Introduction to Abstract Algebra

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.