AN INTRODUCTION TO QUASIGROUPS AND THEIR REPRESENTATIONS

By K. PAURO

CONTENTS 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 1 10 EXERCISES 1 11 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 2 10 NOTES CENTRAL QUASIGROUPS QUASIGROUP CONGRUENCES CENTRALITY NILPOTENCE CENTRAL ISOTOPY CENTRAL PIQUES CENTRAL QUASIGROUPS QUASIGROUPS OF PRIME ORDER STABILITY CONGRUENCES NO-GO THEOREMS 3 10 EXERCISES 3 11 NOTES HOMOGENEOUS SPACES QUASIGROUP HOMOGENEOUS SPACES APPROXIMATE SYMMETRY MACROSCOPIC SYMMETRY REGULARITY LAGRANGEAN PROPERTIES NOTES PERMUTATION REPRESENTATIONS THE CATEGORY IFSQ ACTIONS AS COALGEBRAS IRREDUCIBILITY THE COVARIETY OF Q-SETS THE BURNSIDE ALGEBRA AN EXAMPLE IDEMPOTENTS BURNSIDE’S LEMMA 5 10 PROBLEMS 5 11 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 6 10 EXERCISES 6 11 PROBLEMS 6 12 NOTES COMBINATORIAL CHARACTER THEORY CONGRUENCE LATTICES QUOTIENTS FUSION INDUCTION LINEAR CHARACTERS PROBLEMS SCHEMES AND SUPERSCHEMES SHARP TRANSITIVITY MORE NO-GO THEOREMS SUPERSCHEMES SUPERALGEBRAS TENSOR SQUARES RELATION ALGEBRAS THE RECONSTRUCTION THEOREM 8 10 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 9 10 PROBLEMS 9 11 NOTES 10 MODULES 10 1 ABELIAN GROUPS AND SLICE CATEGORIES 10 2 QUASIGROUP MODULES 10 3 THE FUNDAMENTAL THEOREM 10 4 DI?ERENTIAL 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 POWERS 11 2 EXPONENTS 11 3 STEINER TRIPLE SYSTEMS II 11 4 THE BURNSIDE PROBLEM 11 5 A FREE COMMUTATIVE MOUFANG LOOP 11 6 EXTENSIONS AND COHOMOLOGY 11 7 EXERCISES 11 8 PROBLEMS 11 9 NOTES 12 ANALYTICAL CHARACTER THEORY 12 1 FUNCTIONS ON ?NITE QUASIGROUPS 12 2 PERIODIC FUNCTIONS ON GROUPS 12 3 ANALYTICAL CHARACTER THEORY 12 4 ALMOST PERIODIC FUNCTIONS 12 5 TWISTED TRANSLATION OPERATORS 12 6 PROOF OF THE EXISTENCE THEOREM 12 7 EXERCISES 12 8 PROBLEMS 12 9 NOTES