Preface.... ..7
Chapter 1. Sets and Binary Relations..9
1.1 Sets. 9
1.2 Binary Relations. 1 3
1.3 Functions .. 16
1.4 Order Relations. .. 2 0
1.5 Boolean Matrices and Graphs...26
1.6 Arrowβs Impossibility Theorem. 3 2
Chapter 2. Semigroups and Groups...39
2.1 Semigroups ..40
2.2 Generators and Relations. 4 4
2.3 Greenβs Relations. 5 0
2.4 Blockmodels. . ..55
2.5 Finite State Machines. 6 0
2.6 Recognition of Machine Languages by Finite State
Machines. 6 3
2.7 Groups. .67
2.8 Subgroups.72
2.9 Homomorphisms. 7 6
2.10 Permutation Groups. . ..79
2.11 Systems of Distinct Representatives and Flows on
Networks.83
2.12 Orbits and Conjugations. 9 2
2.13 Symmetries.98
2.14 Polya Enumeration.107
2.15 Kinship Systems. I ll
2.16 Lattices of Subgroups ..114
Chapter 3. Vector Spaces.119
3.1 Vector Spaces.120
3.2 Basis and Dimension.126
3.3 Matrices.131
3.4 Linear Transformations.138
3.5 Determinants and Characteristic Polynomials.143
3.6 Eigenvalues, Eigenvectors, Similarity.149
3.7 Symmetric and Unitary Matrices.157
Chapter 4. Rings.163
4.1 The Integers and Divisibility.164
4.2 Euclidean Domains and Factorization.168
4.3 Ideals and Congruences.173
4.4 Structure of Z n . 1 7 6
4.5 Simple and Semisimple Rings.179
Chapter 5. Group Representations .184
5.1 The Group Ring and Representations.185
5.2 Modules and Representations.190
5.3 Irreducible Representations.194
5.4 Group Characters.197
5.5 Tensor Products.201
Chapter 6. Field Theory.210
6.1 Finite Dimensional Extensions.211
6.2 Applications of Extensions of the Rationals.215
6.3 Finite Fields.221
6.4 Coding Theory.227
6.5 Cyclic Codes.233
6.6 Latin Squares.240
6.7 Projective Planes and Block Designs.246
Open Problems.253
List of Special Symbols.254
References.259
Index
261