Abelian Groups - László Fuchs [Springer]

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264 9 <strong>Groups</strong> of Extensions and Cotorsion <strong>Groups</strong> Proof. The proof relies on the Snake Lemma 2.5 in Chapter 1. We first choose free resolutions 0 ! H 1 ! F 1 ! A ! 0 and 0 ! H 2 ! F 2 ! C ! 0, and note that there is a free resolution 0 ! H 1 ˚ H 2 ! F 1 ˚ F 2 ! B ! 0 (see Sect. 1, Exercise 7 in Chapter 3). These are used in the commutative diagram 0 −−−−−→ Hom(C, G) ⏐ ↓ 0 −−−−−→ Hom(F 2 ,G) ⏐ ↓α 0 −−−−−→ Hom(H 2 ,G) ⏐ ↓ Ext(C, G) ρ −−−−−→ 1 σ Hom(B, G) −−−−−→ 1 Hom(A, G) ⏐ ⏐ ↓ ↓ μ −−−−−→ Hom(F 1 ⊕ F 2 ,G) ⏐ β↓ ν −−−−−→ Hom(F 1 ,G) −−−−−→ 0 ⏐ ↓ γ μ −−−−−→ ′ ν Hom(H 1 ⊕ H 2 ,G) −−−−−→ ′ Hom(H 1 ,A) −−−−−→ 0 ⏐ ⏐ ↓ ↓ ρ 2 −−−−−→ Ext(B, G) σ 2 −−−−−→ Ext(A, G) −−−−−→ 0 where the middle rows are exact, because F i ; H i are free groups, while the columns (bordered with 0’s, not shown) are exact, thanks to Lemma 2.2. By Lemma 2.5 in Chapter 1,(9.7) follows from the diagram. The proof of (9.8) is similar. ut Purity in the Exact Sequence on Ext We complement Theorem 2.3 with the following result where the starting short exact sequence is pure-exact. Lemma 2.4. Suppose 0 ! A ˛!B ˇ!C ! 0 is a pure-exact sequence. Then, for every group G, the induced homomorphisms ˇ W Ext.C; G/ ! Ext.B; G/ and ˛ W Ext.G; A/ ! Ext.G; B/ map onto pure subgroups. Proof. We give a detailed proof for the first part (using Theorem 5.2 below), and leave the rest to the reader as an exercise. In the following commutative diagram, we start from the bottom row and then form the preceding rows by using the maps in the right column as indicated: βσρ : 0 −−−−−→ G −−−−−→ K ′ −−−−−→ C[n] −−−−−→ 0 ⏐ ⏐ ∥ ↓ ↓ ρ βσ : 0 −−−−−→ G −−−−−→ K −−−−−→ B[n] −−−−−→ 0 ⏐ ⏐ ∥ ↓ ↓σ β : 0 −−−−−→ G −−−−−→ H ′ −−−−−→ B −−−−−→ 0 (∈ Im β ∗ ) ⏐ ⏐ ∥ ↓ ↓β : 0 −−−−−→ G −−−−−→ H −−−−−→ C −−−−−→ 0
2 Exact Sequences for Hom and Ext 265 Here the rows are exact, denotes the injection map, and exists, due to the purity of the given exact sequence. The composite map ˇW CŒn ! C will be the natural injection. If the last but one row belongs to n Ext.B; G/, then Theorem 5.2 below shows that the second row splits, and hence so does the top row. This means that the bottom exact sequence belongs to n Ext.C; G/, so the last but one row is in n Im ˇ. This holds for every n 2 N, thus Im ˇ is pure in Ext.B; G/. ut Ext and Direct Sums, Products We have to find out how Ext behaves towards direct sums and products. In view of Theorem 2.3, it should not come as a surprise that Ext imitates Hom in this respect. Theorem 2.5. For all groups A; A i ; C; C i , there exist natural isomorphisms Ext.˚i2I C i ; A/ Š Y i2I Ext.C i ; A/; (9.9) Ext C; Y ! A i Š Y Ext.C; A i /: (9.10) i2I i2I Proof. We prove (9.9), the proof of (9.10) runs dually. We start with free resolutions of the C i , 0 ! H i ! F i ! C i ! 0 with F i free, to obtain the exact sequences Hom.F i ; A/ ! Hom.H i ; A/ ! Ext.C i ; A/ ! Ext.F i ; A/ D 0. The exact sequence 0 ! ˚iH i ! ˚iF i ! ˚iC i ! 0 induces the top exact sequence in the commutative diagram Hom(⊕ i F i ,A) −−−−−→ Hom(⊕ i H i ,A) −−−−−→ Ext(⊕ i C i ,A) −−−−−→ 0 ⏐ ⏐ ⏐ ↓ ↓ ↓ ∏ i Hom(F i,A) −−−−−→ ∏ i Hom(H i,A) −−−−−→ ∏ i Ext(C i,A) −−−−−→ 0 We know from Theorem 1.7 in Chapter 7 that the first two vertical maps represent natural isomorphisms, whence we can conclude that there is a natural isomorphism between the two Exts in the diagram. ut F Notes. The long exact sequence for Hom-Ext generalizes for modules over arbitrary rings. However, the 0 at the right end is then replaced by sequences of higher Exts. The functor Ext does not behave in the same way towards direct and inverse limits as Hom. If fC i .i 2 I/j j i g is a direct system of groups with direct limit C,thenfExt.C i; A/.i 2 I/j Ext. j i ; 1 A/g is an inverse system. The most we can say in general is that there is a natural homomorphism W Ext.C; A/ ! lim Ext.C i ; A/. For the special case when the direct system is indexed by the natural numbers, see Lemma 5.9. In general, the so-called Mittag-Leffler condition guarantees the surjectivity of . Göbel–Prelle [1] ask for groups G with the properties like Ext. Q A i ; G/ Š Q i Ext.A i ; G/ for all choices of the A i , and show that “only G Š Q” is the answer.
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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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180 5 Purity and Basic Subgroups So
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Chapter 6 Algebraically Compact Gro
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1 Algebraic Compactness 185 of the
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1 Algebraic Compactness 187 Na j ;
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1 Algebraic Compactness 189 Here jG
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2 Complete Groups 191 is a Cauchy s
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2 Complete Groups 193 A ⏐ μ A A
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3 The Structure of Algebraically Co
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3 The Structure of Algebraically Co
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4 Pure-Injective Hulls 199 (2) The
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4 Pure-Injective Hulls 201 subgroup
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5 Locally Compact Groups 203 5 Loca
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5 Locally Compact Groups 205 Theore
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6 The Exchange Property 207 Evident
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6 The Exchange Property 209 In the
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6 The Exchange Property 211 to the
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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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