Abelian Groups - László Fuchs [Springer]

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<strong>Springer</strong> Monographs in Mathematics More information about this series at http://www.springer.com/series/3733
Page 1: Springer Monographs in Mathematics
Page 5 and 6: László Fuchs Mathematics Departme
Page 8 and 9: Preface The theory of abelian group
Page 10 and 11: Preface ix of revising it, but deci
Page 12 and 13: Contents 1 Fundamentals ...........
Page 14 and 15: Contents xiii 10 Torsion Groups ...
Page 16: Contents xv 7 Groups of Units in Co
Page 19 and 20: xviii Table of Notations MA: Martin
Page 21 and 22: xx Table of Notations o.a/: order o
Page 23 and 24: Chapter 1 Fundamentals Abstract The
Page 25 and 26: 1 Basic Definitions 3 Example 1.1.
Page 27 and 28: 1 Basic Definitions 5 A subgroup E
Page 29 and 30: 2 Maps and Diagrams 7 A homomorphis
Page 31 and 32: 2 Maps and Diagrams 9 μ A −−
Page 33 and 34: 2 Maps and Diagrams 11 Proof. If W
Page 35 and 36: 2 Maps and Diagrams 13 The 5- and t
Page 37 and 38: 3 Fundamental Examples 15 3 Fundame
Page 39 and 40: 3 Fundamental Examples 17 In an ele
Page 41 and 42: 3 Fundamental Examples 19 Exercises
Page 43 and 44: 4 Sets 21 Ordinals and Cardinals We
Page 45 and 46: 4 Sets 23 The principal filter gene
Page 47 and 48: 5 Families of Subgroups 25 the coun
Page 49 and 50: 5 Families of Subgroups 27 We shall
Page 51 and 52: 5 Families of Subgroups 29 show tha
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6 Categories of Abelian Groups 31 F
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6 Categories of Abelian Groups 33 (
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6 Categories of Abelian Groups 35 E
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7 Linear Topologies 37 Example 7.1.
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8 Modules 39 (6) The group J p is c
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8 Modules 41 (5) Let S ! R be a rin
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44 2 Direct Sums and Direct Product
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76 3 Direct Sums of Cyclic Groups W
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78 3 Direct Sums of Cyclic Groups P
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80 3 Direct Sums of Cyclic Groups (
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82 3 Direct Sums of Cyclic Groups T
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84 3 Direct Sums of Cyclic Groups F
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86 3 Direct Sums of Cyclic Groups (
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88 3 Direct Sums of Cyclic Groups P
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90 3 Direct Sums of Cyclic Groups a
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92 3 Direct Sums of Cyclic Groups M
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98 3 Direct Sums of Cyclic Groups L
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100 3 Direct Sums of Cyclic Groups
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Chapter 4 Divisibility and Injectiv
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1 Divisibility 133 infinite height,
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2 Injective Groups 135 Theorem 2.1
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2 Injective Groups 137 write E D C
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2 Injective Groups 139 F Notes. The
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3 Structure Theorem on Divisible Gr
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4 Systems of Equations 143 (11) A g
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5 Finitely Cogenerated Groups 145 F
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5 Finitely Cogenerated Groups 147 F
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Chapter 5 Purity and Basic Subgroup
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1 Purity 151 called the pure subgro
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1 Purity 153 Honda [1] introduced a
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2 Theorems on Pure Subgroups 155 (1
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2 Theorems on Pure Subgroups 157 Pr
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3 Pure-Exact Sequences 159 (6) (Cut
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3 Pure-Exact Sequences 161 in Theor
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4 Pure-Projectivity and Pure-Inject
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4 Pure-Projectivity and Pure-Inject
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5 Basic Subgroups 167 Lemma 5.1. A
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5 Basic Subgroups 169 of the p-pure
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5 Basic Subgroups 171 A n as the to
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6 Theorems on p-Basic Subgroups 173
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184 6 Algebraically Compact Groups
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214 7 Homomorphism Groups (B) Hom.A
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216 7 Homomorphism Groups Proof. Ap
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218 7 Homomorphism Groups means tha
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220 7 Homomorphism Groups 2 Algebra
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222 7 Homomorphism Groups If we wri
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224 7 Homomorphism Groups Theorem 2
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226 7 Homomorphism Groups (D) The f
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228 7 Homomorphism Groups Exercises
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230 8 Tensor and Torsion Products u
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232 8 Tensor and Torsion Products H
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234 8 Tensor and Torsion Products .
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236 8 Tensor and Torsion Products a
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238 8 Tensor and Torsion Products .
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240 8 Tensor and Torsion Products T
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242 8 Tensor and Torsion Products (
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244 8 Tensor and Torsion Products L
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246 8 Tensor and Torsion Products T
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248 8 Tensor and Torsion Products w
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250 8 Tensor and Torsion Products h
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252 8 Tensor and Torsion Products (
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Chapter 9 Groups of Extensions and
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1 Group Extensions 257 μ −−−
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1 Group Extensions 259 E-morphism e
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2 Exact Sequences for Hom and Ext 2
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3 Basic Properties of Ext 267 Eleme
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3 Basic Properties of Ext 269 Examp
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4 Lemmas on Ext 271 .where is any
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4 Lemmas on Ext 273 Case 1. If is
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5 The Functor Pext 275 F Notes. Lem
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5 The Functor Pext 277 Thus Pext.C;
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5 The Functor Pext 279 Pext and Alg
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5 The Functor Pext 281 (ii) (Schoch
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6 Cotorsion Groups 283 Thus the mid
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6 Cotorsion Groups 285 groups are b
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7 Cotorsion vs. Torsion 287 (8) (Wa
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7 Cotorsion vs. Torsion 289 Theorem
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8 More on Ext 291 8 More on Ext Cot
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8 More on Ext 293 Ext is exactly th
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9 Cotorsion Hull and Torsion-Free C
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9 Cotorsion Hull and Torsion-Free C
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Chapter 10 Torsion Groups Abstract
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1 Preliminaries on p-Groups 301 We
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1 Preliminaries on p-Groups 303 Arb
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1 Preliminaries on p-Groups 305 0
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2 Fully Invariant and Large Subgrou
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2 Fully Invariant and Large Subgrou
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3 Torsion-Complete Groups 311 Separ
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3 Torsion-Complete Groups 313 In pa
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3 Torsion-Complete Groups 315 Corol
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3 Torsion-Complete Groups 317 Exerc
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4 More on Torsion-Complete Groups 3
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5 Pure-Complete and Quasi-Complete
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5 Pure-Complete and Quasi-Complete
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6 Thin and Thick Groups 325 It was
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6 Thin and Thick Groups 327 J 1 ;::
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7 Direct Decompositions of Separabl
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8 Valuated Vector Spaces 335 Catego
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8 Valuated Vector Spaces 337 (C) Ev
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9 Separable p-Groups That Are Deter
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9 Separable p-Groups That Are Deter
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Chapter 11 p-Groups with Elements o
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1 The Ulm-Zippin Theory 345 Lemma 1
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1 The Ulm-Zippin Theory 347 Proof.
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1 The Ulm-Zippin Theory 349 of C i
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1 The Ulm-Zippin Theory 351 There a
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2 Nice Subgroups 353 We can rephras
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3 Simply Presented p-Groups 355 pre
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3 Simply Presented p-Groups 357 whe
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3 Simply Presented p-Groups 359 The
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3 Simply Presented p-Groups 361 8
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4 p-Groups with Nice Systems 363 Pr
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5 Isotypeness, Balancedness, and Ba
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6 Totally Projective p-Groups 371 (
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6 Totally Projective p-Groups 373 L
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6 Totally Projective p-Groups 375 P
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7 Subgroups of Totally Projective p
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8 p -Purity 385 then B has an H./-
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8 p -Purity 387 p B D B \ p A fo
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8 p -Purity 389 satisfies (11.12)f
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8 p -Purity 391 (iii) A=B is a div
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9 The Functor p 393 .c/ A p-group
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9 The Functor p 395 Proof. If 2 N
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10 p !Cn -Projective p-Groups 397 (
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10 p !Cn -Projective p-Groups 399 W
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11 Summable p-Groups 401 (6) (Danch
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12 Elongations of p-Groups 403 (2)
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12 Elongations of p-Groups 405 BŒp
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12 Elongations of p-Groups 407 be s
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Chapter 12 Torsion-Free Groups Abst
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1 Characteristic and Type: Finite R
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2 Balanced Subgroups 417 generating
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2 Balanced Subgroups 419 trivial ma
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2 Balanced Subgroups 421 there is a
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3 Completely Decomposable Groups 42
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4 Indecomposable Groups 431 (7) (Pr
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4 Indecomposable Groups 433 Proof.
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4 Indecomposable Groups 435 Lemma 4
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4 Indecomposable Groups 437 for pur
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5 Pathological Direct Decomposition
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6 Direct Decompositions of Finite R
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7 Substitution Properties 449 The J
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7 Substitution Properties 451 Examp
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7 Substitution Properties 453 1 0 D
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8 Finite Rank p-Local Groups 455 A
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8 Finite Rank p-Local Groups 457 We
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9 Quasi-Isomorphism 459 Actually,
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9 Quasi-Isomorphism 461 It is well
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9 Quasi-Isomorphism 463 ./ If N is
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10 Near-Isomorphism 465 (9) (Murley
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10 Near-Isomorphism 467 if they sat
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10 Near-Isomorphism 469 F Notes. Th
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11 Dualities for Finite Rank Groups
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11 Dualities for Finite Rank Groups
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12 More on Finite Rank Groups 475 E
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12 More on Finite Rank Groups 477 (
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12 More on Finite Rank Groups 479 (
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482 13 Torsion-Free Groups of Infin
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530 14 Butler Groups and .0; 1; 1;0
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532 14 Butler Groups Theorem 1.7 (B
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534 14 Butler Groups Example 1.10.
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536 14 Butler Groups that are ratio
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538 14 Butler Groups the subset of
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540 14 Butler Groups with C torsion
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542 14 Butler Groups Hom.G; T/ ! Ho
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544 14 Butler Groups with exact row
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546 14 Butler Groups 4 Countable Bu
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548 14 Butler Groups it clear: coun
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550 14 Butler Groups The next resul
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552 14 Butler Groups general than t
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554 14 Butler Groups (ii) Let A 1 <
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556 14 Butler Groups A ⏐ ↓ 0
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558 14 Butler Groups (i) there exis
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560 14 Butler Groups 0 −−−−
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562 14 Butler Groups that G D B C S
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564 14 Butler Groups The claim is v
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566 14 Butler Groups t ^ t ) with
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568 14 Butler Groups [3] shows that
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570 14 Butler Groups Superdecomposa
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572 14 Butler Groups in sharpening
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574 15 Mixed Groups Non-Splitting M
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576 15 Mixed Groups The dual proble
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578 15 Mixed Groups Conversely, ass
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580 15 Mixed Groups We start with a
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582 15 Mixed Groups Proof. Necessit
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584 15 Mixed Groups (10) (Megibben)
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586 15 Mixed Groups We denote by W
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588 15 Mixed Groups Thus for each p
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590 15 Mixed Groups 4 Nice, Isotype
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592 15 Mixed Groups is exact. H-exa
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594 15 Mixed Groups We have come to
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596 15 Mixed Groups (2) (Megibben)
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598 15 Mixed Groups (ii) For the pr
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600 15 Mixed Groups Define G to be
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602 15 Mixed Groups Conversely, let
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604 15 Mixed Groups By making use o
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606 15 Mixed Groups Lemma 8.1. The
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608 15 Mixed Groups [1] construct a
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610 15 Mixed Groups f .a/ in G is a
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612 15 Mixed Groups Exercises (1) (
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614 16 Endomorphism Rings The endom
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616 16 Endomorphism Rings In the ne
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618 16 Endomorphism Rings Theorem 1
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620 16 Endomorphism Rings from ˛ D
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622 16 Endomorphism Rings 2 Endomor
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624 16 Endomorphism Rings Consider
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626 16 Endomorphism Rings H.A/ seem
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628 16 Endomorphism Rings rank 1 wi
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630 16 Endomorphism Rings (b) For e
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632 16 Endomorphism Rings are inver
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634 16 Endomorphism Rings 4 Endomor
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638 16 Endomorphism Rings into, and
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640 16 Endomorphism Rings A ! A whi
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642 16 Endomorphism Rings For the s
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644 16 Endomorphism Rings Proof. Th
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646 16 Endomorphism Rings Example 6
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650 16 Endomorphism Rings Vinsonhal
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652 16 Endomorphism Rings The Torsi
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Chapter 17 Automorphism Groups Abst
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1 Automorphism Groups 657 (h) (Baer
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1 Automorphism Groups 659 (j) An in
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2 Automorphism Groups of p-Groups 6
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2 Automorphism Groups of p-Groups 6
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3 Automorphism Groups of Torsion-Fr
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674 18 Groups in Rings and in Field
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708 References A. Mader — [Ma] Al
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710 References K. Benabdallah, J.M.
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712 References A.L.S. Corner, B. Go
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714 References vol. 76 (North Holla
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716 References L. Fuchs, K.H. Hofma
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718 References J.T. Hallett, K.A. H
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720 References A.V. Ivanov — [1]
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722 References A.G. Kurosh — [1]
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724 References A. Mekler, S. Shelah
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726 References groups, in Proceedin
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728 References pp. 384-402. — [3]
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730 References G.V. Wilson — [1]
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732 Author Index Chase, S.U., 66, 6
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734 Author Index Król, M., 512, 51
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736 Author Index Stratton, A.E., 57
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738 Subject Index Balancedexact seq
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740 Subject Index Embedding in alge
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742 Subject Index Injective group,
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744 Subject Index p-nice subgroup,
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746 Subject Index Subsocle, 300 Sub
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Abelian Groups - László Fuchs [Springer]

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