Abelian Groups - László Fuchs [Springer]

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6 1 Fundamentals (8) A superfluous subgroup is contained in every maximal subgroup. (9) (B.H. Neumann) Let a i C B i .i D 1;:::;n/ be cosets of subgroups B i of A such that their set union is all of A. Then one of B i has finite index in A.[Hint: induct on n.] (10) Let B be a subgroup of A.(a)IfB is torsion, then B tA.(b)IfA=B is torsionfree, then tA B. (11) For every n 2 N, wehaven.tA/ D t.nA/. (12) Prove the triangle inequality for the heights: h p .a C b/ minfh p .a/; h p .b/g for the elements a; b of any group A. Equality holds if h p .a/ ¤ h p .b/. (13) (a) Prove the inclusion relations .B \ C/ C .B \ D/ B \ .C C D/ and B C .C \ D/ .B C C/ \ .B C D/ for subgroups B; C; D of a group A. (b) Find examples where proper inclusions hold. 2 Maps and Diagrams Homomorphisms Let A and B be arbitrary groups. A homomorphism ˛ W A ! B (often denoted as A ˛! B) is a function that associates with every element a 2 A a unique element b 2 B, written as ˛.a/ D b (or simply as ˛a D b), such that it preserves addition: ˛.a 1 C a 2 / D ˛.a 1 / C ˛.a 2 / for all a 1 ; a 2 2 A: A is the domain and B is the codomain or range of ˛. If there is no need to name the homomorphism, then we write simply A ! B. A homomorphism ˛ W A ! B gives rise to two subgroups: Ker ˛ A and Im ˛ B. Ker˛ is the kernel of ˛: thesetofalla 2 A with ˛a D 0, while Im ˛, theimage of ˛, consists of all b 2 B such that there is an a 2 A with ˛a D b. The factor group B= Im ˛ is called the cokernel of ˛; notation: Coker ˛. If Im ˛ D B, then˛ is surjective or epic; wealsosaythatitisanepimorphism. If Ker ˛ D 0, ˛ is said to be injective or monic; also,˛ is a monomorphism. If both Im ˛ D B and Ker ˛ D 0, then˛ is a bijection; in this case, it is called an isomorphism. The groups A and B are isomorphic (denoted as A Š B)ifthereisan isomorphism ˛ W A ! B. Then the inverse map ˛ 1 W B ! A exists, and is again an isomorphism. Abstractly, no distinction is made between isomorphic groups, unless they are distinct subgroups of the same larger group under consideration. If G is a subgroup of both A and B, andif˛ W A ! B fixes the elements of G, then˛ is a homomorphism over G.
2 Maps and Diagrams 7 A homomorphism ˛ with Im ˛ D 0 is referred to as a zero homomorphism; it will be denoted by 0. IfA B, then the map that assigns to every a 2 A itself may be regarded as a homomorphism of A into B; itiscalledtheinjection or inclusion map. The injection 0 ! A is the unique homomorphism of the group 0 into A. If ˛ W A ! B and C A, then the restriction D ˛ C has the domain C and codomain B. In this case, ˛ is viewed as an extension of , written ˛. If ˛0 <
Page 1 and 2: Springer Monographs in Mathematics
Page 4 and 5: László Fuchs Abelian Groups 123
Page 6: Dedicated with love to my wife Shul
Page 9 and 10: viii Preface to relevant publicatio
Page 11 and 12: x Preface on abelian groups. The re
Page 13 and 14: xii Contents 4 Divisibility and Inj
Page 15 and 16: xiv Contents 7 Whitehead Groups If
Page 18 and 19: Table of Notations Set Theory ; : s
Page 20 and 21: Table of Notations xix N .p/: Nunk
Page 22 and 23: Table of Notations xxi Q A;;A : di
Page 24 and 25: 2 1 Fundamentals is finite, countab
Page 26 and 27: 4 1 Fundamentals Theorem 1.2. In a
Page 30 and 31: 8 1 Fundamentals A ! A=K acting as
Page 32 and 33: 10 1 Fundamentals φ G ⏐ ↓ η 0
Page 34 and 35: 12 1 Fundamentals Lemma 2.5 (Snake
Page 36 and 37: 14 1 Fundamentals Exercises (1) Ass
Page 38 and 39: 16 1 Fundamentals non-zero subgroup
Page 40 and 41: 18 1 Fundamentals in Sect. 7 in Cha
Page 42 and 43: 20 1 Fundamentals 4 Sets Since the
Page 44 and 45: 22 1 Fundamentals It might be helpf
Page 46 and 47: 24 1 Fundamentals This is an amazin
Page 48 and 49: 26 1 Fundamentals of subgroups, ind
Page 50 and 51: 28 1 Fundamentals If is an infinit
Page 52 and 53: 30 1 Fundamentals For the proof of
Page 54 and 55: 32 1 Fundamentals C3. Identity. For
Page 56 and 57: 34 1 Fundamentals is exact in D; F
Page 58 and 59: 36 1 Fundamentals 7 Linear Topologi
Page 60 and 61: 38 1 Fundamentals Proof. (We need s
Page 62 and 63: 40 1 Fundamentals For two R-modules
Page 64 and 65: Chapter 2 Direct Sums and Direct Pr
Page 66 and 67: 1 Direct Sums and Direct Products 4
Page 72 and 73: 2 Direct Summands 51 Proof. If A D
Page 74 and 75: 2 Direct Summands 53 Exercises (1)
Page 76 and 77: 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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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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