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<h1><strong class="pkg">LAGUNA</strong></h1>


<h2>Lie AlGebras and UNits of group Algebras</h2>

<p>Version 3.9.7</p>

<p>1 July 2024</p>

</div>
<p><b>Victor Bovdi 
          
          
  </b>
<br />Email: <span class="URL"><a href="mailto:vbovdi@science.unideb.hu">vbovdi@science.unideb.hu</a></span>
<br />Address: <br />Institute of Mathematics and Informatics<br /> University of Debrecen<br /> P.O.Box 12, Debrecen, H-4010 Hungary
</p><p><b>Olexandr Konovalov
          
   
           
  </b>
<br />Email: <span class="URL"><a href="mailto:obk1@st-andrews.ac.uk">obk1@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://alex-konovalov.github.io/">https://alex-konovalov.github.io/</a></span>
<br />Address: <br />School of Computer Science<br /> University of St Andrews<br /> Jack Cole Building, North Haugh,<br /> St Andrews, Fife, KY16 9SX, Scotland
</p><p><b>Richard Rossmanith</b>
</p><p><b>
  Csaba Schneider
  
  
           
  </b>
<br />Email: <span class="URL"><a href="mailto:csaba.schneider@sztaki.hu">csaba.schneider@sztaki.hu</a></span>
<br />Homepage: <span class="URL"><a href="http://www.sztaki.hu/~schneider">http://www.sztaki.hu/~schneider</a></span>
<br />Address: <br />Informatics Laboratory<br /> Computer and Automation Research Institute<br /> The Hungarian Academy of Sciences<br /> 1111 Budapest, Lagymanyosi u. 11, Hungary
</p>

<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The title ``<strong class="pkg">LAGUNA</strong>'' stands for ``<strong class="button">L</strong>ie <strong class="button">A</strong>l<strong class="button">G</strong>ebras and <strong class="button">UN</strong>its of group <strong class="button">A</strong>lgebras''. This is the new name of the <strong class="pkg">GAP</strong>4 package <strong class="pkg">LAG</strong>, which is thus replaced by <strong class="pkg">LAGUNA</strong>.</p>

<p><strong class="pkg">LAGUNA</strong> extends the <strong class="pkg">GAP</strong> functionality for computations in group rings. Besides computing some general properties and attributes of group rings and their elements, <strong class="pkg">LAGUNA</strong> 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 <span class="SimpleMath">p</span>-group over the field of <span class="SimpleMath">p</span> elements.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2003-2024 by Victor Bovdi, Olexandr Konovalov, Richard Rossmanith, and Csaba Schneider</p>

<p><strong class="pkg">LAGUNA</strong> 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 https://www.gnu.org/licenses/gpl.html.



<p>If you obtained <strong class="pkg">LAGUNA</strong>, we would be grateful for a short notification sent to one of the authors.</p>

<p>If you publish a result which was partially obtained with the usage of <strong class="pkg">LAGUNA</strong>, please cite it in the following form:</p>

<p>V. Bovdi, O. Konovalov, R. Rossmanith and C. Schneider. <em>LAGUNA --- Lie AlGebras and UNits of group Algebras, Version 3.9.7;</em> 1 July 2024 (<span class="URL"><a href="https://gap-packages.github.io/laguna/">https://gap-packages.github.io/laguna/</a></span>).</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>Some of the features of <strong class="pkg">LAGUNA</strong> were already included in the <strong class="pkg">GAP</strong>4 package <strong class="pkg">LAG</strong> written by the third author, Richard Rossmanith. The three other authors first would like to thank Greg Gamble for maintaining <strong class="pkg">LAG</strong> and for upgrading it from version 2.0 to version 2.1, and Richard Rossmanith for allowing them to update and extend the <strong class="pkg">LAG</strong> 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 <strong class="pkg">LAGUNA</strong> release.</p>

<p>We are all very grateful to the members of the <strong class="pkg">GAP</strong> 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 <strong class="pkg">LAGUNA</strong> and useful suggestions.</p>

<p>A part of the work on upgrading <strong class="pkg">LAG</strong> to <strong class="pkg">LAGUNA</strong> 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.



<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8557083378F2A3B2">1.1 <span class="Heading">General aims</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X790BFE667E3BEC3A">1.2 <span class="Heading">General computations in group rings</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7B7E5E987DBBA237">1.3 <span class="Heading">Computations in the normalized unit group</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X78B33FFF7A5D2FEF">1.4 <span class="Heading">Computing Lie properties of the group algebra </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7DB566D5785B7DBC">1.5 <span class="Heading">Installation and system requirements</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X7AAB1F18838B1DAB">2 <span class="Heading">A sample calculation with <strong class="pkg">LAGUNA</strong></span></a>
</div>
<div class="ContChap"><a href="chap3.html#X7D9FCE3A8526ACBE">3 <span class="Heading">The basic theory behind <strong class="pkg">LAGUNA</strong></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X85C2DE1486FB45FD">3.1 <span class="Heading">Notation and definitions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E26AE6C807D7C07">3.2 <span class="Heading"><span class="SimpleMath">p</span>-modular group algebras</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B74767A849D921E">3.3 <span class="Heading">Polycyclic generating set for <span class="SimpleMath">V</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7DCB12F48752C17E">3.4 <span class="Heading">Computing the canonical form</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X873F4F068759DF5D">3.5 <span class="Heading">Computing a power commutator presentation for <span class="SimpleMath">V</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X79183196837B7CC1">3.6 <span class="Heading">Verifying Lie properties of <span class="SimpleMath">FG</span></span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X7F79B2207E293F2B">4 <span class="Heading"><strong class="pkg">LAGUNA</strong> functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B473F157842958E">4.1 <span class="Heading">General functions for group algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X861A702283A667DF">4.1-1 IsGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82DBCA8F7DB21AC5">4.1-2 IsFModularGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X849599E07C38A739">4.1-3 IsPModularGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C966B177BB62C72">4.1-4 UnderlyingGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8534C18E7EA81CB8">4.1-5 UnderlyingRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X790470C48340E8F7">4.1-6 UnderlyingField</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X81DC99ED7AC03E46">4.2 <span class="Heading">Operations with group algebra elements</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B689C0284AC4296">4.2-1 Support</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8401D79C8678D3FA">4.2-2 CoefficientsBySupport</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81DD298A7C06EC82">4.2-3 TraceOfMagmaRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X780769238600AFD1">4.2-4 Length</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86FA17BE858F2245">4.2-5 Augmentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83A318C887B3E735">4.2-6 PartialAugmentations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81EB2A0A8756372B">4.2-7 Involution</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82D0BA3D7D3994F6">4.2-8 IsSymmetric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7990BD877E89A9EC">4.2-9 IsUnitary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85CBFBAE78DE72E8">4.2-10 IsUnit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82EC4F49877D6EB1">4.2-11 InverseOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FD01F52845445EC">4.2-12 BicyclicUnitOfType1</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8631AD178508D219">4.2-13 BassCyclicUnit</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X841733AB86D30446">4.3 <span class="Heading">Important attributes of group algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F7FD75E84BEE0EF">4.3-1 AugmentationHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B21DB3E7CD80983">4.3-2 AugmentationIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X850C29907A509533">4.3-3 RadicalOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8292BEFC7922E773">4.3-4 WeightedBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84B25D3A812A8A2E">4.3-5 AugmentationIdealPowerSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8567023A821E39A6">4.3-6 AugmentationIdealNilpotencyIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C0D5F3C842302EC">4.3-7 AugmentationIdealOfDerivedSubgroupNilpotencyIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X837294D57E415E37">4.3-8 LeftIdealBySubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X863248708784F94C">4.4 <span class="Heading">Computations with the unit group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X817D5AC78754527F">4.4-1 NormalizedUnitGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D40E42A7B678598">4.4-2 PcNormalizedUnitGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E08F6E87C586A36">4.4-3 NaturalBijectionToPcNormalizedUnitGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83715EF37B9FA94A">4.4-4 NaturalBijectionToNormalizedUnitGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84D64AF17941FA7F">4.4-5 Embedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X853C045B7BA6A580">4.4-6 Units</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E67BDA77E5E6077">4.4-7 PcUnits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F2532888184CB22">4.4-8 IsGroupOfUnitsOfMagmaRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EF9BF297A9412AB">4.4-9 IsUnitGroupOfGroupRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82AC131384191CCE">4.4-10 IsNormalizedUnitGroupOfGroupRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82DCA8A57D0D1114">4.4-11 UnderlyingGroupRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A0FFABC86F89517">4.4-12 UnitarySubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X859F07AE7A4D317B">4.4-13 BicyclicUnitGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A39E44D87CDF7B0">4.4-14 GroupBases</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X783C1A3D86A6656B">4.5 <span class="Heading">The Lie algebra of a group algebra</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7827B4158459DEA0">4.5-1 LieAlgebraByDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7AA7DA6082639B23">4.5-2 IsLieAlgebraByAssociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85F4712B84624DB8">4.5-3 UnderlyingAssociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84050A268514F2EE">4.5-4 NaturalBijectionToLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X821DAFD585EA1DF3">4.5-5 NaturalBijectionToAssociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78C26E3F80E55AC2">4.5-6 IsLieAlgebraOfGroupRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87CCD17A790BE256">4.5-7 UnderlyingGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X833439F7846784B5">4.5-8 Embedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8111F58E7DE3E25C">4.5-9 LieCentre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C95C0057C977747">4.5-10 LieDerivedSubalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F97D08F7B738ADE">4.5-11 IsLieAbelian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X859FF1B3812B8FCC">4.5-12 IsLieSolvable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78452F4E875A62A8">4.5-13 IsLieNilpotent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8504EC257B764BA6">4.5-14 IsLieMetabelian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EEAA09F818CE548">4.5-15 IsLieCentreByMetabelian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C8EBFF5805F8C51">4.5-16 CanonicalBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83026DFF8461B96D">4.5-17 IsBasisOfLieAlgebraOfGroupRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X804ADF0280F67CDC">4.5-18 StructureConstantsTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87E9411284D1A346">4.5-19 LieUpperNilpotencyIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D71DF0579FBE1C9">4.5-20 LieLowerNilpotencyIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E5950517F0CDB47">4.5-21 LieDerivedLength</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B16423A7FBED034">4.6 <span class="Heading">Other commands</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85CE6B407910F768">4.6-1 SubgroupsOfIndexTwo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7AFBD42180787A07">4.6-2 DihedralDepth</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D7CD02F803FFC08">4.6-3 DimensionBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84B7C6A38473722F">4.6-4 LieDimensionSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B2EAF81791D6C68">4.6-5 LieUpperCodimensionSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E5804B881DE4FF2">4.6-6 LAGInfo</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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