Fuchs / Kahane / Robertson Abelian Groups

3. Auflage 2014
ISBN: 978-1-4832-8090-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 12, 368 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Abelian Groups deals with the theory of abelian or commutative groups, with special emphasis on results concerning structure problems. More than 500 exercises of varying degrees of difficulty, with and without hints, are included. Some of the exercises illuminate the theorems cited in the text by providing alternative developments, proofs or counterexamples of generalizations. Comprised of 16 chapters, this volume begins with an overview of the basic facts on group theory such as factor group or homomorphism. The discussion then turns to direct sums of cyclic groups, divisible groups, and direct summands and pure subgroups, as well as Kulikov's basic subgroups. Subsequent chapters focus on the structure theory of the three main classes of abelian groups: the primary groups, the torsion-free groups, and the mixed groups. Applications of the theory are also considered, along with other topics such as homomorphism groups and endomorphism rings; the Schreier extension theory with a discussion of the group of extensions and the structure of the tensor product. In addition, the book examines the theory of the additive group of rings and the multiplicative group of fields, along with Baer's theory of the lattice of subgroups. This book is intended for young research workers and students who intend to familiarize themselves with abelian groups.

Autoren/Hrsg.

Fuchs, L.

Kahane, J. P.

Robertson, A. P.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Abelian Groups;4
3;Copyright Page;5
4;Table of Contents;10
5;PREFACE;6
6;TABLE OF NOTATIONS;13
7;CHAPTER I. BASIC CONCEPTS. THE MOST IMPORTANT GROUPS;14
7.1;1. Notation and terminology;14
7.2;2. Direct sums;18
7.3;3. Cyclic groups;23
7.4;4. Quasicyclic groups;24
7.5;5. The additive group of the rationals;26
7.6;6. The p-adic integers;27
7.7;7. Operator modules;28
7.8;8. Linear independence and rank;30
7.9;Exercises;35
8;CHAPTER II. DIRECT SUM OF CYCLIC GROUPS;38
8.1;9. Free (abelian) groups;38
8.2;10. Finite and finitely generated groups;40
8.3;11. Direct sums of cyclic p-groups;44
8.4;12. Subgroups of direct sums of cyclic groups;46
8.5;13. Two dual criteria for the basis;48
8.6;14. Further criteria for the existence of a basis;51
8.7;Exercises;53
9;CHAPTER III. DIVISIBLE GROUPS;58
9.1;15. Divisibility by integers in groups;58
9.2;16. Homomorphisms into divisible groups;60
9.3;17. Systems of linear equations over divisible groups;61
9.4;18. The direct summand property of divisible groups;63
9.5;19. The structure theorem on divisible groups;65
9.6;20. Embedding in divisible groups;66
9.7;Exercises;68
10;CHAPTER IV. DIRECT SUMMANDS AND PURE SUBGROUPS;72
10.1;21. Direct summands;72
10.2;22. Absolute direct summands;74
10.3;23. Pure subgroups;77
10.4;24. Bounded pure subgroups;80
10.5;25. Factor groups with respect to pure subgroups;82
10.6;26. Algebraically compact groups;84
10.7;27. Generalized pure subgroups;88
10.8;28. Neat subgroups;92
10.9;Exercises;94
11;CHAPTER V. BASIC SUBGROUPS;98
11.1;29. Existence of basic subgroups. The quasibasis;98
11.2;30. Properties of basic subgroups;102
11.3;31. Different basic subgroups of a group;104
11.4;32. The basic subgroup as an endomorphic image;107
11.5;Exercises;109
12;CHAPTER VI. THE STRUCTURE OF p-GROUPS;112
12.1;33. p-groups without elements of infinite height;112
12.2;34. Closed p-groups;115
12.3;35. The Ulm sequence;118
12.4;36. Zippin's theorem;122
12.5;37. Ulm's theorem;124
12.6;38. Construction of groups with a prescribed Ulm sequence;128
12.7;39. Non-isomorphic groups with the same Ulm sequence;135
12.8;40. Some applications;136
12.9;41. Direct decompositions of p-groups;138
12.10;Exercises;142
13;CHAPTER VII. TORSION FREE GROUPS;146
13.1;42. The type of elements. Groups of rank 1;146
13.2;43. Indecomposable groups;151
13.3;44. Torsion free groups over the p-adic integers;155
13.4;45. Countable torsion free groups;158
13.5;46. Completely decomposable groups;163
13.6;47. Complete direct sums of infinite cyclic groups. Slender groups;169
13.7;48. Homogeneous groups;174
13.8;49. Separable groups;177
13.9;Exercises;180
14;CHAPTER VIII. MIXED GROUPS;186
14.1;50. Splitting mixed groups;186
14.2;51. Factor groups of free groups;193
14.3;52. A characterization of arbitrary groups by matrices;197
14.4;53. Groups over the p-adic integers;199
14.5;Exercises;201
15;CHAPTER IX. HOMOMORPHISM GROUPS AND ENDOMORPHISM RINGS;206
15.1;54. Homomorphism groups;206
15.2;55. Endomorphism rings;211
15.3;56. The endomorphism ring of p-groups;215
15.4;57. Endomorphism rings with special properties;219
15.5;58. Automorphism groups;222
15.6;59. Fully invariant subgroups;225
15.7;Exercises;228
16;CHAPTER X. GROUP EXTENSIONS;234
16.1;60. Extensions of groups;234
16.2;61. The group of extensions;237
16.3;62. Induced endomorphisms of the group of extensions;240
16.4;63. Structural properties of the group of extensions;244
16.5;Exercises;248
17;CHAPTER XI. TENSOR PRODUCTS;250
17.1;64. The tensor product;250
17.2;65. The structure of tensor products;255
17.3;Exercises;257
18;CHAPTER XII. THE ADDITIVE GROUP OF RINGS;259
18.1;66. Ideals determined by the additive group;260
18.2;67. Multiplications on a group;262
18.3;68. Rings on direct sums of cyclic groups;264
18.4;69. Torsion rings;266
18.5;70. Torsion free rings;269
18.6;71. Nil groups and quasi nil groups;273
18.7;72. The additive group of Artinian rings;281
18.8;73. Artinian rings without subgroups of type p8;284
18.9;74. The additive group of semi-simple and regular rings;287
18.10;75. The additive group of rings with maximum or restricted minimum condition;289
18.11;Exercises;291
19;CHAPTER XIII. THE MULTIPLICATIVE GROUP OF FIELDS;296
19.1;76. Finite algebraic extensions of prime fields;296
19.2;77. Algebraically and real closed fields;298
19.3;Exercises;299
20;CHAPTER XIV. THE LATTICE OF SUBGROUPS;301
20.1;78. Properties of the subgroup lattice;301
20.2;79. Projectivities. Projectivities of cyclic groups;304
20.3;80. Projectivities of torsion groups;306
20.4;81. Projectivities of torsion free and mixed groups;310
20.5;82. Dualisms;312
20.6;Exercises;313
21;CHAPTER XV. DECOMPOSITIONS INTO DIRECT SUMS OF SUBSETS;316
21.1;83. Decompositions of cyclic groups;316
21.2;84. Decompositions into weakly periodic subsets;319
21.3;85. Decompositions into an infinity of components;325
21.4;Exercises;330
22;CHAPTER XVI. VARIOUS QUESTIONS;333
22.1;86. Hereditarily generating systems;333
22.2;87. Universal homomorphic images;337
22.3;88. Universal subgroups;342
22.4;89. A combinatorial problem;345
22.5;Exercises;351
23;BIBLIOGRAPHY;354
24;AUTHOR INDEX;364
25;SUBJECT INDEX;366
26;ERRATA;369