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<?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 (Loops) - 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="chap7.html" >7</a> <a href="chap8.html" >8</a> <a href="chap9.html" >9</a> <a href="chapA.html" >A</a> <a href="chapB.html" >B</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 >The LOOPS Package</h1 >strong class="pkg" >GAP</strong ></h2>div >br />Email: <span class="URL" ><a href="mailto:nagyg@math.u-szeged.hu" >nagyg@math.u-szeged.hu</a></span >br />Homepage: <span class="URL" ><a href="http://www.math.u-szeged.hu/~nagyg/ " >http://www.math.u-szeged.hu/~nagyg/</a></span >br />Address : <br />Bolyai Institute, University of Szeged<br /> 6725 Szeged, Aradi vertanuk tere 1<br br />br />Email: <span class="URL" ><a href="mailto:petr@math.du.edu" >petr@math.du.edu</a></span >br />Homepage: <span class="URL" ><a href="http://www.math.du.edu/~petr/ " >http://www.math.du.edu/~petr/</a></span >br />Address : <br />Department of Mathematics, University of Denver<br /> 2280 S. Vine Street<br /> Denver, CO 80208<br /> USA<br />"X7AA6C5737B711C89" name="X7AA6C5737B711C89" ></a></p>that integrates standard notions of loop theory with libraries of loops and group-theoretical algorithms of GAP. The package also expands GAP toward nonassociative structures.</p>"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>strong class="pkg" >LOOPS</strong > package is free software; you can redistribute it and/or modify it under the terms of the <span class="URL" ><a href="https://www.fsf.org/licenses/gpl.html " >GNU General Public License</a></span > as published by the Free Software Foundation; either version 2 of the License, or (at your option ) any later version.</p>"X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a></p>tionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Aleš Drápal, Graham Ellis, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Olexandr Konovalov, Frank Lübeck, Jonathan D.H. Smith, David Stanovský and Glen Whitney.</p>ry and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of right Bruck loops of order 27, 81 was obtained jointly with Izabella Stuhl.</p>ed by the 2006 and 2016 University of Denver PROF grants and the Simons Foundation Collaboration Grant 210176.</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#X8360C04082558A12" >1.1 <span class="Heading" >Installation</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X7F4F8D6F7CD6B765" >1.2 <span class="Heading" >Documentation</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X801051CC86594630" >1.3 <span class="Heading" >Test Files</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X79342B4E7E55FD0F" >1.4 <span class="Heading" >Memory Management</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap1.html#X80D704CC7EBFDF7A" >1.5 <span class="Heading" >Feedback</span ></a>span >div >div >div class="ContChap" ><a href="chap2.html#X7EF1B6708069B0C7" >2 <span class="Heading" >Mathematical Background</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X80243DE5826583B8" >2.1 <span class="Heading" >Quasigroups and Loops</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7EC01B437CC2B2C9" >2.2 <span class="Heading" >Translations</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X83EDF04F7952143F" >2.3 <span class="Heading" >Subquasigroups and Subloops</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X869CBCE381E2C422" >2.4 <span class="Heading" >Nilpotence and Solvability</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X7E0849977869E53D" >2.5 <span class="Heading" >Associators and Commutators</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap2.html#X791066ED7DD9F254" >2.6 <span class="Heading" >Homomorphism and Homotopisms</span ></a>span >div >div >div class="ContChap" ><a href="chap3.html#X7A6DF65E826B8CFF" >3 <span class="Heading" >How the Package Works</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X86F02BBD87FEA1C6" >3.1 <span class="Heading" >Representing Quasigroups</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X807D76EF81B9D061" >3.2 <span class="Heading" >Conversions between magmas, quasigroups, loops and groups</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X87E49ED884FA6DC4" >3.3 <span class="Heading" >Calculating with Quasigroups</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7D75C7A6787AF72A" >3.4 <span class="Heading" >Naming, Viewing and Printing Quasigroups and their Elements</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7A7EB1B579273D07" >3.4-1 <span class="Heading" >SetQuasigroupElmName and SetLoopElmName</span ></a>span >div ></div >div >div class="ContChap" ><a href="chap4.html#X7AA4B9C0877550ED" >4 <span class="Heading" >Creating Quasigroups and Loops</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7DE8405B82BC36A9" >4.1 <span class="Heading" >About Cayley Tables</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7827BF877AA87246" >4.2 <span class="Heading" >Testing Cayley Tables</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X81179355869B9DFE" >4.2-1 <span class="Heading" >IsQuasigroupTable and IsQuasigroupCayleyTable</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7AAE48507A471069" >4.2-2 <span class="Heading" >IsLoopTable and IsLoopCayleyTable</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7BA749CA7DB4EA87" >4.3 <span class="Heading" >Canonical and Normalized Cayley Tables</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7971CCB87DAFF7B9" >4.3-1 CanonicalCayleyTable</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7B816D887F46E6B7" >4.3-2 CanonicalCopy</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X821A2F9E85FAD8BF" >4.3-3 NormalizedQuasigroupTable</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7C2372BB8739C5A2" >4.4 <span class="Heading" >Creating Quasigroups and Loops From Cayley Tables</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X860135BB85F2DB19" >4.4-1 <span class="Heading" >QuasigroupByCayleyTable and LoopByCayleyTable</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X849944F17E2B37F8" >4.5 <span class="Heading" >Creating Quasigroups and Loops from a File</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X81A1DB918057933E" >4.5-1 <span class="Heading" >QuasigroupFromFile and LoopFromFile</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X820E67F88319C38B" >4.6 <span class="Heading" >Creating Quasigroups and Loops From Sections</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7F94C8DD7E1A3470" >4.6-1 CayleyTableByPerms</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7EC1EB0D7B8382A1" >4.6-2 <span class="Heading" >QuasigroupByLeftSection and LoopByLeftSection</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X80B436ED7CC0749E" >4.6-3 <span class="Heading" >QuasigroupByRightSection and LoopByRightSection</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X85ABE99E84E5B0E8" >4.7 <span class="Heading" >Creating Quasigroups and Loops From Folders</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X83168E62861F70AB" >4.7-1 <span class="Heading" >QuasigroupByRightFolder and LoopByRightFolder</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X8759431780AC81A9" >4.8 <span class="Heading" >Creating Quasigroups and Loops By Nuclear Extensions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X784733C67AA6B2FA" >4.8-1 NuclearExtension</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X79AEE93E7E15B802" >4.8-2 LoopByExtension</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7AE29A1A7AA5C25A" >4.9 <span class="Heading" >Random Quasigroups and Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8271C0F5786B6FA9" >4.9-1 <span class="Heading" >RandomQuasigroup and RandomLoop</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X817132C887D3FD3A" >4.9-2 RandomNilpotentLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7BC2D8877A943D74" >4.10 <span class="Heading" >Conversions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X84575A4B78CC545E" >4.10-1 IntoQuasigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X79CEA57C850C7070" >4.10-2 PrincipalLoopIsotope</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7A59C36683118E5A" >4.10-3 IntoLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7B5C6C64831B866E" >4.10-4 IntoGroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X79B7327C79029086" >4.11 <span class="Heading" >Products of Quasigroups and Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X861BA02C7902A4F4" >4.11-1 DirectProduct</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7865FC8D7854C2E3" >4.12 <span class="Heading" >Opposite Quasigroups and Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X87B6AED47EE2BCD3" >4.12-1 <span class="Heading" >Opposite, OppositeQuasigroup and OppositeLoop</span ></a>span >div ></div >div >div class="ContChap" ><a href="chap5.html#X7B9F619279641FAA" >5 <span class="Heading" >Basic Methods And Attributes</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X8373A7348161DB23" >5.1 <span class="Heading" >Basic Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X79B130FC7906FB4C" >5.1-1 Elements</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X85457FA27DE7114D" >5.1-2 CayleyTable</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X8129A6877FFD804B" >5.1-3 One</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X858ADA3B7A684421" >5.1-4 Size</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7D44470C7DA59C1C" >5.1-5 Exponent</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X82F2CA4A848ABD2B" >5.2 <span class="Heading" >Basic Arithmetic Operations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7D5956967BCC1834" >5.2-1 <span class="Heading" >LeftDivision and RightDivision</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X804F67C8796A0EB3" >5.2-2 <span class="Heading" >LeftDivisionCayleyTable and RightDivisionCayleyTable</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X810850247ADB4EE9" >5.3 <span class="Heading" >Powers and Inverses</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X805781838020CF44" >5.3-1 <span class="Heading" >LeftInverse, RightInverse and Inverse</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7E0849977869E53D" >5.4 <span class="Heading" >Associators and Commutators</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X82B7448879B91F7B" >5.4-1 Associator</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X7D624A9587FB1FE5" >5.4-2 Commutator</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5.html#X7BD5B55C802805B4" >5.5 <span class="Heading" >Generators</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X83944A777D161D10" >5.5-1 <span class="Heading" >GeneratorsOfQuasigroup and GeneratorsOfLoop</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X82FD78AF7F80A0E2" >5.5-2 GeneratorsSmallest</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5.html#X814DBABC878D5232" >5.5-3 SmallGeneratingSet</a></span >div ></div >div >div class="ContChap" ><a href="chap6.html#X794A04C5854D352B" >6 <span class="Heading" >Methods Based on Permutation Groups</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X8731D818827C08F3" >6.1 <span class="Heading" >Parent of a Quasigroup</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7BC856CC7F116BB0" >6.1-1 Parent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X79975EC6783B4293" >6.1-2 Position</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X832295DE866E44EE" >6.1-3 PosInParent</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X83EDF04F7952143F" >6.2 <span class="Heading" >Subquasigroups and Subloops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7DD511FF864FCDFF" >6.2-1 Subquasigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X84E6744E804AE830" >6.2-2 Subloop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X87AC8B7E80CE9260" >6.2-3 <span class="Heading" >IsSubquasigroup and IsSubloop</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X859B6C8183537E75" >6.2-4 AllSubquasigroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X81EF252585592001" >6.2-5 AllSubloops</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X835F48248571364F" >6.2-6 RightCosets</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X85C65D06822E716F" >6.2-7 RightTransversal</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X78AA3D177CCA49FF" >6.3 <span class="Heading" >Translations and Sections</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7B45B48C7C4D6061" >6.3-1 <span class="Heading" >LeftTranslation and RightTranslation</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7EB9197C80FB4664" >6.3-2 <span class="Heading" >LeftSection and RightSection</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X78ED50F578A88046" >6.4 <span class="Heading" >Multiplication Groups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7AB8C9947C1303E2" >6.4-1 <span class="Heading" >LeftMultiplicationGroup, RightMultiplicationGroup and MultiplicationGroup</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X847256B779E1E7E5" >6.4-2 <span class="Heading" >RelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup and RelativeMultiplicationGroup</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X8740D61178ACD217" >6.5 <span class="Heading" >Inner Mapping Groups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7EE1E78C856C6F7C" >6.5-1 <span class="Heading" >LeftInnerMapping, RightInnerMapping, MiddleInnerMapping</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X79CDA09A7D48BF2B" >6.5-2 <span class="Heading" >LeftInnerMappingGroup, RightInnerMappingGroup, MiddleInnerMappingGroup</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X82513A3B7C3A6420" >6.5-3 InnerMappingGroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X7B45C2AF7C2E28AB" >6.6 <span class="Heading" >Nuclei, Commutant, Center , and Associator Subloop</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X798316F47A47FF63" >6.6-1 <span class="Heading" >LeftNucleus, MiddleNucleus, and RightNucleus</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X84D389677A91C290" >6.6-2 <span class="Heading" >Nuc, NucleusOfQuasigroup and NucleusOfLoop</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7C8428DE791F3CE1" >6.6-3 Commutant</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7C1FBE7A84DD4873" >6.6-4 Center </a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7F7FDE82780EDD7E" >6.6-5 AssociatorSubloop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X85B650D284FE39F3" >6.7 <span class="Heading" >Normal Subloops and Simple Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X838186F9836F678C" >6.7-1 IsNormal</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7BDEA0A98720D1BB" >6.7-2 NormalClosure</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7D8E63A7824037CC" >6.7-3 IsSimple</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X87F66DB383C29A4A" >6.8 <span class="Heading" >Factor Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X83E1953980E2DE2F" >6.8-1 FactorLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X870FCB497AECC730" >6.8-2 NaturalHomomorphismByNormalSubloop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X821F40748401D698" >6.9 <span class="Heading" >Nilpotency and Central Series</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X78A4B93781C96AAE" >6.9-1 IsNilpotent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7D5FC62581A99482" >6.9-2 NilpotencyClassOfLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7E7C2D117B55F6A0" >6.9-3 IsStronglyNilpotent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7ED37AA07BEE79E0" >6.9-4 UpperCentralSeries</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X817BDBC2812992ED" >6.9-5 LowerCentralSeries</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X83A38A6C7EDBCA63" >6.10 <span class="Heading" >Solvability, Derived Series and Frattini Subloop</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X79B10B337A3B1C6E" >6.10-1 IsSolvable</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7A82DC4680DAD67C" >6.10-2 DerivedSubloop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7A9AA1577CEC891F" >6.10-3 DerivedLength</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X85BD2C517FA7A47E" >6.10-4 <span class="Heading" >FrattiniSubloop and FrattinifactorSize</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X855286367A2D5A54" >6.10-5 FrattinifactorSize</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X81F3496578EAA74E" >6.11 <span class="Heading" >Isomorphisms and Automorphisms</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X801067F67E5292F7" >6.11-1 IsomorphismQuasigroups</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7D7B10D6836FCA9F" >6.11-2 IsomorphismLoops</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X82373C5479574F22" >6.11-3 QuasigroupsUpToIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X8308F38283C61B20" >6.11-4 LoopsUpToIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X87677B0787B4461A" >6.11-5 AutomorphismGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7A42812B7B027DD4" >6.11-6 QuasigroupIsomorph</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7BD1AC32851286EA" >6.11-7 LoopIsomorph</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X85B3E22679FD8D81" >6.11-8 IsomorphicCopyByPerm</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X8121DE3A78795040" >6.11-9 IsomorphicCopyByNormalSubloop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7D09D8957E4A0973" >6.11-10 Discriminator</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X812F0DEE7C896E18" >6.11-11 AreEqualDiscriminators</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X7E996BDD81E594F9" >6.12 <span class="Heading" >Isotopisms</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X84C5ADE77F910F63" >6.12-1 IsotopismLoops</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X841E540B7A7EF29F" >6.12-2 LoopsUpToIsotopism</a></span >div ></div >div >div class="ContChap" ><a href="chap7.html#X7910E575825C713E" >7 <span class="Heading" >Testing Properties of Quasigroups and Loops</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X7960E3FB7A7F0F00" >7.1 <span class="Heading" >Associativity, Commutativity and Generalizations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7C83B5A47FD18FB7" >7.1-1 IsAssociative</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X830A4A4C795FBC2D" >7.1-2 IsCommutative</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7D53EA947F1CDA69" >7.1-3 IsPowerAssociative</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X872DCA027E1A4A1D" >7.1-4 IsDiassociative</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X8748BA2187604B24" >7.2 <span class="Heading" >Inverse Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X85EDD10586596458" >7.2-1 <span class="Heading" >HasLeftInverseProperty, HasRightInverseProperty and HasInverseProperty</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X86B93E1B7AEA6EDA" >7.2-2 HasTwosidedInverses</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X793909B780761EA8" >7.2-3 HasWeakInverseProperty</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7F46CE6B7D387158" >7.2-4 HasAutomorphicInverseProperty</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8538D4638232DB51" >7.2-5 HasAntiautomorphicInverseProperty</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X7D8CB6DA828FD744" >7.3 <span class="Heading" >Some Properties of Quasigroups</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X834848ED85F9012B" >7.3-1 IsSemisymmetric</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X834F809B8060B754" >7.3-2 IsTotallySymmetric</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7CB5896082D29173" >7.3-3 IsIdempotent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X83DE7DD77C056C1F" >7.3-4 IsSteinerQuasigroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7CA3DCA07B6CB9BD" >7.3-5 IsUnipotent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7B76FD6E878ED4F1" >7.3-6 <span class="Heading" >IsLeftDistributive, IsRightDistributive, IsDistributive</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7F23D4D97A38D223" >7.3-7 <span class="Heading" >IsEntropic and IsMedial</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X780D907986EBA6C7" >7.4 <span class="Heading" >Loops of Bol Moufang Type</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7988AFE27D06ACB5" >7.4-1 IsExtraLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7F1C151484C97E61" >7.4-2 IsMoufangLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X866F04DC7AE54B7C" >7.4-3 IsCLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X801DAAE8834A1A65" >7.4-4 IsLeftBolLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X79279F9787E72566" >7.4-5 IsRightBolLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X789E0A6979697C4C" >7.4-6 IsLCLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7B03CC577802F4AB" >7.4-7 IsRCLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X819F285887B5EB9E" >7.4-8 IsLeftNuclearSquareLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8474F55681244A8A" >7.4-9 IsMiddleNuclearSquareLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X807B3B21825E3076" >7.4-10 IsRightNuclearSquareLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X796650088213229B" >7.4-11 IsNuclearSquareLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7C32851A7AF1C45F" >7.4-12 IsFlexible</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7DF0196786B9CE08" >7.4-13 IsLeftAlternative</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8416FAD87F148F5D" >7.4-14 IsRightAlternative</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8379356E82DB5DDA" >7.4-15 IsAlternative</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X83A501387E1AC371" >7.5 <span class="Heading" >Power Alternative Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X875C3DF681B3FAE2" >7.5-1 <span class="Heading" >IsLeftPowerAlternative, IsRightPowerAlternative and IsPowerAlternative</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X8176B2C47A4629CD" >7.6 <span class="Heading" >Conjugacy Closed Loops and Related Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X784E08CD7B710AF4" >7.6-1 IsLCCLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7B3016B47A1A8213" >7.6-2 IsRCCLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X878B614479DCB83F" >7.6-3 IsCCLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X8655956878205FC1" >7.6-4 IsOsbornLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X793B22EA8643C667" >7.7 <span class="Heading" >Automorphic Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7F063914804659F1" >7.7-1 IsLeftAutomorphicLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7DFE830584A769E5" >7.7-2 IsMiddleAutomorphicLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7EA9165A87F99E35" >7.7-3 IsRightAutomorphicLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X7899603184CF13FD" >7.7-4 IsAutomorphicLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X878C9D247FB0D56E" >7.8 <span class="Heading" >Additional Varieties of Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X790FA1188087D5C1" >7.8-1 IsCodeLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X793600C9801F4F62" >7.8-2 IsSteinerLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X85F1BD4280E44F5B" >7.8-3 <span class="Heading" >IsLeftBruckLoop and IsLeftKLoop</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X857B373E7B4E0519" >7.8-4 <span class="Heading" >IsRightBruckLoop and IsRightKLoop</span ></a>span >div ></div >div >div class="ContChap" ><a href="chap8.html#X85AFC9C47FD3C03F" >8 <span class="Heading" >Specific Methods</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7990F2F880E717EE" >8.1 <span class="Heading" >Core Methods for Bol Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X8664CA927DD73DBE" >8.1-1 <span class="Heading" >AssociatedLeftBruckLoop and AssociatedRightBruckLoop</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X82FC16F386CE11F1" >8.1-2 IsExactGroupFactorization</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7DCA64807F899127" >8.1-3 RightBolLoopByExactGroupFactorization</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X819F82737C2A860D" >8.2 <span class="Heading" >Moufang Modifications</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7B3165C083709831" >8.2-1 LoopByCyclicModification</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7D7717C587BC2D1E" >8.2-2 LoopByDihedralModification</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7CC6CDB786E9BBA0" >8.2-3 LoopMG2</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X83E73A767D79FAFD" >8.3 <span class="Heading" >Triality for Moufang Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7DB4DE647F6F56F0" >8.3-1 TrialityPermGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X82CC977085DFDFE8" >8.3-2 TrialityPcGroup</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X841ED66B8084AA73" >8.4 <span class="Heading" >Realizing Groups as Multiplication Groups of Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X804F40087DD1225D" >8.4-1 AllLoopTablesInGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7854C8E382DC8E8B" >8.4-2 AllProperLoopTablesInGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7BFFC66A824BA6AA" >8.4-3 OneLoopTableInGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X84C5A76585B335FF" >8.4-4 OneProperLoopTableInGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7E5F1C2879358EEF" >8.4-5 AllLoopsWithMltGroup</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X8266DE05824226E6" >8.4-6 OneLoopWithMltGroup</a></span >div ></div >div >div class="ContChap" ><a href="chap9.html#X7BF3EE6E7953560D" >9 <span class="Heading" >Libraries of Loops</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X874DFEAA79B3377C" >9.1 <span class="Heading" >A Typical Library</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X849865D6786EEF9B" >9.1-1 LibraryLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X78C4B8757902D49F" >9.1-2 MyLibraryLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7A64372E81E713B4" >9.1-3 DisplayLibraryInfo</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7DF21BD685FBF258" >9.2 <span class="Heading" >Left Bol Loops and Right Bol Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7EE99F647C537994" >9.2-1 LeftBolLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8774304282654C58" >9.2-2 RightBolLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X8028D69A86B15897" >9.3 <span class="Heading" >Left Bruck Loops and Right Bruck Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8290B01780F0FCD3" >9.3-1 LeftBruckLoop</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X798DD7CF871F648F" >9.3-2 RightBruckLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7953702D84E60AF4" >9.4 <span class="Heading" >Moufang Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X81E82098822543EE" >9.4-1 MoufangLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7BCA6BCB847F79DC" >9.5 <span class="Heading" >Code Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7DB4D3B27BB4D7EE" >9.5-1 CodeLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X84E941EE7846D3EE" >9.6 <span class="Heading" >Steiner Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X87C235457E859AF4" >9.6-1 SteinerLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X867E5F0783FEB8B5" >9.7 <span class="Heading" >Conjugacy Closed Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X806B2DE67990E42F" >9.7-1 <span class="Heading" >RCCLoop and RightConjugacyClosedLoop</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X80AB8B107D55FB19" >9.7-2 <span class="Heading" >LCCLoop and LeftConjugacyClosedLoop</span ></a>span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X798BC601843E8916" >9.7-3 <span class="Heading" >CCLoop and ConjugacyClosedLoop</span ></a>span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7E3A8F2C790F2CA1" >9.8 <span class="Heading" >Small Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7C6EE23E84CD87D3" >9.8-1 SmallLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X8135C8FD8714C606" >9.9 <span class="Heading" >Paige Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7FCF4D6B7AD66D74" >9.9-1 PaigeLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X86695C577A4D1784" >9.10 <span class="Heading" >Nilpotent Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7A9C960D86E2AD28" >9.10-1 NilpotentLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X793B22EA8643C667" >9.11 <span class="Heading" >Automorphic Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X784FFA9E7FDA9F43" >9.11-1 AutomorphicLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X843BD73F788049F7" >9.12 <span class="Heading" >Interesting Loops</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X87F24AD3811910D3" >9.12-1 InterestingLoop</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X864839227D5C0A90" >9.13 <span class="Heading" >Libraries of Loops Up To Isotopism</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X850C4C01817A098D" >9.13-1 ItpSmallLoop</a></span >div ></div >div >div class="ContChap" ><a href="chapA.html#X7BC4571A79FFB7D0" >A <span class="Heading" >Files</span 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