBijectionToAssociativeAlgebra
      4.5-6 IsLieAlgebraOfGroupRing
      4.5-7 UnderlyingGroup
      4.5-8 Embedding
      4.5-9 LieCentre
      4.5-10 LieDerivedSubalgebra
      4.5-11 IsLieAbelian
      4.5-12 IsLieSolvable
      4.5-13 IsLieNilpotent
      4.5-14 IsLieMetabelian
      4.5-15 IsLieCentreByMetabelian
      4.5-16 CanonicalBasis
      4.5-17 IsBasisOfLieAlgebraOfGroupRing
      4.5-18 StructureConstantsTable
      4.5-19 LieUpperNilpotencyIndex
      4.5-20 LieLowerNilpotencyIndex
      4.5-21 LieDerivedLength
    4.6 Other commands
      4.6-1 SubgroupsOfIndexTwo
      4.6-2 DihedralDepth
      4.6-3 DimensionBasis
      4.6-4 LieDimensionSubgroups
      4.6-5 LieUpperCodimensionSeries
      4.6-6 LAGInfo
      4.6-7 LAGUNABuildManual
      4.6-8 LAGUNABuildManualHTML
  
  
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