Abelian Groups

Abelian Groups

3rd Edition - January 1, 1960

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  • Editors: J. P. Kahane, A. P. Robertson, S. Ulam
  • eBook ISBN: 9781483280905

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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.

Table of Contents

  • 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


    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

Product details

  • No. of pages: 368
  • Language: English
  • Copyright: © Pergamon 1960
  • Published: January 1, 1960
  • Imprint: Pergamon
  • eBook ISBN: 9781483280905

About the Editors

J. P. Kahane

A. P. Robertson

S. Ulam

About the Author

L. Fuchs

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