Abelian Groups - László Fuchs [Springer]

More documents

Recommendations

Info

308 10 Torsion <strong>Groups</strong> Proof. We have noticed above that every subgroup of the form (10.3) is fully invariant. Assume, conversely, that G is a fully invariant subgroup, and define n as the minimum of the heights h.p n g/ with g running over G. The sequence 0 ; 1 ;:::; n ;::: is obviously strictly increasing (except when it reaches 1). To verify the gap condition, suppose k C 1< kC1 for some k. Surely, there exists an x 2 G with h.p k x/ D k , and by definition h.p kC1 x/ kC1 ; thus, this x has a gap at k in its indicator. By Lemma 1.1, A must contain an element of order p and of height k . The inclusion G A. 0 ; 1 ;:::; n ;:::/is obvious. We now verify the existence of a g 2 G such that h.p i g/ D i for i D 0;1;:::;n 1: If there is no gap in the sequence 0 ; 1 ;:::; n 1 ,andifg 2 G satisfies h.p n 1 g/ D n 1 ,then this g is already as desired. If there is a gap in this sequence, and if the first gap appears between j 1 and j , then there is a g j 2 G such that h.p i g j / D i for i D 0; 1; : : : ; j 1: If the second gap lies between k 1 and k .j < k/, then some g 0 2 G exists with h.p i g 0 / D i for i D j;:::;k 1: By Lemma 1.1, A contains a g k such that h.p i g k / maxfh.p i g 0 /; i C 1g for i D 0;1;:::;j 1 and h.p i g k / D h.p i g 0 / for i j. Because of full invariance, and hence full transitivity, as well as Corollary 1.5, g k 2 G. Thus proceeding, we construct elements g j ; g k ;:::;g` 2 G for the gaps in 0 ; 1 ;:::; n 1 , and at the end, g D g j C g k C C g` will satisfy h.p i g/ D i for i D 0;1;:::;n 1: Thus h.g/ h.a/ for every a 2 A. 0 ; 1 ;:::; n ;:::/ of order o.g/. Full transitivity shows that a 2 G, i.e.G is of the form (10.3). If . 0 ; 1 ;:::; n ;:::/ and .0 0;0 1 ;:::;0 n ;:::/ are different sequences, both satisfying the gap condition, then let n be the first index with n ¤ n 0,say, n
2 Fully Invariant and Large Subgroups 309 Pierce condition if for every k 2 N there is an n 2 N such that every a 2 A with o.a/ p k and h.a/ n is contained in G, i.e. p n AŒp k G: Since (B) completely characterizes large subgroups in bounded groups, our focus is on unbounded groups. Theorem 2.3 (Pierce [1]). Let A be an unbounded, reduced p-group. For a fully invariant subgroup G of A, these are equivalent: (i) G is a large subgroup of AI (ii) G D A.r 0 ; r 1 ;:::;r n ;:::/ with a strictly increasing sequence of non-negative integers r n and symbols 1 .satisfying the gap condition/I (iii) the Pierce condition holds for G. Proof. (i) ) (ii) By Theorem 2.2, G D A. 0 ; 1 ;:::; n ;:::/for suitable n , and in view of (E), we have n
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 92 and 93:
Page 94 and 95:
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 104 and 105:
Page 106 and 107:
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 134 and 135:
8 Almost Free Groups 113 (D) Let 0
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 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
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 176 and 177:
Page 178 and 179:
158 5 Purity and Basic Subgroups Th
Page 180 and 181:
160 5 Purity and Basic Subgroups ˛
Page 182 and 183:
Page 184 and 185:
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:
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:
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: 258 9 Groups of Extensions and Coto
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 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 376 and 377:
358 11 p-Groups with Elements of In
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?