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# LAGUNA

## Lie AlGebras and UNits of group Algebras

Version 3.4

February 2007

Victor Bovdi
e-mail: vbovdi@math.klte.hu
Address:
Institute of Mathematics and Informatics
University of Debrecen
P.O.Box 12, Debrecen, H-4010 Hungary

Alexander Konovalov
e-mail: konovalov@member.ams.org
WWW: 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
e-mail: richard.rossmanith@d-fine.de
Address:
d-fine GmbH
Mergenthalerallee 55 65760 Eschborn/Frankfurt
Germany

Csaba Schneider
e-mail: csaba.schneider@sztaki.hu
WWW: http://www.sztaki.hu/~schneider
Address:
Informatics Laboratory
Computer and Automation Research Institute
The Hungarian Academy of Sciences
1111 Budapest, Lagymanyosi u. 11, Hungary

### 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.

### 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).

### 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.

### Contents

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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