<divclass="chlinkprevnexttop"> <ahref="chap0_mj.html">[Top of Book]</a> <ahref="chap0_mj.html#contents">[Contents]</a> <ahref="chap1_mj.html">[Next Chapter]</a> </div>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap2_mj.html#X80243DE5826583B8">2.1 <spanclass="Heading">Quasigroups and Loops</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap2_mj.html#X83EDF04F7952143F">2.3 <spanclass="Heading">Subquasigroups and Subloops</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap2_mj.html#X869CBCE381E2C422">2.4 <spanclass="Heading">Nilpotence and Solvability</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap2_mj.html#X7E0849977869E53D">2.5 <spanclass="Heading">Associators and Commutators</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap2_mj.html#X791066ED7DD9F254">2.6 <spanclass="Heading">Homomorphism and Homotopisms</span></a>
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<divclass="ContChap"><ahref="chap3_mj.html#X7A6DF65E826B8CFF">3 <spanclass="Heading">How the Package Works</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap3_mj.html#X807D76EF81B9D061">3.2 <spanclass="Heading">Conversions between magmas, quasigroups, loops and groups</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap3_mj.html#X87E49ED884FA6DC4">3.3 <spanclass="Heading">Calculating with Quasigroups</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap3_mj.html#X7D75C7A6787AF72A">3.4 <spanclass="Heading">Naming, Viewing and Printing Quasigroups and their Elements</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap3_mj.html#X7A7EB1B579273D07">3.4-1 <spanclass="Heading">SetQuasigroupElmName and SetLoopElmName</span></a>
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<divclass="ContChap"><ahref="chap4_mj.html#X7AA4B9C0877550ED">4 <spanclass="Heading">Creating Quasigroups and Loops</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X81179355869B9DFE">4.2-1 <spanclass="Heading">IsQuasigroupTable and IsQuasigroupCayleyTable</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X7AAE48507A471069">4.2-2 <spanclass="Heading">IsLoopTable and IsLoopCayleyTable</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X7BA749CA7DB4EA87">4.3 <spanclass="Heading">Canonical and Normalized Cayley Tables</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X7C2372BB8739C5A2">4.4 <spanclass="Heading">Creating Quasigroups and Loops From Cayley Tables</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X860135BB85F2DB19">4.4-1 <spanclass="Heading">QuasigroupByCayleyTable and LoopByCayleyTable</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X849944F17E2B37F8">4.5 <spanclass="Heading">Creating Quasigroups and Loops from a File</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X81A1DB918057933E">4.5-1 <spanclass="Heading">QuasigroupFromFile and LoopFromFile</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X820E67F88319C38B">4.6 <spanclass="Heading">Creating Quasigroups and Loops From Sections</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X7EC1EB0D7B8382A1">4.6-2 <spanclass="Heading">QuasigroupByLeftSection and LoopByLeftSection</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X80B436ED7CC0749E">4.6-3 <spanclass="Heading">QuasigroupByRightSection and LoopByRightSection</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X85ABE99E84E5B0E8">4.7 <spanclass="Heading">Creating Quasigroups and Loops From Folders</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X83168E62861F70AB">4.7-1 <spanclass="Heading">QuasigroupByRightFolder and LoopByRightFolder</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X8759431780AC81A9">4.8 <spanclass="Heading">Creating Quasigroups and Loops By Nuclear Extensions</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X7AE29A1A7AA5C25A">4.9 <spanclass="Heading">Random Quasigroups and Loops</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X8271C0F5786B6FA9">4.9-1 <spanclass="Heading">RandomQuasigroup and RandomLoop</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X79B7327C79029086">4.11 <spanclass="Heading">Products of Quasigroups and Loops</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap4_mj.html#X7865FC8D7854C2E3">4.12 <spanclass="Heading">Opposite Quasigroups and Loops</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap4_mj.html#X87B6AED47EE2BCD3">4.12-1 <spanclass="Heading">Opposite, OppositeQuasigroup and OppositeLoop</span></a>
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<divclass="ContChap"><ahref="chap5_mj.html#X7B9F619279641FAA">5 <spanclass="Heading">Basic Methods And Attributes</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap5_mj.html#X7D5956967BCC1834">5.2-1 <spanclass="Heading">LeftDivision and RightDivision</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap5_mj.html#X804F67C8796A0EB3">5.2-2 <spanclass="Heading">LeftDivisionCayleyTable and RightDivisionCayleyTable</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap5_mj.html#X810850247ADB4EE9">5.3 <spanclass="Heading">Powers and Inverses</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap5_mj.html#X805781838020CF44">5.3-1 <spanclass="Heading">LeftInverse, RightInverse and Inverse</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap5_mj.html#X7E0849977869E53D">5.4 <spanclass="Heading">Associators and Commutators</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap5_mj.html#X83944A777D161D10">5.5-1 <spanclass="Heading">GeneratorsOfQuasigroup and GeneratorsOfLoop</span></a>
<divclass="ContChap"><ahref="chap6_mj.html#X794A04C5854D352B">6 <spanclass="Heading">Methods Based on Permutation Groups</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap6_mj.html#X8731D818827C08F3">6.1 <spanclass="Heading">Parent of a Quasigroup</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap6_mj.html#X83EDF04F7952143F">6.2 <spanclass="Heading">Subquasigroups and Subloops</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap6_mj.html#X87AC8B7E80CE9260">6.2-3 <spanclass="Heading">IsSubquasigroup and IsSubloop</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap6_mj.html#X78AA3D177CCA49FF">6.3 <spanclass="Heading">Translations and Sections</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap6_mj.html#X7B45B48C7C4D6061">6.3-1 <spanclass="Heading">LeftTranslation and RightTranslation</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap6_mj.html#X7EB9197C80FB4664">6.3-2 <spanclass="Heading">LeftSection and RightSection</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap6_mj.html#X87302BE983A5FC61">6.4-1 <spanclass="Heading">LeftMutliplicationGroup, RightMultiplicationGroup and MultiplicationGroup</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap6_mj.html#X847256B779E1E7E5">6.4-2 <spanclass="Heading">RelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup and RelativeMultiplicationGroup</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap6_mj.html#X7B45C2AF7C2E28AB">6.6 <spanclass="Heading">Nuclei, Commutant, Center, and Associator Subloop</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap6_mj.html#X7DF536FC85BBD1D2">6.6-1 <spanclass="Heading">LeftNucles, MiddleNucleus, and RightNucleus</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap6_mj.html#X84D389677A91C290">6.6-2 <spanclass="Heading">Nuc, NucleusOfQuasigroup and NucleusOfLoop</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap6_mj.html#X821F40748401D698">6.9 <spanclass="Heading">Nilpotency and Central Series</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap6_mj.html#X83A38A6C7EDBCA63">6.10 <spanclass="Heading">Solvability, Derived Series and Frattini Subloop</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap6_mj.html#X85BD2C517FA7A47E">6.10-4 <spanclass="Heading">FrattiniSubloop and FrattinifactorSize</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap6_mj.html#X81F3496578EAA74E">6.11 <spanclass="Heading">Isomorphisms and Automorphisms</span></a>
<divclass="ContChap"><ahref="chap7_mj.html#X7910E575825C713E">7 <spanclass="Heading">Testing Properties of Quasigroups and Loops</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap7_mj.html#X7960E3FB7A7F0F00">7.1 <spanclass="Heading">Associativity, Commutativity and Generalizations</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap7_mj.html#X85EDD10586596458">7.2-1 <spanclass="Heading">HasLeftInverseProperty, HasRightInverseProperty and HasInverseProperty</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap7_mj.html#X7D8CB6DA828FD744">7.3 <spanclass="Heading">Some Properties of Quasigroups</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap7_mj.html#X7F23D4D97A38D223">7.3-7 <spanclass="Heading">IsEntropic and IsMedial</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap7_mj.html#X780D907986EBA6C7">7.4 <spanclass="Heading">Loops of Bol Moufang Type</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap7_mj.html#X83A501387E1AC371">7.5 <spanclass="Heading">Power Alternative Loops</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap7_mj.html#X875C3DF681B3FAE2">7.5-1 <spanclass="Heading">IsLeftPowerAlternative, IsRightPowerAlternative and IsPowerAlternative</span></a>
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<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap7_mj.html#X8176B2C47A4629CD">7.6 <spanclass="Heading">Conjugacy Closed Loops and Related Properties</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap7_mj.html#X846F363879BAB349">7.8 <spanclass="Heading">Additonal Varieties of Loops</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap7_mj.html#X85F1BD4280E44F5B">7.8-3 <spanclass="Heading">IsLeftBruckLoop and IsLeftKLoop</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap7_mj.html#X857B373E7B4E0519">7.8-4 <spanclass="Heading">IsRightBruckLoop and IsRightKLoop</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap8_mj.html#X7990F2F880E717EE">8.1 <spanclass="Heading">Core Methods for Bol Loops</span></a>
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<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap8_mj.html#X8664CA927DD73DBE">8.1-1 <spanclass="Heading">AssociatedLeftBruckLoop and AssociatedRightBruckLoop</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap8_mj.html#X83E73A767D79FAFD">8.3 <spanclass="Heading">Triality for Moufang Loops</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap8_mj.html#X841ED66B8084AA73">8.4 <spanclass="Heading">Realizing Groups as Multiplication Groups of Loops</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap9_mj.html#X7DF21BD685FBF258">9.2 <spanclass="Heading">Left Bol Loops and Right Bol Loops</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap9_mj.html#X8028D69A86B15897">9.3 <spanclass="Heading">Left Bruck Loops and Right Bruck Loops</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap9_mj.html#X806B2DE67990E42F">9.7-1 <spanclass="Heading">RCCLoop and RightConjugacyClosedLoop</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap9_mj.html#X80AB8B107D55FB19">9.7-2 <spanclass="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a>
<spanclass="ContSS"><br/><spanclass="nocss"> </span><ahref="chap9_mj.html#X798BC601843E8916">9.7-3 <spanclass="Heading">CCLoop and ConjugacyClosedLoop</span></a>
<divclass="ContSect"><spanclass="tocline"><spanclass="nocss"> </span><ahref="chap9_mj.html#X864839227D5C0A90">9.13 <spanclass="Heading">Libraries of Loops Up To Isotopism</span></a>
<divclass="chlinkprevnextbot"> <ahref="chap0_mj.html">[Top of Book]</a> <ahref="chap0_mj.html#contents">[Contents]</a> <ahref="chap1_mj.html">[Next Chapter]</a> </div>
