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• Cover Page
• Half-Title Page
• Series Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Notations
• Preface
• 0 Preliminaries
• Properties of Integers
• Modular Arithmetic
• Complex Numbers
• Mathematical Induction
• Equivalence Relations
• Functions (Mappings)
• Exercises
• 1 Introduction to Groups
• Symmetries of a Square
• The Dihedral Groups
• Biography of Niels Abel
• 2 Groups
• Definition and Examples of Groups
• Elementary Properties of Groups
• Historical Note
• Exercises
• 3 Finite Groups; Subgroups
• Terminology and Notation
• Subgroup Tests
• Examples of Subgroups
• Exercises
• 4 Cyclic Groups
• Properties of Cyclic Groups
• Classification of Subgroups of Cyclic Groups
• Biography of James Joseph Sylvester
• 5 Permutation Groups
• Definitions and Notation
• Cycle Notation
• Properties of Permutations
• A Check-Digit Scheme Based on D_5
• Biography of Augustin Cauchy
• Biography of Alan Turing
• 6 Isomorphisms
• Motivation
• Definition and Examples
• Properties of Isomorphisms
• Automorphisms
• Cayley’s Theorem
• Exercises
• Biography of Arthur Cayley
• 7 Cosets and Lagrange’s Theorem
• Properties of Cosets
• Lagrange’s Theorem and Consequences
• An Application of Cosets to Permutation Groups
• The Rotation Group of a Cube and a Soccer Ball
• An Application of Cosets to the Rubik’s Cube
• Exercises
• Biography of Joseph Lagrange
• 8 External Direct Products
• Definition and Examples
• Properties of External Direct Products
• The Group of Units Modulo n as an External Direct Product
• Applications
• Exercises
• Biography of Leonard Adleman
• 9 Normal Subgroups and Factor Groups
• Normal Subgroups
• Factor Groups
• Applications of Factor Groups
• Internal Direct Products
• Exercises
• Biography of Évariste Galois
• 10 Group Homomorphisms
• Definition and Examples
• Properties of Homomorphisms
• The First Isomorphism Theorem
• Exercises
• Biography of Camille Jordan
• 11 Fundamental Theorem of Finite Abelian Groups
• The Fundamental Theorem
• The Isomorphism Classes of Abelian Groups
• Proof of the Fundamental Theorem
• Exercises
• 12 Introduction to Rings
• Motivation and Definition
• Examples of Rings
• Properties of Rings
• Subrings
• Exercises
• Biography of I. N. Herstein
• 13 Integral Domains
• Definition and Examples
• Fields
• Characteristic of a Ring
• Exercises
• 14 Ideals and Factor Rings
• Ideals
• Factor Rings
• Prime Ideals and Maximal Ideals
• Exercises
• Biography of Richard Dedekind
• Biography of Emmy Noether
• 15 Ring Homomorphisms
• Definition and Examples
• Properties of Ring Homomorphisms
• The Field of Quotients
• Exercises
• 16 Polynomial Rings
• Notation and Terminology
• The Division Algorithm and Consequences
• Exercises
• 17 Factorization of Polynomials
• Reducibility Tests
• Irreducibility Tests
• Unique Factorization in Z[x]
• Weird Dice: An Application of Unique Factorization
• Exercises
• Biography of Serge Lang
• 18 Divisibility in Integral Domains
• Irreducibles, Primes
• Historical Discussion of Fermat’s Last Theorem
• Unique Factorization Domains
• Euclidean Domains
• Exercises
• Biography of Sophie Germain
• Biography of Andrew Wiles
• Biography of Pierre de Fermat
• 19 Extension Fields
• The Fundamental Theorem of Field Theory
• Splitting Fields
• Zeros of an Irreducible Polynomial
• Exercises
• Biography of Leopold Kronecker
• 20 Algebraic Extensions
• Characterization of Extensions
• Finite Extensions
• Properties of Algebraic Extensions
• Exercises
• Biography of Ernst Steinitz
• 21 Finite Fields
• Classification of Finite Fields
• Structure of Finite Fields
• Subfields of a Finite Field
• Exercises
• Biography of L. E. Dickson
• Biography of E. H. Moore
• 22 Geometric Constructions
• Historical Discussion of Geometric Constructions
• Constructible Numbers
• Angle-Trisectors and Circle-Squarers
• Exercises
• 23 Sylow Theorems
• Conjugacy Classes
• The Class Equation
• The Sylow Theorems
• Applications of Sylow Theorems
• Exercises
• Biography of Ludwig Sylow
• 24 Finite Simple Groups
• Historical Background
• Nonsimplicity Tests
• The Simplicity of A_5
• The Fields Medal
• The Cole Prize
• Exercises
• Biography of Michael Aschbacher
• Biography of Daniel Gorenstein
• Biography of John Thompson
• 25 Generators and Relations
• Motivation
• Definitions and Notation
• Free Group
• Generators and Relations
• Classification of Groups of Order Up to 15
• Characterization of Dihedral Groups
• Exercises
• Biography of Marshall Hall, Jr.
• 26 Symmetry Groups
• Isometries
• Classification of Finite Plane Symmetry Groups
• Classification of Finite Groups of Rotations in R^3
• Exercises
• 27 Symmetry and Counting
• Motivation
• Burnside’s Theorem
• Applications
• Group Action
• Exercises
• Biography of William Burnside
• 28 Cayley Digraphs of Groups
• Motivation
• The Cayley Digraph of a Group
• Hamiltonian Circuits and Paths
• Some Applications
• Exercises
• Biography of William Rowan Hamilton
• Biography of Paul Erdős
• 29 Introduction to Algebraic Coding Theory
• Motivation
• Linear Codes
• Parity-Check Matrix Decoding
• Coset Decoding
• Historical Note
• Exercises
• Biography of Richard W. Hamming
• Biography of Jessie MacWilliams
• Biography of Vera Pless
• 30 An Introduction to Galois Theory
• Fundamental Theorem of Galois Theory
• Solvability of Polynomials by Radicals
• Insolvability of a Quintic
• Exercises
• 31 Cyclotomic Extensions
• Motivation
• Cyclotomic Polynomials
• The Constructible Regular n-gons
• Exercises
• Biography of Carl Friedrich Gauss
• Biography of Manjul Bhargava
• Selected Answers
• Index