Book:Thomas A. Whitelaw/An Introduction to Abstract Algebra

Subject Matter

 * Logic
 * Set Theory
 * Mapping Theory
 * Group Theory
 * Ring Theory

Contents

 * Preface


 * Chapter One: Sets and Logic
 * 1. Some very general remarks
 * 2. Introductory remarks on sets
 * 3. Statements and conditions; quantifiers
 * 4. The implies sign
 * 5. Proof by Contradiction
 * 6. Subsets
 * 7. Unions and intersections
 * 8. Cartesian product of sets
 * EXERCISES


 * Chapter Two: Some Properties of $\Z$
 * 9. Introduction
 * 10. The well-ordering principle
 * 11. The division algorithm
 * 12. Highest common factors and Euclid's algorithm
 * 13. The fundamental theorem of arithmetic
 * 14. Congruence modulo $m$ ($m \in \N$)
 * EXERCISES


 * Chapter Three: Equivalence Relations and Equivalence Classes
 * 15. Relations in general
 * 16. Equivalence relations
 * 17. Equivalence classes
 * 18. Congruence classes
 * 19. Properties of $\Z_m$ as an algebraic system
 * EXERCISES


 * Chapter Four: Mappings
 * 20. Introduction
 * 21. The image of a subset of the domain; surjections
 * 22. Injections; bijections; inverse of a bijection
 * 23. Restriction of a Mapping
 * 24. Composition of mappings
 * 25. Some further results and examples on mappings
 * EXERCISES


 * Chapter Five: Semigroups
 * 26. Introduction
 * 27. Binary operations
 * 28. Associativity and commutativity
 * 29. Semigroups: definition and examples
 * 30. Powers of an element in a semigroup
 * 31. Identity elements and inverses
 * 32. Subsemigroups
 * EXERCISES


 * Chapter Six: An Introduction to Groups
 * 33. The definition of a group
 * 34. Examples of groups
 * 35. Elementary consequences of the group axioms
 * 36. Subgroups
 * 37. Some important general examples of subgroups
 * 38. Period of an element
 * 39. Cyclic groups
 * EXERCISES


 * Chapter Seven: Cosets and Lagrange's theorem
 * 40. Introduction
 * 41. Multiplication of the subsets of a group
 * 42. Another approach to cosets
 * 43. Lagrange's theorem
 * 44. Some consequences of Lagrange's theorem
 * EXERCISES


 * Chapter Eight: Homomorphisms, normal subgroups and quotient groups
 * 45. Introduction
 * 46. Isomorphic groups
 * 47. Homomorphisms and their elementary properties
 * 48. Conjugacy
 * 49. Normal subgroups
 * 50. Quotient groups
 * 51. The quotient group $G / Z$
 * 52. The first isomorphism theorem
 * EXERCISES


 * Chapter Nine: Rings
 * 53. Introduction
 * 54. The definition of a ring and its elementary consequences
 * 55. Special types of ring and ring elements
 * 56. Subrings and subfields
 * 57. Ring homomorphisms
 * 58. Ideals
 * 59. Principal ideals in a commutative ring with a one
 * 60. Factor rings
 * 61. Characteristic of an integral domain or field
 * 62. Factorization in an integral domain
 * 63. Construction of fields as factor rings
 * 64. Polynomial rings over an integral domain
 * 65. Some properties of $F \left[{X}\right]$, where $F$ is a field
 * EXERCISES


 * Bibliography
 * Appendix to Exercises
 * Index