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551 lines
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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chapA_mj.html">A</a> <a href="chapB_mj.html">B</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap1_mj.html">[Next Chapter]</a> </div>
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<h1>The <strong class="pkg">LOOPS</strong> Package</h1>
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<h2>Computing with quasigroups and loops in <strong class="pkg">GAP</strong></h2>
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<p>Version 3.3.0</p>
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</div>
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<p><b>Gábor P. Nagy
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</b>
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<br />Email: <span class="URL"><a href="mailto:nagyg@math.u-szeged.hu">nagyg@math.u-szeged.hu</a></span>
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<br />Address: <br />Department of Mathematics, University of Szeged
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</p><p><b>Petr Vojtěchovský
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</b>
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<br />Email: <span class="URL"><a href="mailto:petr@math.du.edu">petr@math.du.edu</a></span>
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<br />Address: <br />Department of Mathematics, University of Denver
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</p>
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<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
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<h3>Copyright</h3>
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<p>© 2016 Gábor P. Nagy and Petr Vojtěchovský.</p>
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<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
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<div class="contents">
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<h3>Contents<a id="contents" name="contents"></a></h3>
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<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X861E5DF986F89AE2">1.1 <span class="Heading">License</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X8360C04082558A12">1.2 <span class="Heading">Installation</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7F4F8D6F7CD6B765">1.3 <span class="Heading">Documentation</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X801051CC86594630">1.4 <span class="Heading">Test Files</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X79342B4E7E55FD0F">1.5 <span class="Heading">Memory Management</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X80D704CC7EBFDF7A">1.6 <span class="Heading">Feedback</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X811B08C07BD79486">1.7 <span class="Heading">Acknowledgment</span></a>
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</span>
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</div>
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</div>
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<div class="ContChap"><a href="chap2_mj.html#X7EF1B6708069B0C7">2 <span class="Heading">Mathematical Background</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X80243DE5826583B8">2.1 <span class="Heading">Quasigroups and Loops</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2 <span class="Heading">Translations</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X83EDF04F7952143F">2.3 <span class="Heading">Subquasigroups and Subloops</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X869CBCE381E2C422">2.4 <span class="Heading">Nilpotence and Solvability</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7E0849977869E53D">2.5 <span class="Heading">Associators and Commutators</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X791066ED7DD9F254">2.6 <span class="Heading">Homomorphism and Homotopisms</span></a>
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</div>
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</div>
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<div class="ContChap"><a href="chap3_mj.html#X7A6DF65E826B8CFF">3 <span class="Heading">How the Package Works</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X86F02BBD87FEA1C6">3.1 <span class="Heading">Representing Quasigroups</span></a>
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</span>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X807D76EF81B9D061">3.2 <span class="Heading">Conversions between magmas, quasigroups, loops and groups</span></a>
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</div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X87E49ED884FA6DC4">3.3 <span class="Heading">Calculating with Quasigroups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7D75C7A6787AF72A">3.4 <span class="Heading">Naming, Viewing and Printing Quasigroups and their Elements</span></a>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7A7EB1B579273D07">3.4-1 <span class="Heading">SetQuasigroupElmName and SetLoopElmName</span></a>
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</span>
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</div></div>
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</div>
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<div class="ContChap"><a href="chap4_mj.html#X7AA4B9C0877550ED">4 <span class="Heading">Creating Quasigroups and Loops</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7DE8405B82BC36A9">4.1 <span class="Heading">About Cayley Tables</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7827BF877AA87246">4.2 <span class="Heading">Testing Cayley Tables</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X81179355869B9DFE">4.2-1 <span class="Heading">IsQuasigroupTable and IsQuasigroupCayleyTable</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7AAE48507A471069">4.2-2 <span class="Heading">IsLoopTable and IsLoopCayleyTable</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7BA749CA7DB4EA87">4.3 <span class="Heading">Canonical and Normalized Cayley Tables</span></a>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7971CCB87DAFF7B9">4.3-1 CanonicalCayleyTable</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B816D887F46E6B7">4.3-2 CanonicalCopy</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X821A2F9E85FAD8BF">4.3-3 NormalizedQuasigroupTable</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7C2372BB8739C5A2">4.4 <span class="Heading">Creating Quasigroups and Loops From Cayley Tables</span></a>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X860135BB85F2DB19">4.4-1 <span class="Heading">QuasigroupByCayleyTable and LoopByCayleyTable</span></a>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X849944F17E2B37F8">4.5 <span class="Heading">Creating Quasigroups and Loops from a File</span></a>
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</span>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X81A1DB918057933E">4.5-1 <span class="Heading">QuasigroupFromFile and LoopFromFile</span></a>
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</span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X820E67F88319C38B">4.6 <span class="Heading">Creating Quasigroups and Loops From Sections</span></a>
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</span>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7F94C8DD7E1A3470">4.6-1 CayleyTableByPerms</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7EC1EB0D7B8382A1">4.6-2 <span class="Heading">QuasigroupByLeftSection and LoopByLeftSection</span></a>
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</span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X80B436ED7CC0749E">4.6-3 <span class="Heading">QuasigroupByRightSection and LoopByRightSection</span></a>
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</span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X85ABE99E84E5B0E8">4.7 <span class="Heading">Creating Quasigroups and Loops From Folders</span></a>
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</span>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83168E62861F70AB">4.7-1 <span class="Heading">QuasigroupByRightFolder and LoopByRightFolder</span></a>
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</span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8759431780AC81A9">4.8 <span class="Heading">Creating Quasigroups and Loops By Nuclear Extensions</span></a>
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</span>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X784733C67AA6B2FA">4.8-1 NuclearExtension</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X79AEE93E7E15B802">4.8-2 LoopByExtension</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7AE29A1A7AA5C25A">4.9 <span class="Heading">Random Quasigroups and Loops</span></a>
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</span>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X8271C0F5786B6FA9">4.9-1 <span class="Heading">RandomQuasigroup and RandomLoop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X817132C887D3FD3A">4.9-2 RandomNilpotentLoop</a></span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7BC2D8877A943D74">4.10 <span class="Heading">Conversions</span></a>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X84575A4B78CC545E">4.10-1 IntoQuasigroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X79CEA57C850C7070">4.10-2 PrincipalLoopIsotope</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7A59C36683118E5A">4.10-3 IntoLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B5C6C64831B866E">4.10-4 IntoGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X79B7327C79029086">4.11 <span class="Heading">Products of Quasigroups and Loops</span></a>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X861BA02C7902A4F4">4.11-1 DirectProduct</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7865FC8D7854C2E3">4.12 <span class="Heading">Opposite Quasigroups and Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X87B6AED47EE2BCD3">4.12-1 <span class="Heading">Opposite, OppositeQuasigroup and OppositeLoop</span></a>
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<div class="ContChap"><a href="chap5_mj.html#X7B9F619279641FAA">5 <span class="Heading">Basic Methods And Attributes</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8373A7348161DB23">5.1 <span class="Heading">Basic Attributes</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79B130FC7906FB4C">5.1-1 Elements</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X85457FA27DE7114D">5.1-2 CayleyTable</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8129A6877FFD804B">5.1-3 One</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X858ADA3B7A684421">5.1-4 Size</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7D44470C7DA59C1C">5.1-5 Exponent</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X82F2CA4A848ABD2B">5.2 <span class="Heading">Basic Arithmetic Operations</span></a>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7D5956967BCC1834">5.2-1 <span class="Heading">LeftDivision and RightDivision</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X804F67C8796A0EB3">5.2-2 <span class="Heading">LeftDivisionCayleyTable and RightDivisionCayleyTable</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X810850247ADB4EE9">5.3 <span class="Heading">Powers and Inverses</span></a>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X805781838020CF44">5.3-1 <span class="Heading">LeftInverse, RightInverse and Inverse</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7E0849977869E53D">5.4 <span class="Heading">Associators and Commutators</span></a>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X82B7448879B91F7B">5.4-1 Associator</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7D624A9587FB1FE5">5.4-2 Commutator</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7BD5B55C802805B4">5.5 <span class="Heading">Generators</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X83944A777D161D10">5.5-1 <span class="Heading">GeneratorsOfQuasigroup and GeneratorsOfLoop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X82FD78AF7F80A0E2">5.5-2 GeneratorsSmallest</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X814DBABC878D5232">5.5-3 SmallGeneratingSet</a></span>
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<div class="ContChap"><a href="chap6_mj.html#X794A04C5854D352B">6 <span class="Heading">Methods Based on Permutation Groups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X8731D818827C08F3">6.1 <span class="Heading">Parent of a Quasigroup</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7BC856CC7F116BB0">6.1-1 Parent</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X79975EC6783B4293">6.1-2 Position</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X832295DE866E44EE">6.1-3 PosInParent</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X83EDF04F7952143F">6.2 <span class="Heading">Subquasigroups and Subloops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7DD511FF864FCDFF">6.2-1 Subquasigroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X84E6744E804AE830">6.2-2 Subloop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87AC8B7E80CE9260">6.2-3 <span class="Heading">IsSubquasigroup and IsSubloop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X859B6C8183537E75">6.2-4 AllSubquasigroups</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X81EF252585592001">6.2-5 AllSubloops</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X835F48248571364F">6.2-6 RightCosets</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85C65D06822E716F">6.2-7 RightTransversal</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X78AA3D177CCA49FF">6.3 <span class="Heading">Translations and Sections</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7B45B48C7C4D6061">6.3-1 <span class="Heading">LeftTranslation and RightTranslation</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7EB9197C80FB4664">6.3-2 <span class="Heading">LeftSection and RightSection</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X78ED50F578A88046">6.4 <span class="Heading">Multiplication Groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87302BE983A5FC61">6.4-1 <span class="Heading">LeftMutliplicationGroup, RightMultiplicationGroup and MultiplicationGroup</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X847256B779E1E7E5">6.4-2 <span class="Heading">RelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup and RelativeMultiplicationGroup</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X8740D61178ACD217">6.5 <span class="Heading">Inner Mapping Groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7EE1E78C856C6F7C">6.5-1 <span class="Heading">LeftInnerMapping, RightInnerMapping, MiddleInnerMapping</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X79CDA09A7D48BF2B">6.5-2 <span class="Heading">LeftInnerMappingGroup, RightInnerMappingGroup, MiddleInnerMappingGroup</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X82513A3B7C3A6420">6.5-3 InnerMappingGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7B45C2AF7C2E28AB">6.6 <span class="Heading">Nuclei, Commutant, Center, and Associator Subloop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7DF536FC85BBD1D2">6.6-1 <span class="Heading">LeftNucles, MiddleNucleus, and RightNucleus</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X84D389677A91C290">6.6-2 <span class="Heading">Nuc, NucleusOfQuasigroup and NucleusOfLoop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7C8428DE791F3CE1">6.6-3 Commutant</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7C1FBE7A84DD4873">6.6-4 Center</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7F7FDE82780EDD7E">6.6-5 AssociatorSubloop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X85B650D284FE39F3">6.7 <span class="Heading">Normal Subloops and Simple Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X838186F9836F678C">6.7-1 IsNormal</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7BDEA0A98720D1BB">6.7-2 NormalClosure</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D8E63A7824037CC">6.7-3 IsSimple</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X87F66DB383C29A4A">6.8 <span class="Heading">Factor Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X83E1953980E2DE2F">6.8-1 FactorLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X870FCB497AECC730">6.8-2 NaturalHomomorphismByNormalSubloop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X821F40748401D698">6.9 <span class="Heading">Nilpotency and Central Series</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X78A4B93781C96AAE">6.9-1 IsNilpotent</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D5FC62581A99482">6.9-2 NilpotencyClassOfLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7E7C2D117B55F6A0">6.9-3 IsStronglyNilpotent</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7ED37AA07BEE79E0">6.9-4 UpperCentralSeries</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X817BDBC2812992ED">6.9-5 LowerCentralSeries</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X83A38A6C7EDBCA63">6.10 <span class="Heading">Solvability, Derived Series and Frattini Subloop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X79B10B337A3B1C6E">6.10-1 IsSolvable</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7A82DC4680DAD67C">6.10-2 DerivedSubloop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7A9AA1577CEC891F">6.10-3 DerivedLength</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85BD2C517FA7A47E">6.10-4 <span class="Heading">FrattiniSubloop and FrattinifactorSize</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X855286367A2D5A54">6.10-5 FrattinifactorSize</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X81F3496578EAA74E">6.11 <span class="Heading">Isomorphisms and Automorphisms</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X801067F67E5292F7">6.11-1 IsomorphismQuasigroups</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D7B10D6836FCA9F">6.11-2 IsomorphismLoops</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8121DE3A78795040">6.11-7 IsomorphicCopyByNormalSubloop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8 Discriminator</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9 AreEqualDiscriminators</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X84C5ADE77F910F63">6.12-1 IsotopismLoops</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X841E540B7A7EF29F">6.12-2 LoopsUpToIsotopism</a></span>
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<div class="ContChap"><a href="chap7_mj.html#X7910E575825C713E">7 <span class="Heading">Testing Properties of Quasigroups and Loops</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7960E3FB7A7F0F00">7.1 <span class="Heading">Associativity, Commutativity and Generalizations</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7C83B5A47FD18FB7">7.1-1 IsAssociative</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X830A4A4C795FBC2D">7.1-2 IsCommutative</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7D53EA947F1CDA69">7.1-3 IsPowerAssociative</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X872DCA027E1A4A1D">7.1-4 IsDiassociative</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X853841C5820BFEA4">7.2 <span class="Heading">Inverse Propeties</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X85EDD10586596458">7.2-1 <span class="Heading">HasLeftInverseProperty, HasRightInverseProperty and HasInverseProperty</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86B93E1B7AEA6EDA">7.2-2 HasTwosidedInverses</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X793909B780761EA8">7.2-3 HasWeakInverseProperty</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F46CE6B7D387158">7.2-4 HasAutomorphicInverseProperty</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8538D4638232DB51">7.2-5 HasAntiautomorphicInverseProperty</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7D8CB6DA828FD744">7.3 <span class="Heading">Some Properties of Quasigroups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X834848ED85F9012B">7.3-1 IsSemisymmetric</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X834F809B8060B754">7.3-2 IsTotallySymmetric</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7CB5896082D29173">7.3-3 IsIdempotent</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X83DE7DD77C056C1F">7.3-4 IsSteinerQuasigroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7CA3DCA07B6CB9BD">7.3-5 IsUnipotent</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7B76FD6E878ED4F1">7.3-6 <span class="Heading">IsLeftDistributive, IsRightDistributive, IsDistributive</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F23D4D97A38D223">7.3-7 <span class="Heading">IsEntropic and IsMedial</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X780D907986EBA6C7">7.4 <span class="Heading">Loops of Bol Moufang Type</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7988AFE27D06ACB5">7.4-1 IsExtraLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F1C151484C97E61">7.4-2 IsMoufangLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X866F04DC7AE54B7C">7.4-3 IsCLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X801DAAE8834A1A65">7.4-4 IsLeftBolLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X79279F9787E72566">7.4-5 IsRightBolLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X789E0A6979697C4C">7.4-6 IsLCLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7B03CC577802F4AB">7.4-7 IsRCLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X819F285887B5EB9E">7.4-8 IsLeftNuclearSquareLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8474F55681244A8A">7.4-9 IsMiddleNuclearSquareLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X807B3B21825E3076">7.4-10 IsRightNuclearSquareLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X796650088213229B">7.4-11 IsNuclearSquareLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7C32851A7AF1C45F">7.4-12 IsFlexible</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X8379356E82DB5DDA">7.4-15 IsAlternative</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X83A501387E1AC371">7.5 <span class="Heading">Power Alternative Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X875C3DF681B3FAE2">7.5-1 <span class="Heading">IsLeftPowerAlternative, IsRightPowerAlternative and IsPowerAlternative</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X8176B2C47A4629CD">7.6 <span class="Heading">Conjugacy Closed Loops and Related Properties</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X784E08CD7B710AF4">7.6-1 IsLCCLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7B3016B47A1A8213">7.6-2 IsRCCLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X793B22EA8643C667">7.7 <span class="Heading">Automorphic Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7F063914804659F1">7.7-1 IsLeftAutomorphicLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X846F363879BAB349">7.8 <span class="Heading">Additonal Varieties of Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X790FA1188087D5C1">7.8-1 IsCodeLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X85F1BD4280E44F5B">7.8-3 <span class="Heading">IsLeftBruckLoop and IsLeftKLoop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X857B373E7B4E0519">7.8-4 <span class="Heading">IsRightBruckLoop and IsRightKLoop</span></a>
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<div class="ContChap"><a href="chap8_mj.html#X85AFC9C47FD3C03F">8 <span class="Heading">Specific Methods</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7990F2F880E717EE">8.1 <span class="Heading">Core Methods for Bol Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1 <span class="Heading">AssociatedLeftBruckLoop and AssociatedRightBruckLoop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X82FC16F386CE11F1">8.1-2 IsExactGroupFactorization</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7DCA64807F899127">8.1-3 RightBolLoopByExactGroupFactorization</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X819F82737C2A860D">8.2 <span class="Heading">Moufang Modifications</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7B3165C083709831">8.2-1 LoopByCyclicModification</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7D7717C587BC2D1E">8.2-2 LoopByDihedralModification</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3 LoopMG2</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X83E73A767D79FAFD">8.3 <span class="Heading">Triality for Moufang Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7DB4DE647F6F56F0">8.3-1 TrialityPermGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X82CC977085DFDFE8">8.3-2 TrialityPcGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X841ED66B8084AA73">8.4 <span class="Heading">Realizing Groups as Multiplication Groups of Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X804F40087DD1225D">8.4-1 AllLoopTablesInGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7854C8E382DC8E8B">8.4-2 AllProperLoopTablesInGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7BFFC66A824BA6AA">8.4-3 OneLoopTableInGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X84C5A76585B335FF">8.4-4 OneProperLoopTableInGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7E5F1C2879358EEF">8.4-5 AllLoopsWithMltGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8266DE05824226E6">8.4-6 OneLoopWithMltGroup</a></span>
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<div class="ContChap"><a href="chap9_mj.html#X7BF3EE6E7953560D">9 <span class="Heading">Libraries of Loops</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X874DFEAA79B3377C">9.1 <span class="Heading">A Typical Library</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X849865D6786EEF9B">9.1-1 LibraryLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X78C4B8757902D49F">9.1-2 MyLibraryLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7A64372E81E713B4">9.1-3 DisplayLibraryInfo</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7DF21BD685FBF258">9.2 <span class="Heading">Left Bol Loops and Right Bol Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7953702D84E60AF4">9.3 <span class="Heading">Moufang Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X81E82098822543EE">9.3-1 MoufangLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.4 <span class="Heading">Code Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1 CodeLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X84E941EE7846D3EE">9.5 <span class="Heading">Steiner Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X87C235457E859AF4">9.5-1 SteinerLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.6 <span class="Heading">Conjugacy Closed Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X806B2DE67990E42F">9.6-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X798BC601843E8916">9.6-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.7 <span class="Heading">Small Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1 SmallLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X8135C8FD8714C606">9.8 <span class="Heading">Paige Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1 PaigeLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X86695C577A4D1784">9.9 <span class="Heading">Nilpotent Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1 NilpotentLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X793B22EA8643C667">9.10 <span class="Heading">Automorphic Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1 AutomorphicLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X843BD73F788049F7">9.11 <span class="Heading">Interesting Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X87F24AD3811910D3">9.11-1 InterestingLoop</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X864839227D5C0A90">9.12 <span class="Heading">Libraries of Loops Up To Isotopism</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X850C4C01817A098D">9.12-1 ItpSmallLoop</a></span>
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<div class="ContChap"><a href="chapA_mj.html#X7BC4571A79FFB7D0">A <span class="Heading">Files</span></a>
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<div class="ContChap"><a href="chapB_mj.html#X84EFA4C07D4277BB">B <span class="Heading">Filters</span></a>
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<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
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