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 --
9. Introduction --
10. The well-ordering principle -- I I. The division algorithm --
12. Highest common factors and Euclid’s algorithm --
13. The fundamental theorem of arithmetic --
14. Congruence modulo m (mE f4J) -- EXERCISES -- Chapter Three EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES --
15. Relations in general --
16. Equivalence relations --
17. Equivalence classes --
18. Congruence classes --
19. Properties of l,, as an algebraic system -- EXERCISES -- Chapter Four MAPPINGS --
20. Introduction --
21. The image of a subset of the domain; surjcctions --
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 ON FINITE GROUPS --
40. Introduction --
41. Multiplication of 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 THE SYMMETRIC GROUP S --
53. Introduction --
54. Cycles --
55. Products of disjoint cycles --
56. Periods of elements of Sft --
57. Conjugacy in S1 --
58. Arrangement of the objects 1,2,...,n --
59. The alternating character, and alternating groups --
60. The simplicity of A5 -- EXERCISES -- Chapter Ten THE FUNDAMENTALS OF RING THEORY --
61. Introduction --
62. The definition of a ring and its elementary consequences --
63. Special types of ring and ring elements --
64. Subrings and subtIelds --
65. Ring homomorphisms --
66. Ideals --
67. Principal ideals in a commutative ring with a one --
68. Factor rings --
69. Characteristic of an integral domain or field --
70. The field of fractions of an integral domain -- EXERCISES -- Chapter Eleven POLYNOMIALS AND FIELDS --
71. Introduction --
72. Polynomial rings --
73. Some properties of F[X], where F is a field -- 74 Generalities on factorization --
75. Further properties of F[XJ, where F is a field --
76. Some matters of notation --
77. Minimal polynomials and the structure of F(c) --
78. Some elementary properties of finite fields --
79. Construction of fields by root adjunction --
80. Degrees of field extensions --
81. Epilogue -- EXERCISES -- BIBLIOGRAPHY -- APPENDIX TO EXERCISES -- INDEX.