<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >title >GAP (HeLP) - Contents</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap0" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1</a> <a href="chap2.html" >2</a> <a href="chap3.html" >3</a> <a href="chap4.html" >4</a> <a href="chap5.html" >5</a> <a href="chap6.html" >6</a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap1.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap0_mj.html" >[MathJax on]</a></p>"X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5" ></a></p>div class="pcenter" >h1 >HeLP</h1 >div >br />Email: <span class="URL" ><a href="mailto:ABachle@gmx.net" >ABachle@gmx.net</a></span >br />Address : <br />Vrije Universiteit Brussel<br /> Vakgroep Wiskunde<br /> Pleinlaan 2<br /> 1050 Brussels<br /> Belgium<br />br />Email: <span class="URL" ><a href="mailto:Leo.Margolis@uam.es" >Leo.Margolis@uam.es</a></span br />Homepage: <span class="URL" ><a href="https://margollo.github.io " >https://margollo.github.io</a></span >br />Address : <br />Universidad Autonoma de Madrid<br /> Campus Cantoblanco<br /> Facultad de Ciencias<br /> Departamento de Matematicas<br /> 28049 Madrid<br /> Spain<br />"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>ublic License Version 2.</p>"X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a></p>h a 4ti2-Interface and a normaliz-Interface. We also would like to thank Gutsche and Söger for many very helpful discussions. We also want to give credits to the developers of the softwares 4ti2 and normaliz. Thanks go to David Avis for writing lrslib and answering our questions about it. We moreover thank Wolfgang Kimmerle for introducing us to the beautiful world of group rings. The development of this package was partially supported by the Research Foundation Flanders (FWO - Vlaanderen) and the DFG priority program SPP 1489 Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory.</p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"contents" name="contents" ></a></h3>div class="ContChap" ><a href="chap1.html#X7DFB63A97E67C0A1" >1 <span class="Heading" >Introduction</span ></a>div >div class="ContChap" ><a href="chap2.html#X7D3DC4ED855DC13C" >2 <span class="Heading" >The main functions</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7C3DBA147B6CF284" >2.1 <span class="Heading" >Zassenhaus Conjecture</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X81AF79A587054306" >2.1-1 HeLP_ZC</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7B12013C7C8A6714" >2.2 <span class="Heading" >Prime Graph Question</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X813A10398218E9EE" >2.2-1 HeLP_PQ</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7A23146485988360" >2.3 <span class="Heading" >Spectrum Problem</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X84F8B60C8016DE7F" >2.3-1 HeLP_SP</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X83B74D498430B5D5" >2.4 <span class="Heading" >Kimmerle Problem</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7CD90F897EEE8670" >2.4-1 HeLP_KP</a></span >div ></div >div >div class="ContChap" ><a href="chap3.html#X85C2B0617ECCA64E" >3 <span class="Heading" >Further functions</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X85F252368293DB34" >3.1 <span class="Heading" >Checks for specific orders</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7F8F6E3D80A23C1D" >3.1-1 HeLP_WithGivenOrder</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7CD0CEF283F13F7B" >3.1-2 HeLP_WithGivenOrderAndPA</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X8186F17681AF25F5" >3.1-3 HeLP_WithGivenOrderAllTables</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X81063633815E39CE" >3.1-4 HeLP_WithGivenOrderAndPAAllTables</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X80D9773D86873CB2" >3.1-5 HeLP_WithGivenOrderAndPAAndSpecificSystem</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7FCD80AC87DD0460" >3.2 <span class="Heading" >Checks for specific orders with s-constant characters</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7CAC647C7D1E95B0" >3.2-1 HeLP_WithGivenOrderSConstant</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7B0FD19084B09AF8" >3.2-2 HeLP_AddGaloisCharacterSums</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X86A4BB01819B625D" >3.3 <span class="Heading" >Checks for all orders</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X8727639883F787C5" >3.3-1 HeLP_AllOrders</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7C00E1567BFF1757" >3.3-2 HeLP_AllOrdersPQ</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X79C24763800E48E9" >3.3-3 HeLP_AllOrdersSP</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X81E3FEE678B0FDD5" >3.3-4 HeLP_AllOrdersKP</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7E81639F8186F858" >3.4 <span class="Heading" >Changing the used Character Table </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7BB9009482784E90" >3.4-1 HeLP_ChangeCharKeepSols</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7C19F3A378AAF294" >3.4-2 HeLP_Reset</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7A05180B7DA5EB77" >3.5 <span class="Heading" >Influencing how the Systems of Inequalities are solved</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7B2326C5813CF36B" >3.5-1 HeLP_Solver</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7A7536D9790C1901" >3.5-2 HeLP_UseRedund</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7F6C4FAD805CD7FC" >3.5-3 HeLP_Change4ti2Precision</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X8490447A857CFD87" >3.5-4 HeLP_Vertices</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7990EC037D6AE938" >3.6 <span class="Heading" >Checking solutions, calculating and checking solutions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7DAA7EF785621D9E" >3.6-1 HeLP_VerifySolution</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X8452B7F58641E7F5" >3.6-2 HeLP_FindAndVerifySolution</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X81E4BAF2815051C4" >3.6-3 HeLP_PossiblePartialAugmentationsOfPowers</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7E37E3767B7085B9" >3.6-4 HeLP_WriteTrivialSolution</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7BA77C9F86ADD546" >3.7 <span class="Heading" >The Wagner test</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X79349D80830FA89B" >3.7-1 HeLP_WagnerTest</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7FCFA1227B958BC0" >3.8 <span class="Heading" >Action of the automorphism group</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X80E976FE781AC904" >3.8-1 HeLP_AutomorphismOrbits</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X84DD5181826CA1C2" >3.9 <span class="Heading" >Output </span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7A5CAEBD801EF192" >3.9-1 HeLP_PrintSolution</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7F88D4667910126A" >3.10 <span class="Heading" >Eigenvalue multiplicities and character values</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X86601BE281C7B8B6" >3.10-1 HeLP_MultiplicitiesOfEigenvalues</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7C4C37B681A5BC7D" >3.10-2 HeLP_CharacterValue</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X8668C68E7C07EF99" >3.11 <span class="Heading" >Check for triviality modulo normal subgroup</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X819973897A847FFE" >3.11-1 HeLP_IsOneModuloN</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X87E2729184EB30B3" >3.11-2 HeLP_ForgetUnderlyingGroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X87C19C44801003CD" >3.12 <span class="Heading" >Check Kimmerle Problem for single units</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7E8B60D97999264E" >3.12-1 HeLP_UnitSatisfiesKP</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X828D02D784AF3DFF" >3.13 <span class="Heading" >Check whether Zassenhaus Conjecture is known from theoretical results</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X80E52D94801193C4" >3.13-1 HeLP_IsZCKnown</a></span >div ></div >div >div class="ContChap" ><a href="chap4.html#X7CDC63A27F7790AA" >4 <span class="Heading" >Extended examples</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7DEAD03D7811F9FA" >4.1 <span class="Heading" >The Character Table Library</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X84ED1F0D7A47B055" >4.2 <span class="Heading" >The behavior of the variable HeLP sol</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7E939D8483F1EE64" >4.3 <span class="Heading" >Saving time </span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X8242093A82FE41FA" >4.4 <span class="Heading" >Using InfoLevels</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X818A647182CA20B3" >4.5 <span class="Heading" >Non-standard characters</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X879DC8C287C41B09" >4.6 <span class="Heading" >A complete example: (PQ) for the MacLaughlin simple group</span ></a>span >div >div >div class="ContChap" ><a href="chap5.html#X84AF2F1D7D4E7284" >5 <span class="Heading" >Background</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7B02D6AE80303BEB" >5.1 <span class="Heading" >The Zassenhaus Conjecture and related questions</span span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7E2BFEC182B09895" >5.2 <span class="Heading" >Partial augmentations and the structure of HeLP sol</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X8663389F87B9CE62" >5.3 <span class="Heading" >The HeLP equations</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7BA77C9F86ADD546" >5.4 <span class="Heading" >The Wagner test</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X85810FF37EB3F4B4" >5.5 <span class="Heading" >s-constant characters</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X79BE759E7F35150E" >5.6 <span class="Heading" >Known results about the Zassenhaus Conjecture and the Prime Graph Question</span ></a>span >div >div >div class="ContChap" ><a href="chap6.html#X7B53A54C823744E9" >6 <span class="Heading" >Remarks on technical problems and the implementation</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X7FBC46BC7A5D28AF" >6.1 <span class="Heading" >Making the HeLP-package run</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X85FD09B67E460537" >6.2 <span class="Heading" >How much 4ti2 and normaliz is really there?</span ></a>span >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 >br />div >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap1.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1</a> <a href="chap2.html" >2</a> <a href="chap3.html" 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