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

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## Thomas A. Whitelaw: *An Introduction to Abstract Algebra*

Published $1978$, **Blackie**

- ISBN 0 216 90488 9.

### Subject Matter

### 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