373:
283:
528:
567:
263:
493:
421:
75:, entirely analogously to the commutator collecting process. Hall words also provide a unique
538:
505:
426:
76:
8:
29:
63:; these can be placed in one-to-one correspondence with words, these being called the
562:
542:
524:
457:
154:
72:
534:
520:
512:
501:
48:
36:
arranged in a certain order. The commutator collecting process was introduced by
122:
92:
41:
461:
556:
60:
17:
546:
239:, and for commutators of any fixed weight some total ordering is chosen.
445:
68:
37:
21:
448:(1934), "A contribution to the theory of groups of prime-power order",
88:
52:
33:
64:
473:
W. Magnus (1937), "Ăśber
Beziehungen zwischen höheren Kommutatoren",
91:, as a similar theorem then holds for any group by writing it as a
71:
are a special case. Hall sets are used to construct a basis for a
56:
44:
in 1937. The process is sometimes called a "collection process".
409:
368:{\displaystyle g=c_{1}^{n_{1}}c_{2}^{n_{2}}\cdots c_{k}^{n_{k}}c}
266:
with a basis consisting of basic commutators of weight
87:
The commutator collecting process is usually stated for
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51:
subset of a free non-associative algebra, that is, a
395:is a product of commutators of weight greater than
367:
554:
450:Proceedings of the London Mathematical Society
387:are the basic commutators of weight at most
32:as a product of generators and their higher
187:are basic commutators whose weights sum to
47:The process can be generalized to define a
179: > 1 are the elements where
153:The basic commutators of weight 1 are the
28:is a method for writing an element of a
511:
555:
142:The basic commutators are elements of
264:finitely generated free abelian group
492:
444:
203: = for basic commutators
13:
14:
579:
235:has weight greater than that of
223:Commutators are ordered so that
175:The basic commutators of weight
149:defined and ordered as follows:
519:(in German), Berlin, New York:
467:
438:
105:is a free group on generators
59:. Members of the Hall set are
1:
432:
26:commutator collecting process
82:
55:; this subset is called the
7:
415:
40:in 1934 and articulated by
10:
584:
568:Combinatorial group theory
486:
121:. Define the descending
77:factorization of monoids
462:10.1112/plms/s2-36.1.29
391:arranged in order, and
422:Hall–Petresco identity
369:
370:
498:The theory of groups
427:Monoid factorisation
284:
273:Then any element of
361:
336:
314:
215: ≤
523:, pp. 90–93,
365:
340:
315:
293:
277:can be written as
95:of a free group.
530:978-3-540-03825-2
163:, ...,
112:, ...,
575:
549:
517:Endliche Gruppen
508:
481:
471:
465:
464:
442:
374:
372:
371:
366:
360:
359:
358:
348:
335:
334:
333:
323:
313:
312:
311:
301:
227: >
195: >
73:free Lie algebra
583:
582:
578:
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574:
573:
572:
553:
552:
531:
521:Springer-Verlag
489:
484:
472:
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354:
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329:
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285:
282:
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261:
251:
171:
162:
148:
137:
120:
111:
104:
85:
49:totally ordered
12:
11:
5:
581:
571:
570:
565:
551:
550:
529:
509:
494:Hall, Marshall
488:
485:
483:
482:
466:
436:
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431:
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417:
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403:
382:
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364:
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256:
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221:
220:
173:
167:
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146:
140:
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132:
123:central series
116:
109:
102:
84:
81:
42:Wilhelm Magnus
20:, a branch of
9:
6:
4:
3:
2:
580:
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561:
560:
558:
548:
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500:, Macmillan,
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428:
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159:
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152:
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138: =
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115:
108:
101:
96:
94:
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78:
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50:
45:
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39:
35:
31:
27:
23:
19:
516:
497:
477:
474:
469:
453:
449:
446:Hall, Philip
440:
404:
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396:
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379:
377:
274:
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247:
243:
241:
236:
232:
228:
224:
222:
216:
212:
208:
204:
200:
196:
192:
191:, such that
188:
184:
180:
176:
168:
164:
157:
143:
141:
133:
129:
117:
113:
106:
99:
97:
86:
69:Lyndon words
61:binary trees
46:
25:
18:group theory
15:
513:Huppert, B.
125:by putting
89:free groups
38:Philip Hall
34:commutators
22:mathematics
557:Categories
480:, 105-115.
433:References
399:, and the
378:where the
155:generators
65:Hall words
53:free magma
475:J. Grelle
456:: 29–95,
338:⋯
83:Statement
563:P-groups
515:(1967),
496:(1959),
416:See also
410:integers
98:Suppose
93:quotient
57:Hall set
539:0224703
506:0103215
487:Reading
250:
199:and if
547:527050
545:
537:
527:
504:
67:; the
24:, the
262:is a
242:Then
211:then
30:group
543:OCLC
525:ISBN
408:are
207:and
183:and
478:177
458:doi
231:if
16:In
559::
541:,
535:MR
533:,
502:MR
454:36
452:,
412:.
270:.
260:+1
136:+1
79:.
460::
405:i
401:n
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393:c
389:m
384:i
380:c
363:c
356:k
352:n
346:k
342:c
331:2
327:n
321:2
317:c
309:1
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299:1
295:c
291:=
288:g
275:F
268:n
258:n
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252:/
248:n
244:F
237:y
233:x
229:y
225:x
219:.
217:y
213:v
209:v
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172:.
169:m
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161:1
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147:1
144:F
134:n
130:F
118:m
114:a
110:1
107:a
103:1
100:F
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