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1 QUASIGROUPS AND LOOPS 1.1 Latin squares 1.2 Equational quasigroups 1.3 Conjugates 1.4 Semisymmetry and homotopy 1.5 Loops and piques 1.6 Steiner triple systems I 1.7 Moufang loops and octonions 1.8 Triality 1.9 Normal forms 1.10 Exercises 1.11 Notes 2 MULTIPLICATION GROUPS 2.1 Combinatorial multiplication groups 2.2 Surjections 2.3 The diagonal action 2.4 Inner multiplication groups of piques 2.5 Loop transversals and right quasigroups 2.6 Loop transversal codes 2.7 Universal multiplication groups 2.8 Universal stabilizers 2.9 Exercises 2.10 Notes 3 CENTRAL QUASIGROUPS 3.1 Quasigroup congruences 3.2 Centrality 3.3 Nilpotence 3.4 Central isotopy 3.5 Central piques 3.6 Central quasigroups 3.7 Quasigroups of prime order 3.8 Stability congruences 3.9 No-go theorems 3.10 Exercises 3.11 Notes 4 HOMOGENEOUS SPACES 4.1 Quasigroup homogeneous spaces 4.2 Approximate symmetry 4.3 Macroscopic symmetry 4.4 Regularity 4.5 Lagrangean prcperties 4.6 Exercises 4.7 Notes 5 PERMUTATION REPRESENTATIONS 5.1 The category ]FSQ 5.2 Actions as coalgebras 5.3 Irreducibility 5.4 The covariety of Q-sets 5.5 The Burnside algebra 5.6 An example 5.7 Idempotents 5.8 Burnside's Lemma 5.9 Exercises 5.10 Problems 5.11 Notes 6 CHARACTER TABLES 6.1 Conjugacy classes 6.2 Class functions 6.3 The centralizer ring 6.4 Convolution of class functions 6.5 Bose-Mcsner and Hecke algebras 6.6 Quasigroup character tables 6.7 Orthogonality relations 6.8 Rank two quasigroups 6.9 Entropy 6.10 Exercises 6.11 Problems 6.12 Netcs 7 COMBINATORIAL CHARACTER THEORY 7.1 Congruence lattices 7.2 Quotients 7.3 Fusion 7.4 Induction 7.5 Linear characters 7.6 Exercises 7.7 Problems 7.8 Notes 8 SCHEMES AND SUPERSCHEMES 8.1 Sharp transitivity 8.2 More no-go theorems 8.3 Superschemes 8.4 Superalgebras 8.5 Tenser squales 8.6 Relation algebras 8.7 The Reconstruction Theorem 8.8 Exercises 8.9 Problems 8.10 Notes 9 PERMUTATION CHARACTERS 9.1 Enveloping algebras 9.2 Structure of enveloping algebras 9.3 The canonical representaticn 9.4 Commutative actions 9.5 Faithful homogeneous spaces 9.6 Characters of homogeneous spaces 9.7 General permutation characters 9.8 The Ising model 9.9 ExeI cises 9.10 Problems 9.11 Nctes 10 MODULES 10.1 Abelian groups and slice categories 10.2 Quasigroup modules 10.3 The Fundamental Theorem 10.4 Differential calculus 10.5 Representations in varieties 10.6 Group representations 10.7 Exercises 10.8 Problems 10.9 Notes 11 APPLICATIONS OF MODULE THEORY 11.1 Nonassociative lowers 11.2 Exponents 11.3 Steincr triple systems ¢ò 11.4 The Burrside Problem 11.5 A free commutative Mcufang loop 11.6 Extensions aid cohomology 11.7 Exercises 11.8 Problems 11.9 Notes 12 ANALYTICAL CHARACTER THEORY 12.1 Functions on finite quasigroups 12.2 Periodic functions on groups 12.3 Analytical character theory 12.4 Ahnost periodic functions 12.5 Twisted translation operators 12.6 Proof of the Existence Theorem 12.7 Exercises 12.8 Problems 12.9 Notes A CATEGORICAL CONCEPTS A.1 Graphs and categories A.2 Natural transformations and functors A.3 Limits and colimits B UNIVERSAL ALGEBRA B.1 Combinatorial universal algebra B.2 Categorical universal algebra C COALGEBRAS C.1 Coalgebras and covarieties C.2 Set functors References Index