533:
130:
to its restriction to the torsion subgroup. There is another covariant functor from the category of abelian groups to the category of torsion-free groups that sends every group to its quotient by its torsion subgroup, and sends every homomorphism to the obvious induced homomorphism (which is easily
350:
438:
211:
255:
609:; this is a torsion group since every element has order 2. Nor need there be an upper bound on the orders of elements in a torsion group if it isn't
512:-torsion subgroups. The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a
376:
150:
and a torsion-free subgroup (but this is not true for all infinitely generated abelian groups). In any decomposition of
595:
Every finite abelian group is a torsion group. Not every torsion group is finite however: consider the direct sum of a
882:
825:
685:
167:
769:
of abelian groups, while taking the quotient by the torsion subgroup makes torsion-free abelian groups into a
900:
91:
is closed under the group operation relies on the commutativity of the operation (see examples section).
166:(but the torsion-free subgroup is not uniquely determined). This is a key step in the classification of
788:
123:
70:
872:
610:
766:
650:
139:
649:
of its torsion-free part is uniquely determined, as is explained in more detail in the article on
546:
542:
478:
103:
536:
The 4-torsion subgroup of the quotient group of the complex numbers under addition by a lattice.
770:
516:
from the category of torsion groups to the product over all prime numbers of the categories of
840:
541:
The torsion subset of a non-abelian group is not, in general, a subgroup. For example, in the
793:
673:
184:
143:
126:
to the category of torsion groups that sends every group to its torsion subgroup and every
46:
8:
856:
631:
is torsion-free, but the converse is not true, as is shown by the additive group of the
628:
596:
127:
878:
821:
783:
119:
868:
513:
78:
524:-torsion groups in isolation tells us everything about torsion groups in general.
860:
719:
632:
589:
585:
345:{\displaystyle A_{T_{p}}=\{a\in A\;|\;\exists n\in \mathbb {N} \;,p^{n}a=0\}.\;}
661:
455:
894:
864:
62:
20:
614:
600:
214:
665:
532:
813:
16:
Subgroup of an abelian group consisting of all elements of finite order
704:
481:. More strongly, any homomorphism between abelian groups sends each
38:
508:-power torsion subgroup, and restricts every homomorphism to the
497:
765:
Taking the torsion subgroup makes torsion abelian groups into a
579:
is a product of two torsion elements, but has infinite order.
433:{\displaystyle A_{T}\cong \bigoplus _{p\in P}A_{T_{p}}.\;}
500:
from the category of abelian groups to the category of
520:-torsion groups. In a sense, this means that studying
855:
379:
258:
187:
504:-power torsion groups that sends every group to its
432:
344:
205:
892:
366:-power torsion subgroups over all prime numbers
527:
485:-power torsion subgroup into the corresponding
877:, Boston, MA: Jones and Bartlett Publishers,
820:(3rd ed.), Addison-Wesley, p. 42,
335:
279:
173:
45:consisting of all elements that have finite
142:and abelian, then it can be written as the
429:
341:
312:
297:
291:
308:
531:
65:(or periodic group) if every element of
362:is isomorphic to the direct sum of its
893:
863:; Holt, Derek F.; Levy, Silvio V. F.;
154:as a direct sum of a torsion subgroup
98:is abelian, then the torsion subgroup
812:
680:is a subgroup of some abelian group
13:
298:
14:
912:
734:= 0. For a general abelian group
168:finitely generated abelian groups
839:See Epstein & Cannon (1992)
684:, then the natural map from the
645:is not finitely generated, the
69:has finite order and is called
833:
806:
293:
200:
188:
1:
849:
722:) kills torsion. That is, if
479:fully characteristic subgroup
158:and a torsion-free subgroup,
104:fully characteristic subgroup
676:, which means that whenever
528:Examples and further results
118:is torsion-free. There is a
7:
777:
710:Tensoring an abelian group
473:-power torsion subgroup of
447:is a finite abelian group,
231:that have order a power of
10:
917:
789:Torsion-free abelian group
584:The torsion elements in a
454:coincides with the unique
131:seen to be well-defined).
124:category of abelian groups
874:Word Processing in Groups
489:-power torsion subgroup.
235:is a subgroup called the
799:
767:coreflective subcategory
726:is a torsion group then
651:rank of an abelian group
613:, as the example of the
599:number of copies of the
177:-power torsion subgroups
146:of its torsion subgroup
543:infinite dihedral group
240:-power torsion subgroup
771:reflective subcategory
738:with torsion subgroup
537:
492:For each prime number
434:
346:
242:or, more loosely, the
207:
181:For any abelian group
81:is of infinite order.
794:Torsion abelian group
535:
435:
355:The torsion subgroup
347:
208:
206:{\displaystyle (A,+)}
110:and the factor group
901:Abelian group theory
869:Thurston, William P.
865:Paterson, Michael S.
857:Epstein, David B. A.
377:
256:
185:
73:if every element of
57:). An abelian group
33:of an abelian group
629:free abelian group
611:finitely generated
538:
496:, this provides a
430:
408:
342:
203:
140:finitely generated
784:Torsion (algebra)
656:An abelian group
393:
247:-torsion subgroup
120:covariant functor
19:In the theory of
908:
887:
861:Cannon, James W.
843:
837:
831:
830:
810:
660:is torsion-free
633:rational numbers
514:faithful functor
439:
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51:torsion elements
25:torsion subgroup
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720:divisible group
608:
590:normal subgroup
586:nilpotent group
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452:
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264:
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227:of elements of
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89:
84:The proof that
31:
17:
12:
11:
5:
914:
904:
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786:
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775:
774:
763:
708:
686:tensor product
662:if and only if
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87:
29:
21:abelian groups
15:
9:
6:
4:
3:
2:
913:
902:
899:
898:
896:
886:
884:0-86720-244-0
880:
876:
875:
870:
866:
862:
858:
854:
853:
842:
836:
829:
827:0-201-55540-9
823:
819:
815:
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805:
795:
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787:
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782:
781:
772:
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764:
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753:
749:
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733:
729:
725:
721:
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713:
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706:
702:
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623:
619:
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602:
598:
594:
591:
587:
583:
582:
578:
574:
573:
569:
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561:
557:
553:
552:
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519:
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499:
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365:
361:
338:
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321:
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269:
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197:
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176:
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145:
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137:
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125:
121:
117:
113:
109:
105:
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97:
92:
90:
82:
80:
76:
72:
68:
64:
63:torsion group
60:
56:
52:
48:
44:
40:
36:
32:
26:
22:
873:
835:
817:
808:
759:
755:
751:
747:
743:
739:
735:
731:
727:
723:
715:
711:
700:
696:
692:
688:
681:
677:
669:
657:
646:
642:
635:
621:
617:
615:factor group
603:
601:cyclic group
576:
575:the element
567:
563:
559:
555:
547:presentation
545:, which has
521:
517:
509:
505:
501:
493:
491:
486:
482:
474:
470:
468:
463:
457:
448:
444:
442:
367:
363:
356:
354:
244:
243:
237:
236:
232:
228:
221:
217:
215:prime number
180:
174:
163:
159:
155:
151:
147:
135:
133:
128:homomorphism
115:
111:
107:
99:
95:
93:
85:
83:
74:
71:torsion-free
66:
61:is called a
58:
54:
50:
42:
34:
27:
24:
18:
814:Serge, Lang
162:must equal
77:except the
850:References
144:direct sum
705:injective
597:countable
460:-subgroup
402:∈
395:⨁
391:≅
305:∈
299:∃
286:∈
122:from the
895:Category
871:(1992),
816:(1993),
778:See also
742:one has
718:(or any
641:Even if
220:the set
213:and any
79:identity
39:subgroup
818:Algebra
588:form a
570:² = 1 ⟩
498:functor
37:is the
881:
841:p. 167
824:
674:module
664:it is
627:Every
624:shows.
456:Sylow
23:, the
800:Notes
714:with
668:as a
477:is a
469:Each
443:When
102:is a
49:(the
47:order
879:ISBN
822:ISBN
666:flat
647:size
566:² =
703:is
695:to
462:of
138:is
134:If
106:of
94:If
53:of
41:of
897::
867:;
859:;
758:⊗
750:≅
746:⊗
730:⊗
699:⊗
691:⊗
577:xy
562:|
558:,
554:⟨
466:.
451:Tp
370::
249::
224:Tp
170:.
773:.
762:.
760:Q
756:T
754:/
752:A
748:Q
744:A
740:T
736:A
732:Q
728:T
724:T
716:Q
712:A
707:.
701:A
697:B
693:A
689:C
682:B
678:C
672:-
670:Z
658:A
653:.
643:A
638:.
636:Q
622:Z
620:/
618:Q
607:2
604:C
592:.
568:y
564:x
560:y
556:x
549::
522:p
518:p
510:p
506:p
502:p
494:p
487:p
483:p
475:A
471:p
464:A
458:p
449:A
445:A
427:.
420:p
416:T
411:A
405:P
399:p
386:T
382:A
368:p
364:p
359:T
357:A
339:.
336:}
333:0
330:=
327:a
322:n
318:p
314:,
309:N
302:n
294:|
289:A
283:a
280:{
277:=
270:p
266:T
261:A
245:p
238:p
233:p
229:A
222:A
218:p
201:)
198:+
195:,
192:A
189:(
175:p
164:T
160:S
156:S
152:A
148:T
136:A
116:T
114:/
112:A
108:A
100:T
96:A
88:T
86:A
75:A
67:A
59:A
55:A
43:A
35:A
30:T
28:A
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