Abelian Groups - László Fuchs [Springer]

More documents

Recommendations

Info

8 Almost Free <strong>Groups</strong> 113 (D) Let 0 D A 0 < < A < < A D A be a smooth chain of groups such that all the factor groups A C1 =A are -free. Then A is also -free. In fact, let X be a pure subgroup of A with jXj
114 3 Direct Sums of Cyclic <strong>Groups</strong> An immediate consequence is the following result that shows that in countable groups free summands can be collected into a single summand. Corollary 8.3 (K. Stein). A countable torsion-free group A can be decomposed as A D F ˚ N where F is a free group, and N has no free factor group. N is uniquely determined by A. Proof. Define N as the intersection of the kernels of all homomorphisms W A ! Z. Then A=N is isomorphic to a countable subgroup of the direct product Q Z,soit is free in view of Theorem 8.2. N is then a summand of A, and we have A D F ˚ N with F free. From the definition of N it is evident that N cannot have a non-trivial map into Z. ut The next two examples show that there exist very large @ 1 -free groups A such that Hom.A; Z/ D 0, and it may also happen that an @ 1 -free group is isomorphic to the countable direct sum and to the countable direct product of itself. Example 8.4 (G. Reid [1]). Let @ 1 be a non-measurable cardinal, and N the subgroup of Z consisting of vectors with countable support. Then A D Z =N is @ 1 -free. To see this, let b n C N .n < !/ be a list of elements in a countable subgroup F of A. Clearly, each b n has uncountable support, and each sum b i C b j is equal to some b k modulo a countable index set. Thus if we change the representatives b n by dropping all the indices in these countably many index sets, then the new representatives form a subgroup F 0 Š F. By Theorem 8.2, F 0 is free, so A is @ 1 -free. A homomorphism W A ! Z maybeviewedasamap W Z ! Z such that .N/ D 0. By Theorem 2.8 in Chapter 13, D 0, which means Hom.A; Z/ D 0. Example 8.5. There exists an @ 1 -free group which is not free and isomorphic both to the direct sum of countably many copies of itself, and to the direct product of countable many copies of itself. See Proposition 4.9 in Chapter 13. Strongly -Free The study of almost free groups brings a stronger version of -freeness into the picture. Let be a regular cardinal. A group A is said to be strongly -free if every subgroup of cardinality
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 28 and 29:
6 1 Fundamentals (8) A superfluous
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 68 and 69:
Page 70 and 71:
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
Page 78 and 79:
4 Direct Limits 57 In this case, A
Page 80 and 81:
4 Direct Limits 59 We now move to t
Page 82 and 83:
5 Inverse Limits 61 for 0 W G ! A
Page 84 and 85: 5 Inverse Limits 63 for every i 2 I
Page 86 and 87: 6 Direct Products vs. Direct Sums 6
Page 88 and 89: 6 Direct Products vs. Direct Sums 6
Page 90 and 91: 7 Completeness in Linear Topologies
Page 96 and 97: Chapter 3 Direct Sums of Cyclic Gro
Page 98 and 99: 1 Freeness and Projectivity 77 Coro
Page 100 and 101: 1 Freeness and Projectivity 79 Let
Page 102 and 103: 2 Finite and Finitely Generated Gro
Page 108 and 109: 3 Factorization of Finite Groups 87
Page 110 and 111: 3 Factorization of Finite Groups 89
Page 112 and 113: 4 Linear Independence and Rank 91 E
Page 114 and 115: 4 Linear Independence and Rank 93 t
Page 116 and 117: 5 Direct Sums of Cyclic Groups 95 P
Page 118 and 119: 5 Direct Sums of Cyclic Groups 97 s
Page 120 and 121: 6 Equivalent Presentations 99 (5) (
Page 122 and 123: 6 Equivalent Presentations 101 Thes
Page 124 and 125: 6 Equivalent Presentations 103 Equi
Page 126 and 127: 7 Chains of Free Groups 105 For Ded
Page 128 and 129: 7 Chains of Free Groups 107 We can
Page 130 and 131: 7 Chains of Free Groups 109 Hill’
Page 132 and 133: 7 Chains of Free Groups 111 which i
Page 136 and 137: 8 Almost Free Groups 115 Proof. (a)
Page 138 and 139: 8 Almost Free Groups 117 be done by
Page 140 and 141: 8 Almost Free Groups 119 with. In t
Page 142 and 143: 9 Shelah’s Singular Compactness T
Page 144 and 145: 10 Groups with Discrete Norm 123 Di
Page 146 and 147: 10 Groups with Discrete Norm 125 Ch
Page 148 and 149: 11 Quasi-Projectivity 127 11 Quasi-
Page 150 and 151: 11 Quasi-Projectivity 129 for diffe
Page 152 and 153: 132 4 Divisibility and Injectivity
Page 170 and 171: 150 5 Purity and Basic Subgroups or
Page 172 and 173: 152 5 Purity and Basic Subgroups Pr
Page 174 and 175: 154 5 Purity and Basic Subgroups ge
Page 178 and 179: 158 5 Purity and Basic Subgroups Th
Page 180 and 181: 160 5 Purity and Basic Subgroups ˛
Page 184 and 185:
164 5 Purity and Basic Subgroups Th
Page 186 and 187:
166 5 Purity and Basic Subgroups 5
Page 188 and 189:
168 5 Purity and Basic Subgroups Le
Page 190 and 191:
170 5 Purity and Basic Subgroups Th
Page 192 and 193:
172 5 Purity and Basic Subgroups F
Page 194 and 195:
174 5 Purity and Basic Subgroups (H
Page 196 and 197:
176 5 Purity and Basic Subgroups Ma
Page 198 and 199:
178 5 Purity and Basic Subgroups Pr
Page 200 and 201:
180 5 Purity and Basic Subgroups So
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 ;
Page 208 and 209:
1 Algebraic Compactness 189 Here jG
Page 210 and 211:
2 Complete Groups 191 is a Cauchy s
Page 212 and 213:
2 Complete Groups 193 A ⏐ μ A A
Page 214 and 215:
3 The Structure of Algebraically Co
Page 216 and 217:
3 The Structure of Algebraically Co
Page 218 and 219:
4 Pure-Injective Hulls 199 (2) The
Page 220 and 221:
4 Pure-Injective Hulls 201 subgroup
Page 222 and 223:
5 Locally Compact Groups 203 5 Loca
Page 224 and 225:
5 Locally Compact Groups 205 Theore
Page 226 and 227:
6 The Exchange Property 207 Evident
Page 228 and 229:
6 The Exchange Property 209 In the
Page 230 and 231:
6 The Exchange Property 211 to the
Page 232 and 233:
Chapter 7 Homomorphism Groups Abstr
Page 234 and 235:
1 Groups of Homomorphisms 215 (and
Page 236 and 237:
1 Groups of Homomorphisms 217 is an
Page 238 and 239:
1 Groups of Homomorphisms 219 F Not
Page 240 and 241:
2 Algebraically Compact Homomorphis
Page 242 and 243:
2 Algebraically Compact Homomorphis
Page 244 and 245:
3 Small Homomorphisms 225 (3) The a
Page 246 and 247:
3 Small Homomorphisms 227 completio
Page 248 and 249:
Chapter 8 Tensor and Torsion Produc
Page 250 and 251:
1 The Tensor Product 231 with the f
Page 252 and 253:
1 The Tensor Product 233 for matchi
Page 254 and 255:
1 The Tensor Product 235 finitely P
Page 256 and 257:
2 The Torsion Product 237 (6) (a) I
Page 258 and 259:
2 The Torsion Product 239 It is pre
Page 260 and 261:
2 The Torsion Product 241 It remain
Page 262 and 263:
3 Theorems on Tensor Products 243 T
Page 264 and 265:
4 Theorems on Torsion Products 245
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
5 Localization 251 (8) (Keef) If A
Page 272 and 273:
5 Localization 253 Exercises (1) If
Page 274 and 275:
256 9 Groups of Extensions and Coto
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
Page 312 and 313:
Page 314 and 315:
Page 316 and 317:
Page 318 and 319:
300 10 Torsion Groups (A) If C is a
Page 320 and 321:
302 10 Torsion Groups the indicator
Page 322 and 323:
304 10 Torsion Groups 0 −−−
Page 324 and 325:
306 10 Torsion Groups is tantamount
Page 326 and 327:
308 10 Torsion Groups Proof. We hav
Page 328 and 329:
310 10 Torsion Groups (3) Let A be
Page 330 and 331:
312 10 Torsion Groups (C) B is a ba
Page 332 and 333:
314 10 Torsion Groups pure-injectiv
Page 334 and 335:
316 10 Torsion Groups Theorem 3.11
Page 336 and 337:
318 10 Torsion Groups 4 More on Tor
Page 338 and 339:
320 10 Torsion Groups (g) The large
Page 340 and 341:
322 10 Torsion Groups that in torsi
Page 342 and 343:
324 10 Torsion Groups subgroup of B
Page 344 and 345:
326 10 Torsion Groups (c) Homomorph
Page 346 and 347:
328 10 Torsion Groups then Ker con
Page 348 and 349:
330 10 Torsion Groups A class more
Page 350 and 351:
332 10 Torsion Groups with basic su
Page 352 and 353:
334 10 Torsion Groups 8 Valuated Ve
Page 354 and 355:
336 10 Torsion Groups V D A ` B for
Page 356 and 357:
338 10 Torsion Groups Proof. Define
Page 358 and 359:
340 10 Torsion Groups When Socles D
Page 360 and 361:
342 10 Torsion Groups Hill [1] gave
Page 362 and 363:
344 11 p-Groups with Elements of In
Page 364 and 365:
Page 366 and 367:
Page 368 and 369:
Page 370 and 371:
Page 372 and 373:
Page 374 and 375:
Page 376 and 377:
Page 378 and 379:
Page 380 and 381:
Page 382 and 383:
Page 384 and 385:
Page 386 and 387:
Page 388 and 389:
Page 390 and 391:
Page 392 and 393:
Page 394 and 395:
Page 396 and 397:
Page 398 and 399:
Page 400 and 401:
Page 402 and 403:
Page 404 and 405:
Page 406 and 407:
Page 408 and 409:
Page 410 and 411:
Page 412 and 413:
Page 414 and 415:
Page 416 and 417:
Page 418 and 419:
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
Page 426 and 427:
Page 428 and 429:
410 12 Torsion-Free Groups is calle
Page 430 and 431:
412 12 Torsion-Free Groups Proof. L
Page 432 and 433:
414 12 Torsion-Free Groups If A is
Page 434 and 435:
416 12 Torsion-Free Groups Exercise
Page 436 and 437:
418 12 Torsion-Free Groups Converse
Page 438 and 439:
420 12 Torsion-Free Groups fa X j a
Page 440 and 441:
422 12 Torsion-Free Groups and 0 !
Page 442 and 443:
424 12 Torsion-Free Groups (ii) If
Page 444 and 445:
426 12 Torsion-Free Groups number o
Page 446 and 447:
428 12 Torsion-Free Groups along B
Page 448 and 449:
430 12 Torsion-Free Groups whenever
Page 450 and 451:
432 12 Torsion-Free Groups If we ha
Page 452 and 453:
434 12 Torsion-Free Groups Then we
Page 454 and 455:
436 12 Torsion-Free Groups (ii) For
Page 456 and 457:
438 12 Torsion-Free Groups (7) (de
Page 458 and 459:
440 12 Torsion-Free Groups it is ro
Page 460 and 461:
442 12 Torsion-Free Groups 1 for `
Page 462 and 463:
444 12 Torsion-Free Groups A D B i
Page 464 and 465:
446 12 Torsion-Free Groups (5) For
Page 466 and 467:
448 12 Torsion-Free Groups Proof. L
Page 468 and 469:
450 12 Torsion-Free Groups local (T
Page 470 and 471:
452 12 Torsion-Free Groups stable r
Page 472 and 473:
454 12 Torsion-Free Groups (3) (War
Page 474 and 475:
456 12 Torsion-Free Groups write x
Page 476 and 477:
458 12 Torsion-Free Groups For thos
Page 478 and 479:
460 12 Torsion-Free Groups A C 1
Page 480 and 481:
462 12 Torsion-Free Groups ˛./.˛
Page 482 and 483:
464 12 Torsion-Free Groups so the s
Page 484 and 485:
466 12 Torsion-Free Groups followed
Page 486 and 487:
468 12 Torsion-Free Groups (i) ther
Page 488 and 489:
470 12 Torsion-Free Groups 11 Duali
Page 490 and 491:
472 12 Torsion-Free Groups Lemma 11
Page 492 and 493:
474 12 Torsion-Free Groups rk p D.A
Page 494 and 495:
476 12 Torsion-Free Groups is proj(
Page 496 and 497:
478 12 Torsion-Free Groups division
Page 498 and 499:
Chapter 13 Torsion-Free Groups of I
Page 500 and 501:
1 Direct Decompositions of Infinite
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
2 Slender Groups 489 (6) A torsion-
Page 508 and 509:
2 Slender Groups 491 Example 2.5. T
Page 510 and 511:
2 Slender Groups 493 Proof. From Th
Page 512 and 513:
2 Slender Groups 495 F Notes. The r
Page 514 and 515:
3 Characterizations of Slender Grou
Page 516 and 517:
3 Characterizations of Slender Grou
Page 518 and 519:
4 Separable Groups 501 Exercises (1
Page 520 and 521:
4 Separable Groups 503 where the G
Page 522 and 523:
4 Separable Groups 505 In the follo
Page 524 and 525:
4 Separable Groups 507 c C g D h 2
Page 526 and 527:
5 Vector Groups 509 (7) Show that t
Page 528 and 529:
5 Vector Groups 511 Isomorphism of
Page 530 and 531:
6 Powers ofZ of Measurable Cardinal
Page 532 and 533:
Page 534 and 535:
Page 536 and 537:
7 Whitehead Groups If V = L 519 (7)
Page 538 and 539:
7 Whitehead Groups If V = L 521 Pro
Page 540 and 541:
7 Whitehead Groups If V = L 523 Sˇ
Page 542 and 543:
8 Whitehead Groups Under Martin’s
Page 544 and 545:
8 Whitehead Groups Under Martin’s
Page 546 and 547:
Chapter 14 Butler Groups Abstract T
Page 548 and 549:
1 Finite Rank Butler Groups 531 and
Page 550 and 551:
1 Finite Rank Butler Groups 533 Pro
Page 552 and 553:
1 Finite Rank Butler Groups 535 Let
Page 554 and 555:
2 Prebalanced and Decent Subgroups
Page 556 and 557:
2 Prebalanced and Decent Subgroups
Page 558 and 559:
3 The Torsion Extension Property 54
Page 560 and 561:
Page 562 and 563:
Page 564 and 565:
4 Countable Butler Groups 547 (ii)
Page 566 and 567:
5 B 1 -andB 2 -Groups 549 (F) Homog
Page 568 and 569:
6 Solid Subgroups 551 understand th
Page 570 and 571:
6 Solid Subgroups 553 fa n j n
Page 572 and 573:
6 Solid Subgroups 555 Again changin
Page 574 and 575:
7 Solid Chains 557 (4) A corank 1 s
Page 576 and 577:
7 Solid Chains 559 B C1 =B admits
Page 578 and 579:
7 Solid Chains 561 Lemma 7.6 (Bican
Page 580 and 581:
8 Butler Groups of Uncountable Rank
Page 582 and 583:
Page 584 and 585:
Page 586 and 587:
9 More on Infinite Butler Groups 56
Page 588 and 589:
9 More on Infinite Butler Groups 57
Page 590 and 591:
Chapter 15 Mixed Groups Abstract Mi
Page 592 and 593:
1 Splitting Mixed Groups 575 For ev
Page 594 and 595:
1 Splitting Mixed Groups 577 so A m
Page 596 and 597:
2 Baer Groups are Free 579 (4) (Opp
Page 598 and 599:
2 Baer Groups are Free 581 We note
Page 600 and 601:
2 Baer Groups are Free 583 One does
Page 602 and 603:
3 Valuated Groups. Height-Matrices
Page 604 and 605:
Page 606 and 607:
Page 608 and 609:
4 Nice, Isotype, and Balanced Subgr
Page 610 and 611:
5 Mixed Groups of Torsion-Free Rank
Page 612 and 613:
5 Mixed Groups of Torsion-Free Rank
Page 614 and 615:
6 Simply Presented Mixed Groups 597
Page 616 and 617:
7 Warfield Groups 599 Exercises (1)
Page 618 and 619:
7 Warfield Groups 601 (i) A=F is to
Page 620 and 621:
7 Warfield Groups 603 Lemma 7.7 (St
Page 622 and 623:
8 The Categories WALK and WARF 605
Page 624 and 625:
8 The Categories WALK and WARF 607
Page 626 and 627:
9 Projective Properties of Warfield
Page 628 and 629:
9 Projective Properties of Warfield
Page 630 and 631:
Chapter 16 Endomorphism Rings Abstr
Page 632 and 633:
1 Endomorphism Rings 615 Lemma 1.5.
Page 634 and 635:
1 Endomorphism Rings 617 For a subg
Page 636 and 637:
1 Endomorphism Rings 619 Before sta
Page 638 and 639:
1 Endomorphism Rings 621 Recently,
Page 640 and 641:
2 Endomorphism Rings of p-Groups 62
Page 642 and 643:
Page 644 and 645:
Page 646 and 647:
3 Endomorphism Rings of Torsion-Fre
Page 648 and 649:
Page 650 and 651:
Page 652 and 653:
4 Endomorphism Rings of Special Gro
Page 654 and 655:
5 Special Endomorphism Rings 637 se
Page 656 and 657:
5 Special Endomorphism Rings 639 Pr
Page 658 and 659:
5 Special Endomorphism Rings 641 Re
Page 660 and 661:
6 Groups as Modules Over Their Endo
Page 662 and 663:
Page 664 and 665:
Page 666 and 667:
Page 668 and 669:
7 Groups with Prescribed Endomorphi
Page 670 and 671:
7 Groups with Prescribed Endomorphi
Page 672 and 673:
656 17 Automorphism Groups Example
Page 674 and 675:
658 17 Automorphism Groups linear g
Page 676 and 677:
660 17 Automorphism Groups Exercise
Page 678 and 679:
662 17 Automorphism Groups Accordin
Page 680 and 681:
664 17 Automorphism Groups (2) If A
Page 682 and 683:
666 17 Automorphism Groups (b) If G
Page 684 and 685:
668 17 Automorphism Groups Then a 7
Page 686 and 687:
670 17 Automorphism Groups for all
Page 688 and 689:
Chapter 18 Groups in Rings and in F
Page 690 and 691:
1 Additive Groups of Rings and Modu
Page 692 and 693:
2 Rings on Groups 677 (11) For a ri
Page 694 and 695:
2 Rings on Groups 679 is a subring,
Page 696 and 697:
2 Rings on Groups 681 Beaumont-Lawv
Page 698 and 699:
3 Additive Groups of Noetherian Rin
Page 700 and 701:
4 Additive Groups of Artinian Rings
Page 702 and 703:
4 Additive Groups of Artinian Rings
Page 704 and 705:
5 Additive Groups of Regular Rings
Page 706 and 707:
5 Additive Groups of Regular Rings
Page 708 and 709:
6 E-Rings 693 (2) A group is isomor
Page 710 and 711:
6 E-Rings 695 (F) Every endomorphis
Page 712 and 713:
7 Groups of Units in Commutative Ri
Page 714 and 715:
7 Groups of Units in Commutative Ri
Page 716 and 717:
8 Multiplicative Groups of Fields 7
Page 718 and 719:
Page 720 and 721:
Page 722 and 723:
References Books D.M. Arnold — [A
Page 724 and 725:
References 709 D.M. Arnold, R. Hunt
Page 726 and 727:
References 711 R. Burkhardt — [1]
Page 728 and 729:
References 713 M. Dugas — [1] Fas
Page 730 and 731:
References 715 A.A. Fomin — [1] T
Page 732 and 733:
References 717 R. Göbel, S. Shelah
Page 734 and 735:
References 719 P. Hill, M. Lane, C.
Page 736 and 737:
References 721 A. Kertész — [1]
Page 738 and 739:
References 723 A. Mader — [1] On
Page 740 and 741:
References 725 L.G. Nongxa — [1]
Page 742 and 743:
References 727 endomorphism rings.
Page 744 and 745:
References 729 G.M. Tsukerman — [
Page 746 and 747:
Author Index A Abel, N.H., vii Albr
Page 748 and 749:
Author Index 733 Grinshpon, S.Ya.,
Page 750 and 751:
Author Index 735 O’Neill, J.D., 3
Page 752 and 753:
Subject Index A Abelian group, 1 Ab
Page 754 and 755:
Subject Index 739 homomorphism, 37,
Page 756 and 757:
Subject Index 741 Full rational gro
Page 758 and 759:
Subject Index 743 Łoś-Eda theorem
Page 760 and 761:
Subject Index 745 Rank 1 torsion-fr
Page 762:
Subject Index 747 vector space, 334
show all

Abelian Groups - László Fuchs [Springer]

Create successful ePaper yourself

Delete template?

Save as template?