Abelian Groups

3rd Edition, Volume 12 - January 1, 1960
Author: L. Fuchs
Editors: J. P. Kahane, A. P. Robertson, S. Ulam
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 8 0 9 0 - 5

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… Read more

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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.

Preface

Table of Contents

Table of Notations

Chapter I. Basic Concepts. The Most Important Groups

§ 1. Notation and Terminology

§ 2. Direct Sums

§ 3. Cyclic Groups

§ 4. Quasicyclic Groups

§ 5. The Additive Group of the Rationals

§ 6. The p-Adic Integers

§ 7. Operator Modules

§ 8. Linear Independence and Rank

Exercises

Chapter II. Direct Sum of Cyclic Groups

§ 9. Free (Abelian) Groups

§ 10. Finite and Finitely Generated Groups

§ 11. Direct Sums of Cyclic p-Groups

§ 12. Subgroups of Direct Sums of Cyclic Groups

§ 13. Two Dual Criteria for the Basis

§ 14. Further Criteria for the Existence of a Basis

Exercises

Chapter III. Divisible Groups

§ 15. Divisibility by Integers in Groups

§ 16. Homomorphisms into Divisible Groups

§ 17. Systems of Linear Equations Over Divisible Groups

§ 18. The Direct Summand Property of Divisible Groups

§ 19. The Structure Theorem on Divisible Groups

§ 20. Embedding in Divisible Groups

Exercises

Chapter IV. Direct Summands and Pure Subgroups

§ 21. Direct Summands

§ 22. Absolute Direct Summands

§ 23. Pure Subgroups

§ 24. Bounded Pure Subgroups

§ 25. Factor Groups with Respect to Pure Subgroups

§ 26. Algebraically Compact Groups

§ 27. Generalized Pure Subgroups

§ 28. Neat Subgroups

Exercises

Chapter V. Basic Subgroups

§ 29. Existence of Basic Subgroups. The Quasibasis

§ 30. Properties of Basic Subgroups

§ 31. Different Basic Subgroups of a Group

§ 32. The Basic Subgroup as an Endomorphic Image

Exercises

Chapter VI. The Structure of p-Groups

§ 33. p-Groups without Elements of Infinite Height

§ 34. Closed p-Groups

§ 35. The Ulm Sequence

§ 36. Zippn's Theorem

§ 37. Ulm's Theorem

§ 38. Construction of Groups with a Prescribed Ulm Sequence

§ 39. Non-Isomorphic Groups with the same Ulm Sequence

§ 40. Some Applications

§ 41. Direct Decompositions of p-Groups

Exercises

Chapter VII. Torsion Free Groups

§ 42. The Type of Elements. Groups of Rank 1

§ 43. Indecomposable Groups

§ 44. Torsion Free Groups Over the p-Adic Integers

§ 45. Countable Torsion Free Groups

§ 46. Completely Decomposable Groups

§ 47. Complete Direct Sums of Infinite Cyclic Groups. Slender Groups

§ 48. Homogeneous Groups

§ 49. Separable Groups

Exercises

Chapter VIII. Mixed Groups

§ 50. Splitting Mixed Groups

§ 51. Factor Groups of Free Groups

§ 52. A Characterization of Arbitrary Groups by Matrices

§ 53. Groups Over the p-Adic Integers

Exercises

Chapter IX. Homomorphism Groups and Endomorphism Rings

§ 54. Homomorphism Groups

§ 55. Endomorphism Rings

§ 56. The Endomorphism Ring of p-Groups

§ 57. Endomorphism Rings with Special Properties

§ 58. Automorphism Groups

§ 59. Fully Invariant Subgroups

Exercises

Chapter X. Group Extensions

§ 60. Extensions of Groups

§ 61. The Group of Extensions

§ 62. Induced Endomorphisms of the Group of Extensions

§ 63. Structural Properties of the Group of Extensions

Exercises

Chapter XI. Tensor Products

§ 64. The Tensor Product

§ 65. The Structure of Tensor Products

Exercises

Chapter XII. The Additive Group of Rings

§ 66. Ideals Determined by the Additive Group

§ 67. Multiplications on a Group

§ 68. Rings on Direct Sums of Cyclic Groups

§ 69. Torsion Rings

§ 70. Torsion Free Rings

§ 71. Nil Groups and Quasi Nil Groups

§ 72. The Additive Group of Artinian Rings

§ 73. Artinian Rings without Subgroups of Type p∞

§ 74. The Additive Group of Semi-Simple and Regular Rings

§ 75. The Additive Group of Rings with Maximum or Restricted Minimum Condition

Exercises

Chapter XIII. The Multiplicative Group of Fields

§ 76. Finite Algebraic Extensions of Prime Fields

§ 77. Algebraically and Real Closed Fields

Exercises

Chapter XIV. The Lattice of Subgroups

§ 78. Properties of the Subgroup Lattice

§ 79. Projectivities. Projectivities of Cyclic Groups

§ 80. Projectivities of Torsion Groups

§ 81. Projectivities of Torsion Free and Mixed Groups

§ 82. Dualisms

Exercises

Chapter XV. Decompositions into Direct Sums of Subsets

§ 83. Decompositions of Cyclic Groups

§ 84. Decompositions into Weakly Periodic Subsets

§ 85. Decompositions into an Infinity of Components

Exercises

Chapter XVI. Various Questions

§ 86. Hereditarily Generating Systems

§ 87. Universal Homomorphic Images

§ 88. Universal Subgroups

§ 89. A Combinatorial Problem

Exercises

Bibliography

Author Index

Subject Index