Abelian Groups - László Fuchs [Springer]

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368 11 p-<strong>Groups</strong> with Elements of Infinite Height .iii/ 0 ! B.u/ ! A.u/ ! C.u/ ! 0 is exact for all increasing sequences u D . 0 ; 1 ;:::; n ;:::/of ordinals and 1; .iv/ .p AŒp/ D p CŒp holds for each . Proof. The equivalence of .i/ and .ii/ has already been established above. To prove .ii/ ) .iii/; note that the map B.u/ ! A.u/ is monic, since h B .p n b/ D h A .p n b/ for all b 2 B and n 0. It is also clear that A.u/ C.u/. Inorderto prove that this is not a proper inclusion, we use induction on the order of c 2 C.u/, and show that p CŒp k .p AŒp k / for k 1. Fork D 1, pickc 2 p CŒp, and argue that there is an a 2 p A with a D c. Nowpa 2 p C1 A \ B D p C1 B implies that pb D pa for some b 2 p B.Since˛ maps a b 2 p AŒp upon c, the reverse inclusion follows. Assume the assertion is true for k 1 and for all . If c 2 p CŒp k .k >1/, then there is x 2 A such that .x/ D c, and by induction hypothesis, some y 2 p C1 AŒp k 1 satisfies .y/ D pc 2 p C1 CŒp k 1 . Choose an a 0 2 p AŒp k with pa 0 D y, and note that .x a 0 / D c .a 0 / 2 p CŒp. Consequently, c .a 0 / D .a 1 / for some a 1 2 p AŒp, andc D .a 0 C a 1 / with a 0 C a 1 2 p AŒp k . Hence .iii/ is exact at C.u/. .iv/ is a special case of .iii/. For the implication .iv/ ) .ii/; it remains to check that B\p A D p B for all . This being trivially true for D 0, suppose it holds for .Toverifyitfor C1,pick a b 2 B \ p C1 A and an a 0 2 p A with pa 0 D b.As.a 0 / 2 p CŒp, we can find an a 1 2 p AŒp such that .a 1 / D .a 0 /.Thena D a 0 a 1 2 B \ p A D p B, whence b D pa 2 p C1 B follows. The induction step for limit ordinals is automatic. ut The Extension Lemma We have come to a crucial lemma about extending maps that do not decrease heights. The idea is borrowed from Hill [11]. Lemma 5.6. Given a commutative diagram 0 −−−−→ N −−−−→ G ⏐ ⏐ ψ↓ ↓φ 0 −−−−→ B −−−−→ A α −−−−→ C −−−−→ 0 with exact rows, suppose that (i) B is a balanced subgroup of A; (ii) does not decrease heights .heights in N are computed in G/; (iii) there is a g 2 G proper with respect to N satisfying pg 2 N. Then can be extended to a map W hN; gi !A which does not decrease heights either, and satisfies ˛ .g/ D .g/.
5 Isotypeness, Balancedness, and Balanced-Projectivity 369 Proof. Using the notation h.g/ D , (i) implies that some a 2 p A satisfies ˛a D g. Visibly, pa .pg/ 2 B \ p C1 A D p C1 B, whence the existence of a b 2 p B with pb D pa .pg/ is immediate. To define , it suffices to specify N D and g D a b 2 A. It is readily seen that this is a homomorphism hN; gi !A such that ˛ g D ˛a D g. Obviously, h.a b/ D h.g/. In order to check that does not decrease heights, it is enough to show that h.g C x/ h.a b C x/ for every x 2 N. From (iii) we infer that, on one hand, h.g C x/ D minfh.g/; h.x/g, and, on the other hand, h.a b C x/ minfh.a b/; h. x/g minf; h.x/g.This completes the proof. ut Balanced-Projective p-<strong>Groups</strong> Fundamental to our treatment is the concept of balanced-projectivity. A p-group G will be called balanced-projective if it has the projective property with respect to balanced-exact sequences of p-groups. Our next aim is to characterize these p-groups, and at the same time to obtain relevant information about groups with nice composition chains. The following two results are preludes to Theorem 5.9 below. (Recall the H denotes the th generalized Prüfer group.) Lemma 5.7 (Nunke [5]). Suppose A is a p-group, and a 2 Aisoforder p n and of height . Then there exists a homomorphism W H Cn ! A such that z D a where hzi Dp H Cn is cyclic of order p n . Proof. The correspondence z 7! a gives rise to a homomorphism W p H Cn !hai that does not decrease heights, computed in H Cn and A, respectively. It remains to refer to Corollary 4.6 to complete the proof. ut Proposition 5.8. Let A be a reduced p-group of length . There exist a group G that is a direct sum of generalized Prüfer groups of lengths , and a balanced-exact sequence 0 ! K ! G ! A ! 0: (11.8) Proof. For a 2 A of order p n and of height , we select a copy G a of H Cn and amap a W G a ! A as described in Lemma 5.7. Define the group G D˚a2A G a , and the map Dr.˚ a / W G ! A. It is evident that .p HŒp/ D p AŒp for every , so that by virtue of Proposition 5.5 we can claim with confidence that Ker D K is balanced in G. ut The next theorem will show that the generalized Prüfer groups are balancedprojective. Consequently, (11.8) may be viewed as a balanced-projective resolution of A. We now have all the tools for the core theorem that completely describes the balanced-projective p-groups. Main portions of this result are due to Hill [7]. Theorem 5.9. For a reduced p-group A, the following conditions are equivalent:
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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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Chapter 7 Homomorphism Groups Abstr
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1 Groups of Homomorphisms 215 (and
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1 Groups of Homomorphisms 217 is an
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1 Groups of Homomorphisms 219 F Not
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2 Algebraically Compact Homomorphis
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2 Algebraically Compact Homomorphis
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3 Small Homomorphisms 225 (3) The a
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3 Small Homomorphisms 227 completio
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Chapter 8 Tensor and Torsion Produc
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1 The Tensor Product 231 with the f
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1 The Tensor Product 233 for matchi
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1 The Tensor Product 235 finitely P
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2 The Torsion Product 237 (6) (a) I
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2 The Torsion Product 239 It is pre
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2 The Torsion Product 241 It remain
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3 Theorems on Tensor Products 243 T
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4 Theorems on Torsion Products 245
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5 Localization 251 (8) (Keef) If A
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5 Localization 253 Exercises (1) If
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256 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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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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