Abelian groups / by L. Fuchs.
Editor: Budapest : Publishing House of the Hungarian Academy of Sciences, 1958Descripción: 367 p. ; cmOtra clasificación: 20TABLE OF CONTENTS Preface [5] Table of contents [9] Table of notations [12] Chapter 1. Basic concepts. The most important groups § 1. Notation and terminology [13] § 2. Direct sums [17] § 3. Cyclic groups [22] § 4. Quasicyclic groups [23] § 5. The additive group of the rationals [25] § 6. The p-adic integers [26] § 7. Operator modules [27] §8. Linear independence and rank [29] Exercises [34] Chapter IL Direct sum. of cyclic groups % 9. Free (abelian) groups [37] § 10. Finite and finitely generated groups [39] § 11, Direct sums of cyclic p-groups [43] § 12, Subgroups of direct sums of cyclic groups [45] § 13. Two dual criteria for the basis [47] § 14, Further criteria for the existence of a basis [50] Exercises [52] Chapter III. Divisible groups § 15, Divisibility by integers in groups [57] i 16, Homomorphisms into divisible groups [59] i 17, Systems of linear equations over divisible groups [60] § 18, The direct summand property of divisible groups [62] 9 19, The structure theorem on divisible groups [64] f 20, Embedding in divisible groups [65] Exercises [67] Chapter iv, Direct summands and pure subgroups Direct summands [71] Absolute direct summands [73] Pure subgroups [76] Bounded pure subgroups [79] Factor groups with respect to pure subgroups [81] Algebraically compact groups [83] Generalized pure subgroups [87] Neat subgroups [91] Exercises [93] Chapter V. Basic subgroups § 29. Existence of basic subgroups. The quasibasis [97] § 30. Properties of basic subgroups [101] §31. Different basic subgroups of a group [103] § 32. The basic subgroup as an endomorphic image [106] Exercises [103] Chapter VI. The structure of p-groups § 33. p-groups without elements of infinite height [111] § 34. Closed p-groups [114] § 35. The Ulm sequence [117] § 36. Zippin’s theorem [121] § 37. Ulm’s theorem [123] § 38. Construction of groups with a prescribed Ulm sequence [127] § 39. Non-isomorphic groups with the same Ulm sequence [134] § 40. Some applications [135] §41. Direct decompositions of p-groups [137] Exercises [141] Chapter VII. Torsion free groups § 42. The type of elements. Groups of rank 1 [145] § 43. Indecomposable groups [150] § 44. Torsion free groups over the p-adic integers [154] § 45. Countable torsion free groups [157] § 46. Completely decomposable groups [162] § 47. Complete direct sums of infinite cyclic groups. Slender groups [168] § 48. Homogeneous groups [173] § 49. Separable groups [176] Exercises [179] Chapter VIII. Mixed groups § 50. Splitting mixed groups [185] §51. Factor groups of free groups [192] § 52. A characterization of arbitrary groups by matrices [196] § 53. Groups over the p-adic integers [198] Exercises [200] Chapter IX. Homomorphism groups and endomorphism rings § 54. Homomorphism groups [205] § 55. Endomorphism rings [210] § 56. The endomorphism ring of p-groups [214] § 57. Endomorphism rings with special properties [218] § 58. Automorphism groups [221] § 59. Fully invariant subgroups [224] Exercises [227] Chapter X. Group extensions § 60. Extensions of groups [233] § 61. The group of extensions [236] § 62. Induced endomorphisms of the group of extensions [239] § 63. Structural properties of the group of extensions [243] Exercises [247] Chapter XI. Tensor products § 64. The tensor product [249] § 65. The structure of tensor products [254] Exercises [256] Chapter XII. The additive group of rings § 66. Ideals determined by the additive group [259] § 67. Multiplications on a group [261] § 68. Rings on direct sums of cyclic groups [263] § 69. Torsion rings [265] § 70. Torsion free rings [268] § 71. Nil groups and quasi nil groups [272] § 72. The additive group of Artinian rings [280] § 73. Artinian rings without subgroups of type p [283] § 74. The additive group of semi-simple and regular rings [286] § 75. The additive group of rings with maximum or restricted minimum condition [288] Exercises [290] Chapter XIII. The multiplicative group of fields § 76. Finite algebraic extensions of prime fields [295] § 77. Algebraically and real closed fields [297] Exercises [298] Chapter XIV. The lattice of subgroups § 78. Properties of the subgroup lattice [300] § 79. Projectivities. Projectivities of cyclic groups [303] § 80. Projectivities of torsion groups [305] §81. Projectivities of torsion free and mixed groups [309] § 82. Dualisms [311] Exercises [312] Chapter XV. Decompositions into direct sums of subsets § 83. Decompositions of cyclic groups [315] § 84. Decompositions into weakly periodic subsets [318] § 85. Decompositions into an infinity of components [324] Exercises [329] Chapter XVI. Various questions § 86. Hereditarily generating systems [332] § 87. Universal homomorphic images [336] § 88. Universal subgroups [341] § 89. A combinatorial problem [344] Exercises [350] Bibliography [353] Author index [363] Subject index [365]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 20 F951 (Browse shelf) | Available | A-434 |
TABLE OF CONTENTS --
Preface [5] --
Table of contents [9] --
Table of notations [12] --
Chapter 1. Basic concepts. The most important groups --
§ 1. Notation and terminology [13] --
§ 2. Direct sums [17] --
§ 3. Cyclic groups [22] --
§ 4. Quasicyclic groups [23] --
§ 5. The additive group of the rationals [25] --
§ 6. The p-adic integers [26] --
§ 7. Operator modules [27] --
§8. Linear independence and rank [29] --
Exercises [34] --
Chapter IL Direct sum. of cyclic groups --
% 9. Free (abelian) groups [37] --
§ 10. Finite and finitely generated groups [39] --
§ 11, Direct sums of cyclic p-groups [43] --
§ 12, Subgroups of direct sums of cyclic groups [45] --
§ 13. Two dual criteria for the basis [47] --
§ 14, Further criteria for the existence of a basis [50] --
Exercises [52] --
Chapter III. Divisible groups --
§ 15, Divisibility by integers in groups [57] --
i 16, Homomorphisms into divisible groups [59] --
i 17, Systems of linear equations over divisible groups [60] --
§ 18, The direct summand property of divisible groups [62] --
9 19, The structure theorem on divisible groups [64] --
f 20, Embedding in divisible groups [65] --
Exercises [67] --
Chapter iv, Direct summands and pure subgroups --
Direct summands [71] --
Absolute direct summands [73] --
Pure subgroups [76] --
Bounded pure subgroups [79] --
Factor groups with respect to pure subgroups [81] --
Algebraically compact groups [83] --
Generalized pure subgroups [87] --
Neat subgroups [91] --
Exercises [93] --
Chapter V. Basic subgroups --
§ 29. Existence of basic subgroups. The quasibasis [97] --
§ 30. Properties of basic subgroups [101] --
§31. Different basic subgroups of a group [103] --
§ 32. The basic subgroup as an endomorphic image [106] --
Exercises [103] --
Chapter VI. The structure of p-groups --
§ 33. p-groups without elements of infinite height [111] --
§ 34. Closed p-groups [114] --
§ 35. The Ulm sequence [117] --
§ 36. Zippin’s theorem [121] --
§ 37. Ulm’s theorem [123] --
§ 38. Construction of groups with a prescribed Ulm sequence [127] --
§ 39. Non-isomorphic groups with the same Ulm sequence [134] --
§ 40. Some applications [135] --
§41. Direct decompositions of p-groups [137] --
Exercises [141] --
Chapter VII. Torsion free groups --
§ 42. The type of elements. Groups of rank 1 [145] --
§ 43. Indecomposable groups [150] --
§ 44. Torsion free groups over the p-adic integers [154] --
§ 45. Countable torsion free groups [157] --
§ 46. Completely decomposable groups [162] --
§ 47. Complete direct sums of infinite cyclic groups. Slender groups [168] --
§ 48. Homogeneous groups [173] --
§ 49. Separable groups [176] --
Exercises [179] --
Chapter VIII. Mixed groups --
§ 50. Splitting mixed groups [185] --
§51. Factor groups of free groups [192] --
§ 52. A characterization of arbitrary groups by matrices [196] --
§ 53. Groups over the p-adic integers [198] --
Exercises [200] --
Chapter IX. Homomorphism groups and endomorphism rings --
§ 54. Homomorphism groups [205] --
§ 55. Endomorphism rings [210] --
§ 56. The endomorphism ring of p-groups [214] --
§ 57. Endomorphism rings with special properties [218] --
§ 58. Automorphism groups [221] --
§ 59. Fully invariant subgroups [224] --
Exercises [227] --
Chapter X. Group extensions --
§ 60. Extensions of groups [233] --
§ 61. The group of extensions [236] --
§ 62. Induced endomorphisms of the group of extensions [239] --
§ 63. Structural properties of the group of extensions [243] --
Exercises [247] --
Chapter XI. Tensor products --
§ 64. The tensor product [249] --
§ 65. The structure of tensor products [254] --
Exercises [256] --
Chapter XII. The additive group of rings --
§ 66. Ideals determined by the additive group [259] --
§ 67. Multiplications on a group [261] --
§ 68. Rings on direct sums of cyclic groups [263] --
§ 69. Torsion rings [265] --
§ 70. Torsion free rings [268] --
§ 71. Nil groups and quasi nil groups [272] --
§ 72. The additive group of Artinian rings [280] --
§ 73. Artinian rings without subgroups of type p [283] --
§ 74. The additive group of semi-simple and regular rings [286] --
§ 75. The additive group of rings with maximum or restricted minimum condition [288] --
Exercises [290] --
Chapter XIII. The multiplicative group of fields --
§ 76. Finite algebraic extensions of prime fields [295] --
§ 77. Algebraically and real closed fields [297] --
Exercises [298] --
Chapter XIV. The lattice of subgroups --
§ 78. Properties of the subgroup lattice [300] --
§ 79. Projectivities. Projectivities of cyclic groups [303] --
§ 80. Projectivities of torsion groups [305] --
§81. Projectivities of torsion free and mixed groups [309] --
§ 82. Dualisms [311] --
Exercises [312] --
Chapter XV. Decompositions into direct sums of subsets --
§ 83. Decompositions of cyclic groups [315] --
§ 84. Decompositions into weakly periodic subsets [318] --
§ 85. Decompositions into an infinity of components [324] --
Exercises [329] --
Chapter XVI. Various questions --
§ 86. Hereditarily generating systems [332] --
§ 87. Universal homomorphic images [336] --
§ 88. Universal subgroups [341] --
§ 89. A combinatorial problem [344] --
Exercises [350] --
Bibliography [353] --
Author index [363] --
Subject index [365] --
MR, 21 #5672
Click on an image to view it in the image viewer
There are no comments on this title.