614:
17:
186:
683:
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
558:
498:
389:
333:
454:
434:
779:
839:
578:
all subgroups of the group are finitely generated. A group such that all its subgroups are finitely generated is called
456:
are the numbers of generators of the two finitely generated subgroups then their intersection is generated by at most
227:
205:
625:
159:
863:
391:
on two generators is an example of a subgroup of a finitely generated group that is not finitely generated.
169:
173:
684:
707:
572:
413:
51:
21:
698:
of a generating set for the group. By definition, the rank of a finitely generated group is finite.
561:
507:
345:
712:
672:
416:
of two finitely generated subgroups of a free group is again finitely generated. Furthermore, if
644:
459:
402:
394:
On the other hand, all subgroups of a finitely generated abelian group are finitely generated.
858:
165:
25:
61:
can be written as the combination (under the group operation) of finitely many elements of
800:
594:
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568:
367:
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215:
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117:
112:
is finitely generated; the quotient group is generated by the images of the generators of
8:
680:
656:
398:
361:
41:
439:
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341:
251:
219:
750:
733:
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studies the connections between algebraic properties of finitely generated groups and
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128:
788:
777:
Howson, Albert G. (1954). "On the intersection of finitely generated free groups".
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Subgroups of a finitely generated abelian group are themselves finitely generated.
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66:
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A subgroup of a finitely generated group need not be finitely generated. The
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85:
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but countable groups need not be finitely generated. The additive group of
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A group such that every finitely generated subgroup is finite is called
155:
48:
401:
in a finitely generated group is always finitely generated, and the
652:
648:
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on a finite set is finitely generated by the elements of that set (
146:
95:
is an example of a countable group that is not finitely generated.
16:
33:
500:
generators. This upper bound was then significantly improved by
185:
349:
334:
fundamental theorem of finitely generated abelian groups
123:
A group that is generated by a single element is called
84:
itself. Every infinite finitely generated group must be
336:
states that a finitely generated abelian group is the
510:
462:
442:
422:
370:
348:
and a finite abelian group, each of which are unique
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gives a bound on the number of generators required.
180:
601:, every periodic abelian group is locally finite.
552:
492:
448:
428:
383:
807:
758:
850:
738:Proceedings of the American Mathematical Society
24:requires two generators, as represented by this
145:is a group in which every finitely generated
780:Journal of the London Mathematical Society
749:
731:
184:
15:
851:
776:
675:for a finitely generated group is the
734:"A note on finitely generated groups"
829:
813:
764:
694:is often defined to be the smallest
608:
13:
666:
14:
875:
751:10.1090/S0002-9939-1967-0215904-3
593:, i.e., every element has finite
181:Finitely generated abelian groups
127:. Every infinite cyclic group is
612:
589:. Every locally finite group is
228:finitely generated abelian group
206:Finitely generated abelian group
604:
770:
725:
541:
529:
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108:of a finitely generated group
1:
823:
553:{\displaystyle 2(m-1)(n-1)+1}
131:to the additive group of the
76:is finitely generated, since
732:Gregorac, Robert J. (1967).
355:
7:
701:
98:
10:
880:
685:algebraically closed group
203:
708:Finitely generated module
573:ascending chain condition
571:of a group satisfies the
493:{\displaystyle 2mn-m-n+1}
57:so that every element of
22:dihedral group of order 8
832:A Course on Group Theory
793:10.1112/jlms/s1-29.4.428
718:
562:Hanna Neumann conjecture
176:) is finitely generated.
170:finitely presented group
38:finitely generated group
830:Rose, John S. (2012) .
713:Presentation of a group
834:. Dover Publications.
659:on which these groups
645:Geometric group theory
554:
494:
450:
430:
403:Schreier index formula
385:
246:, every group element
201:
29:
555:
495:
451:
431:
397:A subgroup of finite
386:
384:{\displaystyle F_{2}}
254:of these generators,
200:under multiplication.
188:
72:By definition, every
19:
864:Properties of groups
569:lattice of subgroups
508:
460:
440:
420:
368:
250:can be written as a
143:locally cyclic group
118:canonical projection
362:commutator subgroup
80:can be taken to be
69:of such elements.
624:. You can help by
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490:
446:
426:
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364:of the free group
342:free abelian group
252:linear combination
202:
30:
841:978-0-486-68194-8
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641:
449:{\displaystyle n}
429:{\displaystyle m}
214:can be seen as a
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756:
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729:
677:decision problem
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551:
499:
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452:
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412:showed that the
410:Albert G. Howson
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230:with generators
90:rational numbers
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692:rank of a group
679:of whether two
669:
667:Related notions
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622:needs expansion
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787:(4): 428–434.
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744:(4): 756–758.
723:
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717:
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703:
700:
668:
665:
655:properties of
640:
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633:September 2017
619:
617:
606:
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587:locally finite
576:if and only if
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321:
314:
310:with integers
308:
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204:Main article:
194:roots of unity
182:
179:
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121:
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97:
52:generating set
47:that has some
9:
6:
4:
3:
2:
876:
865:
862:
860:
857:
856:
854:
843:
837:
833:
828:
827:
816:, p. 75.
815:
810:
802:
798:
794:
790:
786:
782:
781:
773:
767:, p. 55.
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623:
620:This section
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588:
583:
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517:
511:
503:
502:Hanna Neumann
487:
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466:
463:
443:
423:
415:
411:
406:
404:
400:
395:
392:
376:
372:
363:
353:
352:isomorphism.
351:
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343:
339:
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305:
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292:
285:
278:
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257:
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249:
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233:
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217:
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212:abelian group
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56:
53:
50:
46:
43:
39:
35:
27:
26:cycle diagram
23:
18:
859:Group theory
831:
809:
784:
778:
772:
760:
741:
737:
727:
689:
673:word problem
670:
643:
630:
626:adding to it
621:
605:Applications
584:
566:
414:intersection
407:
396:
393:
359:
331:
328:
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262:
258:
247:
242:
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231:
223:
222:of integers
209:
198:cyclic group
189:The six 6th
135:
113:
109:
92:
81:
77:
74:finite group
71:
62:
58:
54:
44:
37:
31:
814:Rose (2012)
765:Rose (2012)
696:cardinality
649:topological
226:, and in a
853:Categories
824:References
599:Conversely
580:Noetherian
344:of finite
338:direct sum
166:A fortiori
156:free group
149:is cyclic.
129:isomorphic
116:under the
653:geometric
536:−
521:−
479:−
473:−
408:In 1954,
356:Subgroups
218:over the
174:§Examples
160:§Examples
86:countable
702:See also
591:periodic
289:+ ... +
168:, every
147:subgroup
133:integers
106:quotient
99:Examples
67:inverses
801:0065557
317:, ...,
237:, ...,
196:form a
191:complex
65:and of
34:algebra
838:
799:
657:spaces
560:; see
216:module
210:Every
125:cyclic
104:Every
49:finite
719:Notes
681:words
595:order
399:index
350:up to
340:of a
42:group
40:is a
836:ISBN
690:The
671:The
651:and
567:The
436:and
346:rank
332:The
220:ring
154:The
36:, a
20:The
789:doi
746:doi
661:act
628:.
504:to
32:In
855::
797:MR
795:.
785:29
783:.
742:18
740:.
736:.
687:.
663:.
597:.
582:.
564:.
326:.
275:+
261:=
162:).
141:A
138:.
844:.
803:.
791::
754:.
748::
635:)
631:(
548:1
545:+
542:)
539:1
533:n
530:(
527:)
524:1
518:m
515:(
512:2
488:1
485:+
482:n
476:m
470:n
467:m
464:2
444:n
424:m
377:2
373:F
323:n
319:α
315:1
312:α
304:n
300:x
298:â‹…
295:n
291:α
287:2
284:x
282:â‹…
280:2
277:α
273:1
270:x
268:â‹…
266:1
263:α
259:x
248:x
243:n
239:x
235:1
232:x
224:Z
172:(
136:Z
120:.
114:G
110:G
93:Q
82:G
78:S
63:S
59:G
55:S
45:G
28:.
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