Infinite Abelian Groups (eBook)

Front Cover 1
Infinite Abelian Groups 4
Copyright Page 5
Contents 10
Preface 8
Chapter I. Preliminaries 14
1. Definitions 14
2. Maps and Diagrams 19
3. The Most Important Types of Groups 27
4. Modules 32
5. Categories of Abelian Groups 34
6. Functorial Subgroups and Quotient Groups 38
7. Topologies in Groups 42
Notes 47
Chapter II. Direct Sums 49
8. Direct Sums and Direct Products 49
9. Direct Summands 59
10. Pullback and Pushout Diagrams 64
11. Direct Limits 66
12. Inverse Limits 72
13. Completeness and Completions 78
Notes 83
Chapter III. Direct Sums of Cyclic Groups 85
14. Free Abelian Groups—Defining Relations 85
15. Finitely Generated Groups 90
16. Linear Independence and Rank 96
17. Direct Sums of Cyclic p-Groups 100
18. Subgroups of Direct Sums of Cyclic Groups 104
19. Countable Free Groups 106
Notes 108
Chapter IV. Divisible Groups 110
20. Divisibility 110
21. Injective Groups 112
22. Systems of Equations 114
23. The Structure of Divisible Groups 117
24. The Divisible Hull 119
25. Finitely Cogenerated Groups 122
Notes 125
Chapter V. Pure Subgroups 126
26. Purity 126
27. Bounded Pure Subgroups 130
28. Quotient Groups Modulo Pure Subgroups 133
29. Pure-Exact Sequences 135
30. Pure-Projectivity and Pure-Injectivity 138
31. Generalizations of Purity 141
Notes 146
Chapter VI. Basic Subgroups 148
32. p-Basic Subgroups 148
33. Basic Subgroups of p-Groups 152
34. Further Results on p-Basic Subgroups 156
35. Different p-Basic Subgroups 160
36. Basic Subgroups Are Endomorphic Images 165
37. The Ulm Sequence 166
Notes 171
Chapter VII. Algebraically Compact Groups 172
38. Algebraic Compactness 172
39. Complete Groups 175
40. The Structure of Algebraically Compact Groups 180
41. Pure-Essential Extensions 183
42. More about Algebraically Compact Groups 187
Notes 191
Chapter VIII. Homomorphism Groups 193
43. Groups of Homomorphisms 193
44. Exact Sequences for Hom 197
45. Certain Subgroups of Hom 201
46. Homomorphism Groups of Torsion Groups 206
47. Character Groups 212
48. Duality between Discrete Torsion and 0-Dimensional Compact Groups 216
Notes 220
Chapter IX. Groups of Extensions 222
49. Group Extensions 222
50. Extensions as Short Exact Sequences 225
51. Exact Sequences for Ext 230
52. Elementary Properties of Ext 234
53. The Functor Pext 239
54. Cotorsion Groups 245
55. The Structure of Cotorsion Groups 250
56. The Ulm Factors of Cotorsion Groups 253
57. Applications to Ext 256
58. Injective Properties of Cotorsion Groups 260
Notes 263
Chapter X. Tensor and Torsion Products 265
59. The Tensor Product 265
60. Exact Sequences for Tensor Products 270
61. The Structure of Tensor Products 274
62. The Torsion Product 277
63. Exact Sequences for Tor 281
64. The Structure of Torsion Products 284
Notes 287
Bibliography 288
Table of Notations 294
Author Index 298
Subject Index 300
Pure and Applied Mathematics 304