QUASIGROUPS AND LOOPS Latin squares Equational quasigroups Conjugates Semisymmetry and homotopy Loops and piques Steiner triple systems I Moufang loops and octonions Triality Normal forms Exercises Notes MULTIPLICATION GROUPS Combinatorial multiplication groups Surjections The diagonal action Inner multiplication groups of piques Loop transversals and right quasigroups Loop transversal codes Universal multiplication groups Universal stabilizers Exercises Notes CENTRAL QUASIGROUPS Quasigroup congruences Centrality Nilpotence Central isotopy Central piques Central quasigroups Quasigroups of prime order Stability congruences No-go theorems Exercises Notes HOMOGENEOUS SPACES Quasigroup homogeneous spaces Approximate symmetry Macroscopic symmetry Regularity Lagrangean properties Exercises Notes PERMUTATION REPRESENTATIONS The category IFSQ Actions as coalgebras Irreducibility The covariety of Q-sets The Burnside algebra An example Idempotents Burnside''s lemma Exercises Problems Notes CHARACTER TABLES Conjugacy classes Class functions The centralizer ring Convolution of class functions Bose-Mesner and Hecke algebras Quasigroup character tables Orthogonality relations Rank two quasigroups Entropy Exercises Problems Notes COMBINATORIAL CHARACTER THEORY Congruence lattices Quotients Fusion Induction Linear characters Exercises Problems Notes SCHEMES AND SUPERSCHEMES Sharp transitivity More no-go theorems Superschemes Superalgebras Tensor squares Relation algebras The reconstruction theorem Exercises Problems Notes PERMUTATION CHARACTERS Enveloping algebras Structure of enveloping algebras The canonical representation Commutative actions Faithful homogeneous spaces Characters of homogeneous spaces General pxercises Notes HOMOGENEOUS SPACES Quasigroup homogeneous spaces Approximate symmetry Macroscopic symmetry Regularity Lagrangean properties Exercises Notes PERMUTATION REPRESENTATIONS The category IFSQ Actions as coalgebras Irreducibility The covariety of Q-sets The Burnside algebra An example Idempotents Burnside''s lemma Exercises Problems Notes CHARACTER TABLES Conjugacy classes Class functions The centralizer ring Convolution of class functions Bose-Mesner and Hecke algebras Quasigroup character tables Orthogonality relations Rank two quasigroups Entropy Exercises Problems Notes COMBINATORIAL CHARACTER THEORY Congruence lattices Quotients Fusion Induction Linear characters Exercises Problems Notes SCHEMES AND SUPERSCHEMES Sharp transitivity More no-go theorems Superschemes Superalgebras Tensor squares Relation algebras The reconstruction theorem Exercises Problems Notes PERMUTATION CHARACTERS Enveloping algebras Structure of enveloping algebras The canonical representation Commutative actions Faithful homogeneous spaces Characters of homogeneous spaces General pAL CHARACTER THEORY Congruence lattices Quotients Fusion Induction Linear characters Exercises Problems Notes SCHEMES AND SUPERSCHEMES Sharp transitivity More no-go theorems Superschemes Superalgebras Tensor squares Relation algebras The reconstruction theorem Exercises Problems Notes PERMUTATION CHARACTERS Enveloping algebras Structure of enveloping algebras The canonical representation Commutative actions Faithful homogeneous spaces Characters of homogeneous spaces General permutation characters The Ising model Exercises Problems Notes MODULES Abelian groups and slice categories Quasigroup modules The fundamental theorem Differential calculus Representations in varieties Group representations Exercises Problems Notes APPLICATIONS OF MODULE THEORY Nonassociative powers Exponents Steiner triple systems II The Burnside problem A free commutative Moufang loop Extensions and cohomology Exercises Problems Notes ANALYTICAL CHARACTER THEORY Functions on finite quasigroups Periodic functions on groups Analytical character theory Almost periodic functions Twisted translation operators Proof of the existence theorem Exercises Problems Notes APPENDIX A: CATEGORICAL CONCEPTS Graphs and categories Natural transformations and functors Limits and colimits APPENDIX B: UNIVERSAL ALGEBRA Combinatorial universal algebra Categorical universal algebra APPENDIX C: COALGEBRAS Coalgebras and covarieties Set functors REFERENCES INDEXrcises Problems Notes APPENDIX A: CATEGORICAL CONCEPTS Graphs and categories Natural transformations and functors Limits and colimits APPENDIX B: UNIVERSAL ALGEBRA Combinatorial universal algebra Categorical universal algebra APPENDIX C: COALGEBRAS Coalgebras and covarieties Set functors REFERENCES INDEX.