## Quasigroups and loops: introduction |

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### Contents

Quasigroups and loops | 1 |

Quasigroups and geometry | 34 |

Isotopy theory for quasigroups | 57 |

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### Common terms and phrases

abelian group automorphism autotopism autotopism of Q bijection binary operation Bol loop Bruck called carrier set Cayley table commutative Moufang loop Complete the proof Corollary decomposition modulo defined Definition denote di-associative distributive quasigroup element for G entropic quasigroup equivalence relation Example Exercise G-loop groupoid G idempotent identical relation identity element implies isomorphic isotopic invariant isotopism of G Lemma Let G Let Q loop G loop with identity loop-isotope loops of order Moufang identity nilpotency normal subgroup normal subloop normal subquasigroup nucleus operation table parastrophs permutations planar ternary ring principal isotope proof of Theorem prove pseudo-automorphism with companion quasigroups and loops right companion right identity right pseudo-automorphism satisfies Section set G Steiner triple systems subloop of G subset surjective T.S. quasigroup Theorem totally symmetric quasigroup unique W.I.P. loop x o y x,y G G