Abelian Groups - László Fuchs [Springer]

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1 The Tensor Product 231 with the following property: if g W A C ! G is a bilinear function into any group G, then there is a unique homomorphism W A ˝ C ! G making the above triangle commute. This property determines A ˝ C up to isomorphism. Proof. It remains to prove the last statement. Assume there are a group B, anda bilinear function f W A C ! B with the same properties as A ˝ C and e. Bythe first part of the theorem, there is a unique homomorphism W A ˝ C ! B with e D f , and by the hypothesis on B, there is a unique W B ! A ˝ C with f D e. Hence e D e; f D f , and in view of the uniqueness property, we conclude that D 1 A˝C and D 1 B ,soA ˝ C Š B, as claimed. ut As mentioned above, A ˝ C is called the tensor product of A and C, and we will refer to the map e W A C ! A ˝ C as the tensor map. From the symmetry of the roles of A and C in the definition it follows right away that the correspondence a ˝ c 7! c ˝ a induces a natural isomorphism A ˝ C Š C ˝ A: It is easy to check, for arbitrary groups A; B; C; the associative law .A ˝ B/ ˝ C Š A ˝ .B ˝ C/ either by the universal property or by the equality of generators. Basic Properties Before stating some relevant properties of tensor products, we record a simple lemma that is instrumental in exploring tensor products. Lemma 1.2. Let a 2 A; c 2 C. (a) If m; n 2 Z such that ma D 0 D nc, then d.a ˝ c/ D 0 where d D gcdfm; ng; (b) if mja and mc D 0,thena˝ c D 0; (c) if mja and njc, then mnj.a ˝ c/. Proof. (a) If s; t 2 Z such that sm C tn D d,thend.a ˝ c/ D sma ˝ c C tna ˝ c D sma ˝ c C a ˝ tnc D 0. (b) If a D ma 0 .a 0 2 A/,thena ˝ c D ma 0 ˝ c D a 0 ˝ mc D 0. (c) If also c D nc 0 .c 0 2 C/, thena ˝ c D mn.a 0 ˝ c 0 /. ut Here is a short list of the most useful properties of tensor products. (Note how important an ingredient the universal property is in the proofs of (A)-(B).) The proofs of (C)-(H) are straightforward, and will be left to the reader. (A) There is a natural isomorphism Z ˝ C Š C for every group C. The elements of Z ˝ C may be brought to the simpler form P i .n i ˝ c i / D P i .1 ˝ n ic i / D 1 ˝ c for c D P i n ic i 2 C. Thus the map W c 7! 1 ˝ c is an epimorphism C ! Z ˝ C. Clearly, .m; c/ 7! mc is a bilinear map Z C ! C, so the universal property implies that there is a unique W Z ˝ C ! C acting as .1 ˝ c/ D c. Consequently, and are inverse to each other. (B) There is a natural isomorphism Z.m/ ˝ C Š C=mC for every integer m >0 and group C.
232 8 Tensor and Torsion Products Here again, W c 7! N1 ˝ c is an epimorphism C ! Z.m/ ˝ C where N1 D 1 C mZ. WehavemC Ker , since1 ˝ mc D m ˝ c D 0 ˝ c D 0. Now .n; c/ 7! nc C mC is a bilinear map Z.m/ C ! C=mC, so by the universal property, there is a homomorphism W Z.m/ ˝ C ! C=mC such that is the canonical map C ! C=mC. Thus Ker D mC. (C) The heights satisfy h p .a ˝ c/ h p .a/ C h p .c/ .a 2 A; c 2 C/. Thus if either A or C is p-divisible (resp. divisible), then so is A ˝ C. (D) If h p .a/ ! for a 2 A and C is a p-group, then a ˝ c D 0 for every c 2 C. Thus if A is p-divisible and C is a p-group, then A ˝ C D 0. (E) If a 2 mA and c 2 CŒm for some m 2 Z, thena˝ c D 0. (F) If either A or C is a p-group .torsion group/,thensoisA˝ C. (G) If A is a p-group and C is a q-group for different primes p; q, then A ˝ C D 0. Example 1.3. We have the following natural isomorphisms: Z ˝ Z.n/ Š Z.n/, Z.p n / ˝ Z.p m / Š Z.p k / with k D minfm; ng, andZ.n/ ˝ Z.m/ Š Z.d/ where d D gcdfm; ng. Example 1.4. Suppose A is a rational group. Then every x 2 A ˝ C can be written in the form x D a˝c with some a 2 A; c 2 C. In fact, as always, x D P n iD1 .a i˝c i / holds with a i 2 A; c i 2 C, but in the present case A is locally cyclic, i.e. there exists an a 2 A such that each a i is an integral multiple of a, saya i D m i a .m i 2 Z/. Thus x D X .m i a ˝ c i / D X .a ˝ m i c i / D a ˝ . X m i c i / where P m i c i D c 2 C. This form is not unique: if a D ma 0 ,thenalsox D a 0 ˝ mc. (H) It is very important to keep in mind that if B is a subgroup of A, then B ˝ C need not be a subgroup of A ˝ C. For instance, Z < Q,butZ ˝ Z.p/ Š Z.p/ is not a subgroup of Q ˝ Z.p/ D 0;or,Z.p/
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Springer Monographs in Mathematics
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László Fuchs Abelian Groups 123
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Dedicated with love to my wife Shul
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viii Preface to relevant publicatio
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x Preface on abelian groups. The re
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xii Contents 4 Divisibility and Inj
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xiv Contents 7 Whitehead Groups If
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Table of Notations Set Theory ; : s
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Table of Notations xix N .p/: Nunk
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Table of Notations xxi Q A;;A : di
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2 1 Fundamentals is finite, countab
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4 1 Fundamentals Theorem 1.2. In a
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6 1 Fundamentals (8) A superfluous
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8 1 Fundamentals A ! A=K acting as
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10 1 Fundamentals φ G ⏐ ↓ η 0
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12 1 Fundamentals Lemma 2.5 (Snake
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14 1 Fundamentals Exercises (1) Ass
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16 1 Fundamentals non-zero subgroup
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18 1 Fundamentals in Sect. 7 in Cha
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20 1 Fundamentals 4 Sets Since the
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22 1 Fundamentals It might be helpf
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24 1 Fundamentals This is an amazin
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26 1 Fundamentals of subgroups, ind
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28 1 Fundamentals If is an infinit
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30 1 Fundamentals For the proof of
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32 1 Fundamentals C3. Identity. For
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34 1 Fundamentals is exact in D; F
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36 1 Fundamentals 7 Linear Topologi
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38 1 Fundamentals Proof. (We need s
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40 1 Fundamentals For two R-modules
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Chapter 2 Direct Sums and Direct Pr
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1 Direct Sums and Direct Products 4
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2 Direct Summands 51 Proof. If A D
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2 Direct Summands 53 Exercises (1)
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3 Pull-Back and Push-Out Diagrams 5
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4 Direct Limits 57 In this case, A
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4 Direct Limits 59 We now move to t
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5 Inverse Limits 61 for 0 W G ! A
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5 Inverse Limits 63 for every i 2 I
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6 Direct Products vs. Direct Sums 6
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6 Direct Products vs. Direct Sums 6
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7 Completeness in Linear Topologies
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Chapter 3 Direct Sums of Cyclic Gro
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1 Freeness and Projectivity 77 Coro
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1 Freeness and Projectivity 79 Let
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2 Finite and Finitely Generated Gro
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3 Factorization of Finite Groups 87
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3 Factorization of Finite Groups 89
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4 Linear Independence and Rank 91 E
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4 Linear Independence and Rank 93 t
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5 Direct Sums of Cyclic Groups 95 P
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5 Direct Sums of Cyclic Groups 97 s
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6 Equivalent Presentations 99 (5) (
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6 Equivalent Presentations 101 Thes
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6 Equivalent Presentations 103 Equi
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7 Chains of Free Groups 105 For Ded
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7 Chains of Free Groups 107 We can
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7 Chains of Free Groups 109 Hill’
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7 Chains of Free Groups 111 which i
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8 Almost Free Groups 113 (D) Let 0
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8 Almost Free Groups 115 Proof. (a)
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8 Almost Free Groups 117 be done by
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8 Almost Free Groups 119 with. In t
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9 Shelah’s Singular Compactness T
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10 Groups with Discrete Norm 123 Di
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10 Groups with Discrete Norm 125 Ch
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11 Quasi-Projectivity 127 11 Quasi-
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11 Quasi-Projectivity 129 for diffe
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132 4 Divisibility and Injectivity
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150 5 Purity and Basic Subgroups or
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152 5 Purity and Basic Subgroups Pr
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154 5 Purity and Basic Subgroups ge
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158 5 Purity and Basic Subgroups Th
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160 5 Purity and Basic Subgroups ˛
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166 5 Purity and Basic Subgroups 5
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168 5 Purity and Basic Subgroups Le
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172 5 Purity and Basic Subgroups F
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174 5 Purity and Basic Subgroups (H
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176 5 Purity and Basic Subgroups Ma
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Page 202 and 203: Chapter 6 Algebraically Compact Gro
Page 204 and 205: 1 Algebraic Compactness 185 of the
Page 206 and 207: 1 Algebraic Compactness 187 Na j ;
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Page 210 and 211: 2 Complete Groups 191 is a Cauchy s
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Page 218 and 219: 4 Pure-Injective Hulls 199 (2) The
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Page 226 and 227: 6 The Exchange Property 207 Evident
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Page 232 and 233: Chapter 7 Homomorphism Groups Abstr
Page 234 and 235: 1 Groups of Homomorphisms 215 (and
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Page 238 and 239: 1 Groups of Homomorphisms 219 F Not
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Page 248 and 249: Chapter 8 Tensor and Torsion Produc
Page 252 and 253: 1 The Tensor Product 233 for matchi
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Page 262 and 263: 3 Theorems on Tensor Products 243 T
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Page 270 and 271: 5 Localization 251 (8) (Keef) If A
Page 272 and 273: 5 Localization 253 Exercises (1) If
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282 9 Groups of Extensions and Coto
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300 10 Torsion Groups (A) If C is a
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302 10 Torsion Groups the indicator
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304 10 Torsion Groups 0 −−−
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306 10 Torsion Groups is tantamount
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308 10 Torsion Groups Proof. We hav
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310 10 Torsion Groups (3) Let A be
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312 10 Torsion Groups (C) B is a ba
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314 10 Torsion Groups pure-injectiv
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316 10 Torsion Groups Theorem 3.11
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318 10 Torsion Groups 4 More on Tor
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320 10 Torsion Groups (g) The large
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322 10 Torsion Groups that in torsi
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324 10 Torsion Groups subgroup of B
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326 10 Torsion Groups (c) Homomorph
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328 10 Torsion Groups then Ker con
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330 10 Torsion Groups A class more
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332 10 Torsion Groups with basic su
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334 10 Torsion Groups 8 Valuated Ve
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336 10 Torsion Groups V D A ` B for
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338 10 Torsion Groups Proof. Define
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340 10 Torsion Groups When Socles D
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342 10 Torsion Groups Hill [1] gave
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344 11 p-Groups with Elements of In
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410 12 Torsion-Free Groups is calle
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412 12 Torsion-Free Groups Proof. L
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414 12 Torsion-Free Groups If A is
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416 12 Torsion-Free Groups Exercise
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418 12 Torsion-Free Groups Converse
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420 12 Torsion-Free Groups fa X j a
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422 12 Torsion-Free Groups and 0 !
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424 12 Torsion-Free Groups (ii) If
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426 12 Torsion-Free Groups number o
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428 12 Torsion-Free Groups along B
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430 12 Torsion-Free Groups whenever
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432 12 Torsion-Free Groups If we ha
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434 12 Torsion-Free Groups Then we
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436 12 Torsion-Free Groups (ii) For
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438 12 Torsion-Free Groups (7) (de
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440 12 Torsion-Free Groups it is ro
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442 12 Torsion-Free Groups 1 for `
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444 12 Torsion-Free Groups A D B i
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446 12 Torsion-Free Groups (5) For
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448 12 Torsion-Free Groups Proof. L
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450 12 Torsion-Free Groups local (T
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452 12 Torsion-Free Groups stable r
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454 12 Torsion-Free Groups (3) (War
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456 12 Torsion-Free Groups write x
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458 12 Torsion-Free Groups For thos
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460 12 Torsion-Free Groups A C 1
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462 12 Torsion-Free Groups ˛./.˛
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464 12 Torsion-Free Groups so the s
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466 12 Torsion-Free Groups followed
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468 12 Torsion-Free Groups (i) ther
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470 12 Torsion-Free Groups 11 Duali
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472 12 Torsion-Free Groups Lemma 11
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474 12 Torsion-Free Groups rk p D.A
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476 12 Torsion-Free Groups is proj(
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478 12 Torsion-Free Groups division
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Chapter 13 Torsion-Free Groups of I
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1 Direct Decompositions of Infinite
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2 Slender Groups 489 (6) A torsion-
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2 Slender Groups 491 Example 2.5. T
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2 Slender Groups 493 Proof. From Th
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2 Slender Groups 495 F Notes. The r
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3 Characterizations of Slender Grou
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3 Characterizations of Slender Grou
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4 Separable Groups 501 Exercises (1
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4 Separable Groups 503 where the G
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4 Separable Groups 505 In the follo
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4 Separable Groups 507 c C g D h 2
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5 Vector Groups 509 (7) Show that t
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5 Vector Groups 511 Isomorphism of
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6 Powers ofZ of Measurable Cardinal
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7 Whitehead Groups If V = L 519 (7)
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7 Whitehead Groups If V = L 521 Pro
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7 Whitehead Groups If V = L 523 Sˇ
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8 Whitehead Groups Under Martin’s
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8 Whitehead Groups Under Martin’s
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Chapter 14 Butler Groups Abstract T
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1 Finite Rank Butler Groups 531 and
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1 Finite Rank Butler Groups 533 Pro
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1 Finite Rank Butler Groups 535 Let
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2 Prebalanced and Decent Subgroups
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2 Prebalanced and Decent Subgroups
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3 The Torsion Extension Property 54
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4 Countable Butler Groups 547 (ii)
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5 B 1 -andB 2 -Groups 549 (F) Homog
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6 Solid Subgroups 551 understand th
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6 Solid Subgroups 553 fa n j n
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6 Solid Subgroups 555 Again changin
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7 Solid Chains 557 (4) A corank 1 s
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7 Solid Chains 559 B C1 =B admits
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7 Solid Chains 561 Lemma 7.6 (Bican
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8 Butler Groups of Uncountable Rank
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9 More on Infinite Butler Groups 56
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9 More on Infinite Butler Groups 57
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Chapter 15 Mixed Groups Abstract Mi
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1 Splitting Mixed Groups 575 For ev
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1 Splitting Mixed Groups 577 so A m
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2 Baer Groups are Free 579 (4) (Opp
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2 Baer Groups are Free 581 We note
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2 Baer Groups are Free 583 One does
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3 Valuated Groups. Height-Matrices
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4 Nice, Isotype, and Balanced Subgr
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5 Mixed Groups of Torsion-Free Rank
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5 Mixed Groups of Torsion-Free Rank
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6 Simply Presented Mixed Groups 597
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7 Warfield Groups 599 Exercises (1)
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7 Warfield Groups 601 (i) A=F is to
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7 Warfield Groups 603 Lemma 7.7 (St
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8 The Categories WALK and WARF 605
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8 The Categories WALK and WARF 607
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9 Projective Properties of Warfield
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9 Projective Properties of Warfield
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Chapter 16 Endomorphism Rings Abstr
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1 Endomorphism Rings 615 Lemma 1.5.
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1 Endomorphism Rings 617 For a subg
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1 Endomorphism Rings 619 Before sta
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1 Endomorphism Rings 621 Recently,
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2 Endomorphism Rings of p-Groups 62
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3 Endomorphism Rings of Torsion-Fre
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4 Endomorphism Rings of Special Gro
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5 Special Endomorphism Rings 637 se
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5 Special Endomorphism Rings 639 Pr
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5 Special Endomorphism Rings 641 Re
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6 Groups as Modules Over Their Endo
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7 Groups with Prescribed Endomorphi
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7 Groups with Prescribed Endomorphi
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656 17 Automorphism Groups Example
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658 17 Automorphism Groups linear g
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660 17 Automorphism Groups Exercise
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662 17 Automorphism Groups Accordin
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664 17 Automorphism Groups (2) If A
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666 17 Automorphism Groups (b) If G
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668 17 Automorphism Groups Then a 7
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670 17 Automorphism Groups for all
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Chapter 18 Groups in Rings and in F
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1 Additive Groups of Rings and Modu
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2 Rings on Groups 677 (11) For a ri
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2 Rings on Groups 679 is a subring,
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2 Rings on Groups 681 Beaumont-Lawv
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3 Additive Groups of Noetherian Rin
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4 Additive Groups of Artinian Rings
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4 Additive Groups of Artinian Rings
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5 Additive Groups of Regular Rings
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5 Additive Groups of Regular Rings
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6 E-Rings 693 (2) A group is isomor
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6 E-Rings 695 (F) Every endomorphis
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7 Groups of Units in Commutative Ri
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7 Groups of Units in Commutative Ri
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8 Multiplicative Groups of Fields 7
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References Books D.M. Arnold — [A
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References 709 D.M. Arnold, R. Hunt
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References 711 R. Burkhardt — [1]
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References 713 M. Dugas — [1] Fas
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References 715 A.A. Fomin — [1] T
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References 717 R. Göbel, S. Shelah
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References 719 P. Hill, M. Lane, C.
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References 721 A. Kertész — [1]
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References 723 A. Mader — [1] On
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References 725 L.G. Nongxa — [1]
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References 727 endomorphism rings.
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References 729 G.M. Tsukerman — [
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Author Index A Abel, N.H., vii Albr
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Author Index 733 Grinshpon, S.Ya.,
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Author Index 735 O’Neill, J.D., 3
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Subject Index A Abelian group, 1 Ab
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Subject Index 739 homomorphism, 37,
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Subject Index 741 Full rational gro
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Subject Index 743 Łoś-Eda theorem
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Subject Index 745 Rank 1 torsion-fr
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Subject Index 747 vector space, 334
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Abelian Groups - László Fuchs [Springer]

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