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

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

Published $\text {1978}$, Blackie

ISBN 0 216 90488 9

### 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

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