[1X[5XLAGUNA[105X[101X [1XLie AlGebras and UNits of group Algebras[101X Version 3.9.5 27 April 2022 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://alex-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© 2003-2022 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 3.9.5;[113X 27 April 2022 ([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
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