  
  
                                     [1X[5XLAGUNA[105X[101X
  
  
                    [1XLie AlGebras and UNits of group Algebras[101X
  
  
                                 Version 3.10.1
  
  
                                 14 August 2026
  
  
                                  Victor Bovdi
  
                               Olexandr Konovalov
  
                               Richard Rossmanith
  
                                Csaba Schneider
  
  
  
  Victor Bovdi
      Email:    [7Xmailto:vbovdi@science.unideb.hu[107X
      Address:  [33X[0;14YInstitute of Mathematics and Informatics[133X
                [33X[0;14YUniversity of Debrecen[133X
                [33X[0;14YP.O.Box 12, Debrecen, H-4010 Hungary[133X
  
  
  Olexandr Konovalov
      Email:    [7Xmailto:obk1@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://olexandr-konovalov.github.io/[107X
      Address:  [33X[0;14YSchool of Computer Science[133X
                [33X[0;14YUniversity of St Andrews[133X
                [33X[0;14YJack Cole Building, North Haugh,[133X
                [33X[0;14YSt Andrews, Fife, KY16 9SX, Scotland[133X
  
  
  Csaba Schneider
      Email:    [7Xmailto:csaba.schneider@sztaki.hu[107X
      Homepage: [7Xhttp://www.sztaki.hu/~schneider[107X
      Address:  [33X[0;14YInformatics Laboratory[133X
                [33X[0;14YComputer and Automation Research Institute[133X
                [33X[0;14YThe Hungarian Academy of Sciences[133X
                [33X[0;14Y1111 Budapest, Lagymanyosi u. 11, Hungary[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe   title  ``[5XLAGUNA[105X''  stands  for  ``[12XL[112Xie  [12XA[112Xl[12XG[112Xebras  and  [12XUN[112Xits  of  group
  [12XA[112Xlgebras''.  This  is  the  new  name of the [5XGAP[105X4 package [5XLAG[105X, which is thus
  replaced by [5XLAGUNA[105X.[133X
  
  [33X[0;0Y[5XLAGUNA[105X  extends  the  [5XGAP[105X  functionality  for  computations  in group rings.
  Besides  computing some general properties and attributes of group rings and
  their  elements,  [5XLAGUNA[105X  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 [22Xp[122X-group over the field of [22Xp[122X
  elements.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©  2003-2026  by  Victor  Bovdi, Olexandr Konovalov, Richard Rossmanith, and
  Csaba Schneider[133X
  
  [33X[0;0Y[5XLAGUNA[105X  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
  [7Xhttps://www.gnu.org/licenses/gpl.html[107X.[133X
  
  [33X[0;0YIf  you  obtained [5XLAGUNA[105X, we would be grateful for a short notification sent
  to one of the authors.[133X
  
  [33X[0;0YIf  you  publish  a  result  which  was partially obtained with the usage of
  [5XLAGUNA[105X, please cite it in the following form:[133X
  
  [33X[0;0YV.  Bovdi,  O.  Konovalov,  R.  Rossmanith  and C. Schneider. [13XLAGUNA --- Lie
  AlGebras  and  UNits  of  group  Algebras,  Version  3.10.1;[113X  14 August 2026
  ([7Xhttps://gap-packages.github.io/laguna/[107X).[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YSome of the features of [5XLAGUNA[105X were already included in the [5XGAP[105X4 package [5XLAG[105X
  written  by  the  third  author, Richard Rossmanith. The three other authors
  first  would like to thank Greg Gamble for maintaining [5XLAG[105X and for upgrading
  it from version 2.0 to version 2.1, and Richard Rossmanith for allowing them
  to  update  and  extend  the  [5XLAG[105X  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 [5XLAGUNA[105X
  release.[133X
  
  [33X[0;0YWe  are  all  very  grateful  to the members of the [5XGAP[105X 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 [5XLAGUNA[105X and useful suggestions.[133X
  
  [33X[0;0YA  part  of  the  work  on  upgrading  [5XLAG[105X to [5XLAGUNA[105X was done in 2002 during
  Olexandr   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.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (LAGUNA)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YGeneral aims[133X
    1.2 [33X[0;0YGeneral computations in group rings[133X
    1.3 [33X[0;0YComputations in the normalized unit group[133X
    1.4 [33X[0;0YComputing Lie properties of the group algebra[133X
    1.5 [33X[0;0YInstallation and system requirements[133X
  2 [33X[0;0YA sample calculation with [5XLAGUNA[105X[133X
  3 [33X[0;0YThe basic theory behind [5XLAGUNA[105X[133X
    3.1 [33X[0;0YNotation and definitions[133X
    3.2 [33X[0;0Y[22Xp[122X-modular group algebras[133X
    3.3 [33X[0;0YPolycyclic generating set for [22XV[122X[133X
    3.4 [33X[0;0YComputing the canonical form[133X
    3.5 [33X[0;0YComputing a power commutator presentation for [22XV[122X[133X
    3.6 [33X[0;0YVerifying Lie properties of [22XFG[122X[133X
  4 [33X[0;0Y[5XLAGUNA[105X functions[133X
    4.1 [33X[0;0YGeneral functions for group algebras[133X
      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 [33X[0;0YOperations with group algebra elements[133X
      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
      4.2-13 BassCyclicUnit
    4.3 [33X[0;0YImportant attributes of group algebras[133X
      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 [33X[0;0YComputations with the unit group[133X
      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 GroupBases
    4.5 [33X[0;0YThe Lie algebra of a group algebra[133X
      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 [33X[0;0YOther commands[133X
      4.6-1 SubgroupsOfIndexTwo
      4.6-2 DihedralDepth
      4.6-3 DimensionBasis
      4.6-4 LieDimensionSubgroups
      4.6-5 LieUpperCodimensionSeries
      4.6-6 LAGInfo
  
  
  [32X
