  
  
                                     [1m[4m[31m[1mLAGUNA[1m[4m[31m[0m
  
  
                    [1m[4m[31mLie AlGebras and UNits of group Algebras[0m
  
  
                                 Version 3.2.2
  
  
                                   April 2004
  
  
                                  Victor Bovdi
  
                              Alexander Konovalov
  
                               Richard Rossmanith
  
                                Csaba Schneider
  
  
  
  Victor Bovdi
      Email:    [34mmailto:vbovdi@math.klte.hu[0m
      Address:  Institute of Mathematics and Informatics
                University of Debrecen
                P.O.Box 12, Debrecen, H-4010 Hungary
  
  
  Alexander Konovalov
      Email:    [34mmailto:konovalov@member.ams.org[0m
      Homepage: [34mhttp://ukrgap.exponenta.ru/konoval.htm[0m
      Address:  Department of Mathematics
                Zaporozhye State University
                P.O.Box 1317, Central Post Office
                Zaporozhye, 69000 Ukraine
  
  
  Richard Rossmanith
      Email:    [34mmailto:richard.rossmanith@d-fine.de[0m
      Address:  d-fine GmbH
                Mergenthalerallee 55 65760 Eschborn/Frankfurt
                Germany
  
  
  Csaba Schneider
      Email:    [34mmailto:csaba@maths.uwa.edu.au[0m
      Homepage: [34mhttp://www.maths.uwa.edu.au/~csaba/[0m
      Address:  School of Mathematics and Statistics
                The University of Western Australia
                35 Stirling Highway
                6009 Crawley Western Australia
  
  
  
  -------------------------------------------------------
  [1m[4m[31mAbstract[0m
  The   title  ``[1mLAGUNA[0m''  stands  for  ``[1m[46mL[0mie  [1m[46mA[0ml[1m[46mG[0mebras  and  [1m[46mUN[0mits  of  group
  [1m[46mA[0mlgebras''.  This  is  the  new  name of the [1mGAP[0m4 package [1mLAG[0m, which is thus
  replaced by [1mLAGUNA[0m.
  
  [1mLAGUNA[0m  extends  the  [1mGAP[0m  functionality  for  computations  in group rings.
  Besides  computing some general properties and attributes of group rings and
  their  elements,  [1mLAGUNA[0m  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.
  
  
  -------------------------------------------------------
  [1m[4m[31mCopyright[0m
  (C) 2003 by Victor Bovdi, Alexander Konovalov, Richard Rossmanith, and Csaba
  Schneider
  
  We  adopt  the copyright regulations of [1mGAP[0m from the copyright notice in the
  [1mGAP[0m manual.
  
  If  you  obtained [1mLAGUNA[0m, 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
  [1mLAGUNA[0m, please cite it in the following form:
  
  V.  Bovdi,  A.  Konovalov,  R.  Rossmanith  and C. Schneider. [22m[36mLAGUNA --- Lie
  AlGebras    and    UNits    of   group   Algebras,   Version   3.2.2;[0m   2003
  ([34mhttp://ukrgap.exponenta.ru/laguna.htm[0m).
  
  
  -------------------------------------------------------
  [1m[4m[31mAcknowledgements[0m
  Some of the features of [1mLAGUNA[0m were already included in the [1mGAP[0m4 package [1mLAG[0m
  written  by  the  third  author, Richard Rossmanith. The three other authors
  first  would like to thank Greg Gamble for maintaining [1mLAG[0m and for upgrading
  it from version 2.0 to version 2.1, and Richard Rossmanith for allowing them
  to  update  and  extend  the  [1mLAG[0m  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 [1mLAGUNA[0m
  release.
  
  We  are  all  very  grateful  to the members of the [1mGAP[0m 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 [1mLAGUNA[0m and useful suggestions.
  
  A  part  of  the  work  on  upgrading  [1mLAG[0m to [1mLAGUNA[0m 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.
  
  
  -------------------------------------------------------
  
  
  [1m[4m[31mContent (LAGUNA)[0m
  
  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 [1mLAGUNA[0m
  3. The basic theory behind [1mLAGUNA[0m
    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. [1mLAGUNA[0m 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 Involution
      4.2-7 IsSymmetric
      4.2-8 IsUnitary
      4.2-9 IsUnit
      4.2-10 InverseOp
    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
    4.4 Computations with the unit group
      4.4-1 NormalizedUnitGroup
      4.4-2 PcNormalizedUnitGroup
      4.4-3 NaturalBijectionToPcNormalizedUnitGroup
      4.4-4 NaturalBijectionToNormalizedUnitGroup
      4.4-5 Units
      4.4-6 PcUnits
      4.4-7 IsGroupOfUnitsOfMagmaRing
      4.4-8 IsUnitGroupOfGroupRing
      4.4-9 IsNormalizedUnitGroupOfGroupRing
      4.4-10 UnderlyingGroupRing
      4.4-11 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
    4.6 Other commands
      4.6-1 SubgroupsOfIndexTwo
      4.6-2 DihedralDepth
      4.6-3 DimensionBasis
      4.6-4 LieDimensionSubgroups
      4.6-5 LAGInfo
      4.6-6 LAGUNABuildManual
      4.6-7 LAGUNABuildManualHTML
  
  
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