625:
28:
197:
694:
in the generators of the group represent the same element. The word problem for a given finitely generated group is solvable if and only if the group can be embedded in every
569:
509:
400:
344:
465:
445:
17:
790:
850:
589:
all subgroups of the group are finitely generated. A group such that all its subgroups are finitely generated is called
467:
are the numbers of generators of the two finitely generated subgroups then their intersection is generated by at most
238:
216:
636:
170:
874:
402:
on two generators is an example of a subgroup of a finitely generated group that is not finitely generated.
180:
184:
695:
718:
583:
424:
62:
32:
709:
of a generating set for the group. By definition, the rank of a finitely generated group is finite.
572:
518:
356:
723:
683:
427:
of two finitely generated subgroups of a free group is again finitely generated. Furthermore, if
655:
470:
413:
405:
On the other hand, all subgroups of a finitely generated abelian group are finitely generated.
869:
176:
36:
72:
can be written as the combination (under the group operation) of finitely many elements of
811:
605:
597:
579:
378:
348:
226:
153:
128:
123:
is finitely generated; the quotient group is generated by the images of the generators of
8:
691:
667:
409:
372:
52:
450:
430:
352:
262:
230:
761:
744:
658:
studies the connections between algebraic properties of finitely generated groups and
143:
846:
139:
799:
788:
Howson, Albert G. (1954). "On the intersection of finitely generated free groups".
756:
687:
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Subgroups of a finitely generated abelian group are themselves finitely generated.
807:
702:
100:
77:
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116:
863:
512:
371:
A subgroup of a finitely generated group need not be finitely generated. The
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96:
671:
208:
135:
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but countable groups need not be finitely generated. The additive group of
84:
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A group such that every finitely generated subgroup is finite is called
166:
59:
412:
in a finitely generated group is always finitely generated, and the
663:
659:
169:
on a finite set is finitely generated by the elements of that set (
157:
106:
is an example of a countable group that is not finitely generated.
27:
44:
511:
generators. This upper bound was then significantly improved by
196:
360:
345:
fundamental theorem of finitely generated abelian groups
134:
A group that is generated by a single element is called
95:
itself. Every infinite finitely generated group must be
347:
states that a finitely generated abelian group is the
521:
473:
453:
433:
381:
359:
and a finite abelian group, each of which are unique
416:
gives a bound on the number of generators required.
191:
612:, every periodic abelian group is locally finite.
563:
503:
459:
439:
394:
818:
769:
861:
749:Proceedings of the American Mathematical Society
35:requires two generators, as represented by this
156:is a group in which every finitely generated
791:Journal of the London Mathematical Society
760:
742:
195:
26:
14:
862:
787:
686:for a finitely generated group is the
745:"A note on finitely generated groups"
840:
824:
775:
705:is often defined to be the smallest
619:
24:
677:
25:
886:
762:10.1090/S0002-9939-1967-0215904-3
604:, i.e., every element has finite
192:Finitely generated abelian groups
138:. Every infinite cyclic group is
623:
600:. Every locally finite group is
239:finitely generated abelian group
217:Finitely generated abelian group
615:
781:
736:
552:
540:
537:
525:
119:of a finitely generated group
13:
1:
834:
564:{\displaystyle 2(m-1)(n-1)+1}
142:to the additive group of the
87:is finitely generated, since
743:Gregorac, Robert J. (1967).
366:
7:
712:
109:
18:Finitely generated subgroup
10:
891:
696:algebraically closed group
214:
719:Finitely generated module
584:ascending chain condition
582:of a group satisfies the
504:{\displaystyle 2mn-m-n+1}
68:so that every element of
33:dihedral group of order 8
843:A Course on Group Theory
804:10.1112/jlms/s1-29.4.428
729:
573:Hanna Neumann conjecture
187:) is finitely generated.
181:finitely presented group
49:finitely generated group
841:Rose, John S. (2012) .
724:Presentation of a group
845:. Dover Publications.
670:on which these groups
656:Geometric group theory
565:
505:
461:
441:
414:Schreier index formula
396:
257:, every group element
212:
40:
566:
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462:
442:
408:A subgroup of finite
397:
395:{\displaystyle F_{2}}
265:of these generators,
211:under multiplication.
199:
83:By definition, every
30:
875:Properties of groups
580:lattice of subgroups
519:
471:
451:
431:
379:
261:can be written as a
154:locally cyclic group
129:canonical projection
373:commutator subgroup
91:can be taken to be
80:of such elements.
635:. You can help by
561:
501:
457:
437:
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375:of the free group
353:free abelian group
263:linear combination
213:
41:
852:978-0-486-68194-8
653:
652:
460:{\displaystyle n}
440:{\displaystyle m}
225:can be seen as a
16:(Redirected from
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766:
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688:decision problem
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510:
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466:
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423:showed that the
421:Albert G. Howson
401:
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241:with generators
101:rational numbers
21:
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703:rank of a group
690:of whether two
680:
678:Related notions
649:
643:
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633:needs expansion
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798:(4): 428–434.
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755:(4): 756–758.
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721:
714:
711:
679:
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666:properties of
651:
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644:September 2017
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598:locally finite
587:if and only if
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332:
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321:with integers
319:
318:
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297:
290:
283:
276:
252:
245:
215:Main article:
205:roots of unity
193:
190:
189:
188:
174:
163:
162:
161:
132:
111:
108:
63:generating set
58:that has some
9:
6:
4:
3:
2:
887:
876:
873:
871:
868:
867:
865:
854:
848:
844:
839:
838:
827:, p. 75.
826:
821:
813:
809:
805:
801:
797:
793:
792:
784:
778:, p. 55.
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634:
631:This section
629:
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621:
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611:
607:
603:
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594:
592:
588:
585:
581:
576:
574:
558:
555:
549:
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543:
534:
531:
528:
522:
514:
513:Hanna Neumann
498:
495:
492:
489:
486:
483:
480:
477:
474:
454:
434:
426:
422:
417:
415:
411:
406:
403:
387:
383:
374:
364:
363:isomorphism.
362:
358:
354:
350:
346:
341:
338:
335:
331:
324:
316:
312:
307:
303:
296:
289:
282:
275:
271:
268:
267:
266:
264:
260:
255:
251:
244:
240:
236:
232:
228:
224:
223:abelian group
218:
210:
206:
203:
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186:
182:
178:
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151:
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114:
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107:
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90:
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81:
79:
75:
71:
67:
64:
61:
57:
54:
50:
46:
38:
37:cycle diagram
34:
29:
19:
870:Group theory
842:
820:
795:
789:
783:
771:
752:
748:
738:
700:
684:word problem
681:
654:
641:
637:adding to it
632:
616:Applications
595:
577:
425:intersection
418:
407:
404:
370:
342:
339:
333:
329:
322:
320:
314:
310:
305:
301:
294:
287:
280:
273:
269:
258:
253:
249:
242:
234:
233:of integers
220:
209:cyclic group
200:The six 6th
146:
124:
120:
103:
92:
88:
85:finite group
82:
73:
69:
65:
55:
48:
42:
825:Rose (2012)
776:Rose (2012)
707:cardinality
660:topological
237:, and in a
864:Categories
835:References
610:Conversely
591:Noetherian
355:of finite
349:direct sum
177:A fortiori
167:free group
160:is cyclic.
140:isomorphic
127:under the
664:geometric
547:−
532:−
490:−
484:−
419:In 1954,
367:Subgroups
229:over the
185:§Examples
171:§Examples
97:countable
713:See also
602:periodic
300:+ ... +
179:, every
158:subgroup
144:integers
117:quotient
110:Examples
78:inverses
812:0065557
328:, ...,
248:, ...,
207:form a
202:complex
76:and of
45:algebra
849:
810:
668:spaces
571:; see
227:module
221:Every
136:cyclic
115:Every
60:finite
730:Notes
692:words
606:order
410:index
361:up to
351:of a
53:group
51:is a
847:ISBN
701:The
682:The
662:and
578:The
447:and
357:rank
343:The
231:ring
165:The
47:, a
31:The
800:doi
757:doi
672:act
639:.
515:to
43:In
866::
808:MR
806:.
796:29
794:.
753:18
751:.
747:.
698:.
674:.
608:.
593:.
575:.
337:.
286:+
272:=
173:).
152:A
149:.
855:.
814:.
802::
765:.
759::
646:)
642:(
559:1
556:+
553:)
550:1
544:n
541:(
538:)
535:1
529:m
526:(
523:2
499:1
496:+
493:n
487:m
481:n
478:m
475:2
455:n
435:m
388:2
384:F
334:n
330:α
326:1
323:α
315:n
311:x
309:â‹…
306:n
302:α
298:2
295:x
293:â‹…
291:2
288:α
284:1
281:x
279:â‹…
277:1
274:α
270:x
259:x
254:n
250:x
246:1
243:x
235:Z
183:(
147:Z
131:.
125:G
121:G
104:Q
93:G
89:S
74:S
70:G
66:S
56:G
39:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
