products/Sources/formale Sprachen/GAP/pkg/congruence/doc/chap0.html
<?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 (Congruence) - 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="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 ><strong class="pkg" >Congruence</strong ></h1 >span class="SimpleMath" >SL_2(ℤ)</span ></h2>div >br />Email: <span class="URL" ><a href="mailto:andooms@vub.ac.be" >andooms@vub.ac.be</a></span >br />Homepage: <span class="URL" ><a href="http://homepages.vub.ac.be/~andooms " >http://homepages.vub.ac.be/~andooms</a></span >br />Address : <br />Department of Mathematics, Vrije Universiteit Brussel<br /> Pleinlaan 2, Brussels, B-1050 Belgiumbr />Email: <span class="URL" ><a href="mailto:efjesper@vub.ac.be" >efjesper@vub.ac.be</a></span >br />Homepage: <span class="URL" ><a href="http://homepages.vub.ac.be/~efjesper " >http://homepages.vub.ac.be/~efjesper</a></span >br />Address : <br />Department of Mathematics, Vrije Universiteit Brussel<br /> Pleinlaan 2, Brussels, B-1050 Belgiumbr />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://olexandr-konovalov.github.io/ " >https://olexandr-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, Scotlandbr />Email: <span class="URL" ><a href="mailto:verrill@math.lsu.edu" >verrill@math.lsu.edu</a></span >br />Homepage: <span class="URL" ><a href="http://www.math.lsu.edu/~verrill/ " >http://www.math.lsu.edu/~verrill/</a></span >br />Address : <br />Department of Mathematics<br /> Louisiana State University<br /> Baton Rouge, Louisiana, 70803-4918<br /> USA"X7AA6C5737B711C89" name="X7AA6C5737B711C89" ></a></p>strong class="pkg" >GAP</strong > package <strong class="pkg" >Congruence</strong > provides functionality to work with congruence subgroups of <span class="SimpleMath" >SL_2(ℤ)</span >.</p>"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>strong class="pkg" >Congruence</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 strong class="pkg" >Congruence</strong >, we would be grateful for a short notification sent to one of the authors.</p>strong class="pkg" >Congruence</strong >, please cite it in the following form :</p>em >Congruence --- Congruence subgroups of <span class="SimpleMath" >SL_2(ℤ)</span >, Version 1.2.7;</em > 2024 (<span class="URL" ><a href="https://gap-packages.github.io/congruence/ " >https://gap-packages.github.io/congruence/</a></span >).</p>"X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a></p>ded while implementing algorithms from <a href="chapBib.html#biBLLT-Algorithm" >[LLT95a]</a> and <a href="chapBib.html#biBLLT-Hecke" >[LLT95b]</a>, and to Francqui Stichting (Belgium) for the support of the third author.</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 class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X80AE633F82C4D9BF" >1.1 <span class="Heading" >General aims of <strong class="pkg" >Congruence</strong > package</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X7DB566D5785B7DBC" >1.2 <span class="Heading" >Installation and system requirements</span ></a>span >div >div >div class="ContChap" ><a href="chap2.html#X7B010EE67FACF45E" >2 <span class="Heading" >Construction of congruence subgroups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7B010EE67FACF45E" >2.1 <span class="Heading" >Construction of congruence subgroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7A61F693873F7136" >2.1-1 PrincipalCongruenceSubgroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7B8DB77B81BE58D7" >2.1-2 CongruenceSubgroupGamma0</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7B4FBED17ECE2A7F" >2.1-3 CongruenceSubgroupGammaUpper0</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7CFDC47279AC0E85" >2.1-4 CongruenceSubgroupGamma1</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7C3FCDD878FE57ED" >2.1-5 CongruenceSubgroupGammaUpper1</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7FE839377D7F45EB" >2.1-6 IntersectionOfCongruenceSubgroups</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X8267F261874959E5" >2.2 <span class="Heading" >Properties of congruence subgroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X828F7E08787650DC" >2.2-1 IsPrincipalCongruenceSubgroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X85124A697E826AB4" >2.2-2 IsCongruenceSubgroupGamma0</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7A03633C83A286F5" >2.2-3 IsCongruenceSubgroupGammaUpper0</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X8262396080F3B0DD" >2.2-4 IsCongruenceSubgroupGamma1</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7D731035834CF878" >2.2-5 IsCongruenceSubgroupGammaUpper1</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X83B4E4FA7F4DFB97" >2.2-6 IsIntersectionOfCongruenceSubgroups</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X8664A60E875EA5DE" >2.3 <span class="Heading" >Attributes of congruence subgroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7D5696F584970D21" >2.3-1 LevelOfCongruenceSubgroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X87302F8A7E44D67B" >2.3-2 IndexInSL2Z</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7BF57D157824FFC8" >2.3-3 DefiningCongruenceSubgroups</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7B15B49583DC9EF5" >2.4 <span class="Heading" >Operations for congruence subgroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X8146AC8587C65DEE" >2.4-1 Random</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X87BDB89B7AAFE8AD" ><code >2.4-2 \in</code ></a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X7FC5BF527931FF4C" >2.4-3 CanEasilyCompareCongruenceSubgroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X79CA175481F8105F" >2.4-4 IsSubset</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap2.html#X83A0356F839C696F" >2.4-5 Index</a></span >div ></div >div >div class="ContChap" ><a href="chap3.html#X85CABB30818CD99C" >3 <span class="Heading" >Farey symbols and their properties</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7B7B81E584CCA30C" >3.1 <span class="Heading" >Construction of Farey symbols</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7F8F5919870A46FE" >3.1-1 FareySymbolByData</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X845F9BA182F4E73B" >3.1-2 IsValidFareySymbol</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X8016C45082AEC784" >3.2 <span class="Heading" >Properties of Farey symbols</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X8245766978F02751" >3.2-1 GeneralizedFareySequence</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X80BB58E58492D103" >3.2-2 NumeratorOfGFSElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X87477604878BCD42" >3.2-3 DenominatorOfGFSElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X83C941047D486000" >3.2-4 LabelsOfFareySymbol</a></span >div ></div >div >div class="ContChap" ><a href="chap4.html#X831C60277F7D80B2" >4 <span class="Heading" >Farey symbols for congruence subgroups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7F43DB8B803F313F" >4.1 <span class="Heading" >Computation of the Farey symbol for a finite index subgroup</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8594896287DCFE8D" >4.1-1 FareySymbol</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X80AE179D869BEE90" >4.2 <span class="Heading" >Computation of generators of a finite index subgroup from its Farey symbol</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8790C1498107A39A" >4.2-1 MatrixByEvenInterval</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X78779BDF7A1DB4AE" >4.2-2 MatrixByOddInterval</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7F792846795E3A63" >4.2-3 MatrixByFreePairOfIntervals</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7905B050800E4416" >4.2-4 GeneratorsByFareySymbol</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X79C44528864044C5" >4.2-5 GeneratorsOfGroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7C5AB1D786207745" >4.3 <span class="Heading" >Other properties derived from Farey symbols</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X80EED34183408106" >4.3-1 IndexInPSL2ZByFareySymbol</a></span >div ></div >div >div class="ContChap" ><a href="chap5.html#X82C56A367A418E7C" >5 <span class="Heading" >Service functions of the <strong class="pkg" >Congruence</strong > package</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X86D04EE08437C320" >5.1 <span class="Heading" >Additional information displayed by <strong class="pkg" >Congruence</strong > algorithms</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X83B2A8607C2E6A38" >5.1-1 InfoCongruence</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 >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" >3</a> <a href="chap4.html" >4</a> <a href="chap5.html" >5</a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >hr />"foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " 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