475:
489:
Given a solvable group, the lower
Fitting series is a "coarser" division than the lower central series: the lower Fitting series gives a series for the whole group, while the lower central series descends only from the whole group to the first term of the Fitting series.
174:. These correspond to the center and the commutator subgroup (for upper and lower central series, respectively). These do not hold for infinite groups, so for the sequel, assume all groups to be finite.
385:
The lower
Fitting series descends most quickly amongst all Fitting chains, and the upper Fitting series ascends most quickly amongst all Fitting chains. Explicitly: For every Fitting chain, 1 =
486:
do for nilpotent groups, Fitting series do for solvable groups. A group has a central series if and only if it is nilpotent, and a
Fitting series if and only if it is solvable.
478:
Combining the lower
Fitting series and lower central series on a solvable group yields a series with coarse and fine divisions, like the coarse and fine marks on a ruler.
458:
For a solvable group, the length of the lower
Fitting series is equal to length of the upper Fitting series, and this common length is the Fitting length of the group.
684:
567:
Proceeding in this way (lifting the lower central series for each quotient of the
Fitting series) yields a subnormal series:
155:) is the maximal normal nilpotent subgroup, while the minimal normal subgroup such that the quotient by it is nilpotent is
738:
720:
733:
715:
367:
The lower
Fitting series is a Fitting chain if and only if it eventually reaches the trivial subgroup, if and only if
641:
The successive quotients are abelian, showing the equivalence between being solvable and having a
Fitting series.
322:
754:
241:
167:
130:
126:
710:
728:
114:
80:
374:
The upper
Fitting series is a Fitting chain if and only if it eventually reaches the whole group,
171:
698:
680:
329:
145:
102:
of the previous one, and such that the quotients of successive terms are nilpotent groups.
84:
52:
694:
690:
676:
668:
336:
99:
88:
44:
650:
483:
361:
134:
91:
40:
748:
655:
474:
48:
28:
117:
is defined to be the smallest possible length of a
Fitting chain, if one exists.
20:
702:
98:
including both the whole group and the trivial group, such that each is a
95:
339:
on five or more points has no Fitting chain at all, not being solvable.
318:
A nontrivial group has Fitting length 1 if and only if it is nilpotent.
24:
635:
473:
346:
copies of the symmetric group on three points has Fitting length 2
550:
and is a lift of the lower central series for the first quotient
305:)). It is a descending nilpotent series, at each step taking the
225:)). It is an ascending nilpotent series, at each step taking the
181:
of a finite group is the sequence of characteristic subgroups
137:, there are analogous series extremal among nilpotent series.
518:
while the lower central series subdivides the first step,
470:
Connection between central series and Fitting series
360:A group has a Fitting chain if and only if it is
16:Measurement in group theory algebra mathematics
8:
51:, due to his investigations of nilpotent
634:like the coarse and fine divisions on a
94:. In other words, a finite sequence of
463:
7:
166:), the intersection of the (finite)
493:The lower Fitting series proceeds:
462:More information can be found in (
14:
675:(in German), Berlin, New York:
342:The iterated wreath product of
323:symmetric group on three points
330:symmetric group on four points
121:Upper and lower Fitting series
23:, specifically in the area of
1:
47:. The concept is named after
734:Encyclopedia of Mathematics
727:Turull, Alexandre (2001) ,
716:Encyclopedia of Mathematics
709:Turull, Alexandre (2001) ,
771:
242:characteristic subgroups
564:, which is nilpotent.
479:
170:, which is called the
477:
332:has Fitting length 3.
325:has Fitting length 2.
39:) measures how far a
234:lower Fitting series
179:upper Fitting series
168:lower central series
131:lower central series
127:upper central series
309:possible subgroup.
240:is the sequence of
229:possible subgroup.
140:For a finite group
133:are extremal among
480:
236:of a finite group
172:nilpotent residual
686:978-3-540-03825-2
466:, Kap. III, §4).
378:, if and only if
762:
741:
723:
711:"Fitting length"
705:
673:Endliche Gruppen
146:Fitting subgroup
111:nilpotent length
85:subnormal series
77:
76:
75:nilpotent series
53:normal subgroups
37:nilpotent length
770:
769:
765:
764:
763:
761:
760:
759:
755:Subgroup series
745:
744:
729:"Fitting chain"
726:
708:
687:
677:Springer-Verlag
667:
664:
647:
629:
622:
615:
608:
601:
594:
587:
580:
563:
556:
545:
538:
531:
513:
506:
472:
454:
437:
420:
412:, one has that
407:
398:
391:
357:
337:symmetric group
315:
300:
291:
280:
262:
251:
161:
123:
100:normal subgroup
74:
73:
61:
17:
12:
11:
5:
768:
766:
758:
757:
747:
746:
743:
742:
724:
706:
685:
663:
660:
659:
658:
653:
651:Central series
646:
643:
632:
631:
627:
620:
613:
606:
599:
592:
585:
578:
561:
554:
548:
547:
543:
536:
529:
516:
515:
511:
504:
484:central series
471:
468:
460:
459:
456:
446:
433:
416:
403:
396:
389:
383:
372:
365:
356:
353:
352:
351:
340:
333:
326:
319:
314:
311:
296:
289:
275:
260:
256:) defined by
247:
159:
135:central series
122:
119:
107:Fitting length
69:Fitting series
60:
57:
43:is from being
41:solvable group
33:Fitting length
15:
13:
10:
9:
6:
4:
3:
2:
767:
756:
753:
752:
750:
740:
736:
735:
730:
725:
722:
718:
717:
712:
707:
704:
700:
696:
692:
688:
682:
678:
674:
670:
666:
665:
661:
657:
654:
652:
649:
648:
644:
642:
639:
637:
626:
619:
612:
605:
598:
591:
584:
577:
573:
570:
569:
568:
565:
560:
553:
542:
535:
528:
524:
521:
520:
519:
510:
503:
499:
496:
495:
494:
491:
487:
485:
476:
469:
467:
465:
457:
453:
449:
445:
441:
436:
432:
428:
424:
419:
415:
411:
406:
402:
395:
388:
384:
381:
377:
373:
370:
366:
363:
359:
358:
354:
349:
345:
341:
338:
334:
331:
327:
324:
320:
317:
316:
312:
310:
308:
304:
299:
295:
288:
284:
278:
274:
270:
266:
259:
255:
250:
246:
243:
239:
235:
230:
228:
224:
220:
216:
212:
208:
204:
200:
196:
192:
189:) defined by
188:
184:
180:
175:
173:
169:
165:
158:
154:
150:
147:
143:
138:
136:
132:
128:
120:
118:
116:
112:
108:
103:
101:
97:
93:
90:
86:
82:
78:
70:
66:
65:Fitting chain
58:
56:
54:
50:
46:
42:
38:
34:
30:
26:
22:
732:
714:
672:
656:3-step group
640:
633:
624:
617:
610:
603:
596:
589:
582:
575:
571:
566:
558:
551:
549:
540:
533:
526:
522:
517:
508:
501:
497:
492:
488:
481:
464:Huppert 1967
461:
451:
447:
443:
439:
434:
430:
426:
422:
417:
413:
409:
404:
400:
393:
386:
382:is solvable.
379:
375:
371:is solvable.
368:
347:
343:
306:
302:
297:
293:
286:
282:
276:
272:
268:
264:
257:
253:
248:
244:
237:
233:
231:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
176:
163:
156:
152:
148:
141:
139:
125:Just as the
124:
110:
106:
104:
72:
68:
64:
62:
49:Hans Fitting
36:
32:
29:group theory
18:
669:Huppert, B.
197:) = 1, and
21:mathematics
662:References
355:Properties
59:Definition
739:EMS Press
721:EMS Press
96:subgroups
92:quotients
89:nilpotent
45:nilpotent
27:known as
749:Category
671:(1967),
645:See also
514:⊵ ⋯ ⊵ 1,
429:), and
362:solvable
313:Examples
79:) for a
695:0224703
307:minimal
227:maximal
25:algebra
703:527050
701:
693:
683:
623:⊵ ⋯ ⊵
609:⊵ ⋯ ⊵
588:⊵ ⋯ ⊵
539:⊵ ⋯ ⊵
399:⊲ … ⊲
271:, and
144:, the
31:, the
636:ruler
482:What
115:group
113:of a
87:with
83:is a
81:group
699:OCLC
681:ISBN
630:= 1,
442:) ≤
335:The
328:The
321:The
285:) =
267:) =
232:The
213:) =
177:The
129:and
105:The
67:(or
35:(or
621:2,1
607:1,2
600:1,1
423:Fit
219:Fit
217:(G/
215:Fit
207:Fit
199:Fit
191:Fit
183:Fit
149:Fit
109:or
71:or
19:In
751::
737:,
731:,
719:,
713:,
697:,
691:MR
689:,
679:,
638:.
616:=
602:⊵
595:=
581:⊵
574:=
532:⊵
525:=
507:⊵
500:=
421:≤
408:=
392:⊲
279:+1
205:)/
63:A
55:.
628:n
625:F
618:F
614:2
611:F
604:F
597:F
593:1
590:F
586:2
583:G
579:1
576:G
572:G
562:1
559:F
557:/
555:0
552:F
546:,
544:1
541:F
537:2
534:G
530:1
527:G
523:G
512:1
509:F
505:0
502:F
498:G
455:.
452:i
450:−
448:n
444:H
440:G
438:(
435:i
431:F
427:G
425:(
418:i
414:H
410:G
405:n
401:H
397:1
394:H
390:0
387:H
380:G
376:G
369:G
364:.
350:.
348:n
344:n
303:G
301:(
298:n
294:F
292:(
290:∞
287:γ
283:G
281:(
277:n
273:F
269:G
265:G
263:(
261:0
258:F
254:G
252:(
249:n
245:F
238:G
223:G
221:(
211:G
209:(
203:G
201:(
195:G
193:(
187:G
185:(
164:H
162:(
160:∞
157:γ
153:H
151:(
142:H
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