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Intro; Preface; Contents; Part I Introduction; 1 Latin Squares Based on Groups; 1.1 Latin Squares; 1.1.1 Latin Squares and Quasigroups; 1.1.2 Isotopism; 1.2 Orthogonality; 1.2.1 MOLS; 1.2.2 The Number of Squares in a Set of MOLS; 1.2.3 Transversals; 1.2.4 Complete Mappings; 1.3 Difference Matrices and Orthomorphisms; 1.3.1 Difference Matrices; 1.3.2 Orthomorphisms; 1.3.3 Maximal Sets of MOLS; 1.3.4 Quotient Group Constructions; 1.4 Incidence Structures Related to MOLS; 1.4.1 Nets and Transversal Designs; 1.4.2 Affine Planes; 1.4.3 Projective Planes; 2 When Is a Latin Square Based on a Group?
2.1 The Quadrangle Criterion2.1.1 Loops Isotopic to Groups; 2.1.2 The Quadrangle Criterion for Loops; 2.1.3 Aczél's Test; 2.2 The Thomsen Condition; 2.3 The Rectangle Rule; 2.4 Tests Based on Row and Column Permutations; 2.4.1 Light's Test; 2.4.2 The Row Composition Rule; 2.5 Keedwell's Criteria; 2.5.1 The Frolov Property; 2.5.2 The F*-Property; 2.5.3 The Fa*-Property; Part II Admissible Groups; 3 The Existence Problem for Complete Mappings: The Hall-Paige Conjecture; 3.1 Introduction; 3.1.1 Cyclic Groups; 3.1.2 Groups of Odd Order; 3.1.3 Infinite Groups
3.2 The Admissibility of Finite Abelian Groups3.2.1 Paige's Theorem; 3.2.2 Hall's Theorem; 3.2.3 Complete Primitive Residue Sets; 3.2.4 Carlitz's Proof; 3.3 Some Admissibility Criteria; 3.3.1 Euler's Conjecture and Cayley Tables of Groups; 3.3.2 Products of All Group Elements; 3.3.3 Some Nonadmissibility Criteria; 3.3.4 The Hall-Paige Theorem and Conjecture; 4 Some Classes of Admissible Groups; 4.1 A Construction of Hall and Paige; 4.1.1 HP-Systems; 4.1.2 The Symmetric and Alternating Groups; 4.1.3 Factorable Groups; 4.2 Solvable Groups; 4.2.1 Some Properties of 2-Groups
4.2.2 Admissible 2-Groups4.2.3 Admissible Solvable Groups; 4.3 The Admissibility of the Mathieu Groups and Suzuki Groups; 4.3.1 Mathieu Groups; 4.3.2 Suzuki Groups; 4.4 Unitary Groups and Groups with Trivially Intersecting Sylow 2-Subgroups; 4.4.1 Unitary Groups; 4.4.2 Groups with Trivially Intersecting Sylow 2-Subgroups; 5 The Groups GL(n,q), SL(n,q), PGL(n,q), and PSL(n,q); 5.1 Complete Mappings of SL(2,q); 5.1.1 Partitions and Dual Systems of Coset Representatives; 5.1.2 SL(2,q), q Even; 5.1.3 SL(2,q), q=5, 7, and 11; 5.1.4 SL(2,q), q1 8mu(mod6mu4); 5.1.5 SL(2,q), q3 8mu(mod6mu4)
2.1 The Quadrangle Criterion2.1.1 Loops Isotopic to Groups; 2.1.2 The Quadrangle Criterion for Loops; 2.1.3 Aczél's Test; 2.2 The Thomsen Condition; 2.3 The Rectangle Rule; 2.4 Tests Based on Row and Column Permutations; 2.4.1 Light's Test; 2.4.2 The Row Composition Rule; 2.5 Keedwell's Criteria; 2.5.1 The Frolov Property; 2.5.2 The F*-Property; 2.5.3 The Fa*-Property; Part II Admissible Groups; 3 The Existence Problem for Complete Mappings: The Hall-Paige Conjecture; 3.1 Introduction; 3.1.1 Cyclic Groups; 3.1.2 Groups of Odd Order; 3.1.3 Infinite Groups
3.2 The Admissibility of Finite Abelian Groups3.2.1 Paige's Theorem; 3.2.2 Hall's Theorem; 3.2.3 Complete Primitive Residue Sets; 3.2.4 Carlitz's Proof; 3.3 Some Admissibility Criteria; 3.3.1 Euler's Conjecture and Cayley Tables of Groups; 3.3.2 Products of All Group Elements; 3.3.3 Some Nonadmissibility Criteria; 3.3.4 The Hall-Paige Theorem and Conjecture; 4 Some Classes of Admissible Groups; 4.1 A Construction of Hall and Paige; 4.1.1 HP-Systems; 4.1.2 The Symmetric and Alternating Groups; 4.1.3 Factorable Groups; 4.2 Solvable Groups; 4.2.1 Some Properties of 2-Groups
4.2.2 Admissible 2-Groups4.2.3 Admissible Solvable Groups; 4.3 The Admissibility of the Mathieu Groups and Suzuki Groups; 4.3.1 Mathieu Groups; 4.3.2 Suzuki Groups; 4.4 Unitary Groups and Groups with Trivially Intersecting Sylow 2-Subgroups; 4.4.1 Unitary Groups; 4.4.2 Groups with Trivially Intersecting Sylow 2-Subgroups; 5 The Groups GL(n,q), SL(n,q), PGL(n,q), and PSL(n,q); 5.1 Complete Mappings of SL(2,q); 5.1.1 Partitions and Dual Systems of Coset Representatives; 5.1.2 SL(2,q), q Even; 5.1.3 SL(2,q), q=5, 7, and 11; 5.1.4 SL(2,q), q1 8mu(mod6mu4); 5.1.5 SL(2,q), q3 8mu(mod6mu4)