Abelian Groups - 3rd Edition - ISBN: 9780080092065, 9781483280905

Abelian Groups, Volume 12

3rd Edition

Editors: J. P. Kahane A. P. Robertson S. Ulam
Authors: L. Fuchs
eBook ISBN: 9781483280905
Imprint: Pergamon
Published Date: 1st January 1960
Page Count: 368
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Table of Contents


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


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


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


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


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


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


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


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


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


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


Chapter XI. Tensor Products

§ 64. The Tensor Product

§ 65. The Structure of Tensor Products


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


Chapter XIII. The Multiplicative Group of Fields

§ 76. Finite Algebraic Extensions of Prime Fields

§ 77. Algebraically and Real Closed Fields


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


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


Chapter XVI. Various Questions

§ 86. Hereditarily Generating Systems

§ 87. Universal Homomorphic Images

§ 88. Universal Subgroups

§ 89. A Combinatorial Problem



Author Index

Subject Index


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.


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© Pergamon 1960
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Ratings and Reviews

About the Editors

J. P. Kahane Editor

A. P. Robertson Editor

S. Ulam Editor

About the Authors

L. Fuchs Author