Book:László Fuchs/Infinite Abelian Groups/Volume I

László Fuchs: Infinite Abelian Groups, Volume $\text { I }$

Published $\text {1970}$, New York: Academic Press

Continued in Infinite Abelian Groups, Volume II.

Subject Matter

• Infinite Abelian Groups

Contents

Preface

I. Preliminaries

1. Definitions

2. Maps and Diagrams

3. The Most Important Types of Groups

4. Modules

5. Categories of Abelian Groups

6. Functorial Subgroups and Quotient Groups

7. Topologies in Groups

Notes

II. Direct Sums

8. Direct Sums and Direct Products

9. Direct Summands

10. Pullback and Pushout Diagrams

11. Direct Limits

12. Inverse Limits

13. Completeness and Completions

Notes

III. Direct Sums of Cyclic Groups

14. Free Abelian Groups—Defining Relations

15. Finitely Generated Groups

16. Linear Independence and Rank

17. Direct Sums of Cyclic $p$-Groups

18. Subgroups of Direct Sums of Cyclic Groups

19. Countable Free Groups

Notes

IV. Divisible Groups

20. Divisibility

21. Injective Groups

22. Systems of Equations

23. The Structure of the Divisible Groups

24. The Divisible Hull

25. Finitely Cogenerated Groups

Notes

V. Pure Subgroups

26. Purity

27. Bounded Pure Subgroups

28. Quotient Groups Modulo Pure Subgroups

29. Pure-Exact Sequences

30. Pure-Projectivity and Pure-Injectivity

31. Generalizations of Purity

Notes

VI. Basic Subgroups

32. $p$-Basic Subgroups

33. Basic Subgroups of $p$-Groups

34. Further Results on $p$-Basic Subgroups

35. Different $p$-Basic Subgroups

36. Basic Subgroups are Endomorphic Images

37. The Ulm Sequence

Notes

VII. Algebraically Compact Groups

38. Algebraic Compactness

39. Complete Groups

40. The Structure of Algebraically Compact Groups

41. Pure-Essential Extensions

42. More about Algebraically Compact Groups

Notes

VIII. Homomorphism Groups

43. Groups of Homomorphisms

44. Exact Sequences for Hom

45. Certain Subgroups for Hom

46. Homomorphism Groups of Torsion Groups

47. Character Groups

48. Duality between Discrete Torsion and $0$-Dimensional Compact Groups

Notes

IX. Groups of Extensions

49. Group Extensions

50. Extensions as Short Exact Sequences

51. Exact Sequences for Ext

52. Elementary Properties of Ext

53. The Functor Pext

54. Cotorsion Groups

55. The Structure of Cotorsion Groups

56. The Ulm Factors of Cotorsion Groups

57. Applications to Ext

58. Injective Properties of Cotorsion Groups

Notes

X. Tensor and Torsion Products

59. The Tensor Product

60. Exact Sequences for Tensor Products

61. The Structure of Tensor Products

62. The Torsion Product

63. Exact Sequences for Tor

64. The Structure of Torsion Products

Notes

Bibliography

Table of Notations

Author Index

Subject Index