Applied Abstract Algebra

Description

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 Arrow’s Impossibility Theorem. 3 2 Chapter 2. Semigroups and Groups...39 2.1 Semigroups ..40 2.2 Generators and Relations. 4 4 2.3 Green’s 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