540:
if every element acts as an inner automorphism on every chief factor. The generalized
Fitting subgroup is the unique largest subnormal quasi-nilpotent subgroup, and is equal to the set of all elements which act as inner automorphisms on every chief factor of the whole group
719:
is a finite solvable group, then the
Fitting subgroup contains its own centralizer. The centralizer of the Fitting subgroup is the center of the Fitting subgroup. In this case, the generalized Fitting subgroup is equal to the Fitting subgroup. More generally, if
842:-local subgroups usually have components in the generalized Fitting subgroup, though there are many exceptions for groups that have small rank, are defined over small fields, or are sporadic. This is used to classify the finite simple groups, because if a
652:
310:
535:
The generalized
Fitting subgroup can also be viewed as a generalized centralizer of chief factors. A nonabelian semisimple group cannot centralize itself, but it does act on itself as inner automorphisms. A group is said to be
371:) of a group is the subgroup generated by all components. Any two components of a group commute, so the layer is a perfect central extension of a product of simple groups, and is the largest normal subgroup of
208:
is centralized by every element. Relaxing the condition somewhat, and taking the subgroup of elements of a general finite group which centralize every chief factor, one simply gets the
Fitting subgroup again
383:) is the subgroup generated by the layer and the Fitting subgroup. The layer commutes with the Fitting subgroup, so the generalized Fitting subgroup is a central extension of a product of
853:
The analysis of finite simple groups by means of the structure and embedding of the generalized
Fitting subgroups of their maximal subgroups was originated by Helmut Bender (
551:
219:
401:
This definition of the generalized
Fitting subgroup can be motivated by some of its intended uses. Consider the problem of trying to identify a normal subgroup
724:
is a finite group, then the generalized
Fitting subgroup contains its own centralizer. This means that in some sense the generalized Fitting subgroup controls
532:
contains its own centralizer. The generalized
Fitting subgroup is the smallest subgroup that contains the Fitting subgroup and all normal semisimple subgroups.
820:
966:
887:
823:, this allows one to guess over which field a simple group should be defined. Note that a few groups are of characteristic
195:
1003:
1034:
914:
647:{\displaystyle \operatorname {Fit} ^{*}(G)=\bigcap \{HC_{G}(H/K):H/K{\text{ a chief factor of }}G\}.}
100:, the Fitting subgroup is defined to be the subgroup generated by the nilpotent normal subgroups of
764:). In particular there are only a finite number of groups with given generalized Fitting subgroup.
349:
131:
is again a normal nilpotent subgroup. It may also be explicitly constructed as the product of the
85:
998:, Grundlehren der Mathematischen Wissenschaften, vol. 243, Berlin-New York: Springer-Verlag,
426:
305:{\displaystyle \operatorname {Fit} (G)=\bigcap \{C_{G}(H/K):H/K{\text{ a chief factor of }}G\}.}
154:
is a finite non-trivial solvable group then the
Fitting subgroup is always non-trivial, i.e. if
1029:
1013:
976:
943:
783:
and exert a great deal of control over the structure of the group (allowing what is called
124:
8:
846:-local subgroup has a known component, it is often possible to identify the whole group (
896:
875:
862:
858:
335:
198:
205:
60:. Intuitively, it represents the smallest subgroup which "controls" the structure of
999:
980:
962:
931:
883:
812:
338:
127:
which says that the product of a finite collection of normal nilpotent subgroups of
923:
21:
394:
The layer is also the maximal normal semisimple subgroup, where a group is called
1009:
991:
972:
958:
950:
939:
191:
120:
50:
47:
784:
183:
69:
899:; Seitz, Gary M. (1976), "On groups with a standard component of known type",
1023:
935:
346:
861:. It is especially effective in the exceptional cases where components or
465:
388:
132:
108:
43:
36:
25:
984:
834:
If a simple group is not of Lie type over a field of given characteristic
186:. Since the Fitting subgroup of a finite solvable group contains its own
496:) is a product of non-abelian simple groups then the derived subgroup of
316:
187:
17:
516:
contains the
Fitting subgroup and all normal semisimple subgroups, then
927:
398:
if it is a perfect central extension of a product of simple groups.
104:. For infinite groups, the Fitting subgroup is not always nilpotent.
912:
Bender, Helmut (1970), "On groups with abelian Sylow 2-subgroups",
53:
190:, this gives a method of understanding finite solvable groups as
409:
that contains its own centralizer and the Fitting group. If
123:
of the Fitting subgroup of a finite group is guaranteed by
182:
is not itself nilpotent, giving rise to the concept of
375:
with this structure. The generalized Fitting subgroup
554:
222:
107:
The remainder of this article deals exclusively with
83:, which is generated by the Fitting subgroup and the
325:
646:
304:
76:is not solvable, a similar role is played by the
1021:
990:
542:
96:For an arbitrary (not necessarily finite) group
895:
847:
500:is a normal semisimple subgroup mapping onto
776:-subgroups of a finite group are called the
740:) is contained in the automorphism group of
638:
583:
296:
244:
444:, which is the same as the intersection of
874:
166:)≠1. Similarly the Fitting subgroup of
949:
815:defined over a field of characteristic
468:as it is characteristically simple. If
210:
114:
1022:
911:
854:
821:classification of finite simple groups
480:) is a product of cyclic groups then
787:). A finite group is said to be of
484:must be in the Fitting subgroup. If
545:, Chapter X, Theorem 5.4, p. 126):
13:
14:
1046:
326:The generalized Fitting subgroup
957:(in German), Berlin, New York:
767:
679:) if and only if there is some
882:, Cambridge University Press,
857:) and has come to be known as
772:The normalizers of nontrivial
613:
599:
574:
568:
271:
257:
235:
229:
1:
868:
811:-local subgroup, because any
710:
632: a chief factor of
290: a chief factor of
994:; Blackburn, Norman (1982),
543:Huppert & Blackburn 1982
464:) is a product of simple or
213:, Kap.VI, Satz 5.4, p.686):
204:In a nilpotent group, every
158:≠1 is finite solvable, then
78:generalized Fitting subgroup
20:, especially in the area of
7:
848:Aschbacher & Seitz 1976
819:has this property. In the
10:
1051:
748:), and the centralizer of
732:modulo the centralizer of
915:Mathematische Zeitschrift
425:. If not, pick a minimal
352:of a simple group.) The
178:) will be nontrivial if
46:, is the unique largest
827:type for more than one
427:characteristic subgroup
194:of nilpotent groups by
139:over all of the primes
648:
528:) must be trivial, so
417:we want to prove that
413:is the centralizer of
341:subgroup. (A group is
315:The generalization to
306:
143:dividing the order of
649:
307:
201:of nilpotent groups.
1035:Functional subgroups
865:are not applicable.
687:such that for every
552:
220:
115:The Fitting subgroup
996:Finite groups. III.
897:Aschbacher, Michael
880:Finite Group Theory
876:Aschbacher, Michael
863:signalizer functors
322:groups is similar.
199:automorphism groups
928:10.1007/BF01109839
756:) is contained in
644:
302:
968:978-3-540-03825-2
889:978-0-521-78675-1
813:group of Lie type
807:-group for every
633:
440:is the center of
350:central extension
291:
125:Fitting's theorem
1042:
1016:
992:Huppert, Bertram
987:
955:Endliche Gruppen
946:
908:
892:
781:-local subgroups
657:Here an element
653:
651:
650:
645:
634:
631:
626:
609:
598:
597:
564:
563:
421:is contained in
334:of a group is a
311:
309:
308:
303:
292:
289:
284:
267:
256:
255:
30:Fitting subgroup
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1049:
1045:
1044:
1043:
1041:
1040:
1039:
1020:
1019:
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969:
959:Springer-Verlag
890:
871:
859:Bender's method
789:characteristic
770:
713:
670:
630:
622:
605:
593:
589:
559:
555:
553:
550:
549:
538:quasi-nilpotent
328:
288:
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263:
251:
247:
221:
218:
217:
117:
12:
11:
5:
1048:
1038:
1037:
1032:
1018:
1017:
1004:
988:
967:
947:
909:
901:Osaka J. Math.
893:
888:
870:
867:
785:local analysis
769:
766:
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184:Fitting length
116:
113:
42:, named after
9:
6:
4:
3:
2:
1047:
1036:
1033:
1031:
1030:Finite groups
1028:
1027:
1025:
1015:
1011:
1007:
1005:3-540-10633-2
1001:
997:
993:
989:
986:
982:
978:
974:
970:
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956:
952:
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933:
929:
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623:
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495:
491:
487:
483:
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467:
466:cyclic groups
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459:
455:
451:
447:
443:
439:
435:
431:
428:
424:
420:
416:
412:
408:
404:
399:
397:
392:
390:
389:simple groups
386:
382:
378:
374:
370:
366:
362:
358:
355:
351:
348:
344:
340:
337:
333:
323:
321:
319:
299:
293:
285:
281:
277:
274:
268:
264:
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252:
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241:
238:
232:
226:
223:
216:
215:
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207:
202:
200:
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185:
181:
177:
173:
169:
165:
161:
157:
153:
148:
146:
142:
138:
134:
130:
126:
122:
112:
110:
109:finite groups
105:
103:
99:
94:
92:
88:
87:
82:
79:
75:
71:
67:
63:
59:
55:
52:
49:
45:
41:
38:
34:
31:
27:
23:
19:
995:
954:
919:
913:
907:(3): 439–482
904:
900:
879:
852:
843:
839:
835:
833:
828:
824:
816:
808:
804:
800:
796:
790:
788:
778:
777:
773:
771:
768:Applications
761:
757:
753:
749:
745:
741:
737:
733:
729:
725:
721:
716:
714:
704:
700:
696:
692:
688:
684:
680:
676:
672:
667:
662:
658:
656:
537:
534:
529:
525:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
469:
461:
457:
453:
449:
445:
441:
437:
433:
429:
422:
418:
414:
410:
406:
402:
400:
395:
393:
387:-groups and
384:
380:
376:
372:
368:
364:
360:
356:
353:
342:
331:
329:
317:
314:
211:Huppert 1967
206:chief factor
203:
179:
175:
171:
167:
163:
159:
155:
151:
149:
144:
140:
136:
128:
118:
106:
101:
97:
95:
90:
84:
80:
77:
73:
65:
61:
57:
44:Hans Fitting
39:
37:finite group
32:
29:
26:group theory
15:
951:Huppert, B.
922:: 164–176,
855:Bender 1970
838:, then the
345:if it is a
343:quasisimple
339:quasisimple
188:centralizer
18:mathematics
1024:Categories
869:References
728:, because
711:Properties
396:semisimple
320:-nilpotent
192:extensions
121:nilpotency
86:components
936:0025-5874
581:⋂
566:
561:∗
512:). So if
336:subnormal
332:component
242:⋂
227:
51:nilpotent
24:known as
953:(1967),
878:(2000),
452:. Then
436:, where
196:faithful
70:solvable
54:subgroup
1014:0650245
977:0224703
944:0288180
803:) is a
347:perfect
133:p-cores
72:. When
22:algebra
1012:
1002:
985:527050
983:
975:
965:
942:
934:
886:
661:is in
434:C/Z(H)
430:M/Z(H)
48:normal
28:, the
363:) or
354:layer
64:when
35:of a
1000:ISBN
981:OCLC
963:ISBN
932:ISSN
884:ISBN
793:type
703:mod
448:and
438:Z(H)
119:The
924:doi
920:117
850:).
795:if
715:If
691:in
683:in
557:Fit
432:of
405:of
224:Fit
150:If
135:of
93:.
89:of
68:is
56:of
16:In
1026::
1010:MR
1008:,
979:,
973:MR
971:,
961:,
940:MR
938:,
930:,
918:,
905:13
903:,
831:.
707:.
699:≡
695:,
391:.
330:A
147:.
111:.
926::
844:p
840:p
836:p
829:p
825:p
817:p
809:p
805:p
801:G
799:(
797:F
791:p
779:p
774:p
762:G
760:(
758:F
754:G
752:(
750:F
746:G
744:(
742:F
738:G
736:(
734:F
730:G
726:G
722:G
717:G
705:K
701:x
697:x
693:H
689:x
685:H
681:h
677:K
675:/
673:H
671:(
668:G
665:C
663:H
659:g
642:.
639:}
636:G
628:K
624:/
620:H
617::
614:)
611:K
607:/
603:H
600:(
595:G
591:C
587:H
584:{
578:=
575:)
572:G
569:(
541:(
530:H
526:H
524:(
522:Z
520:/
518:M
514:H
510:H
508:(
506:Z
504:/
502:M
498:M
494:H
492:(
490:Z
488:/
486:M
482:M
478:H
476:(
474:Z
472:/
470:M
462:H
460:(
458:Z
456:/
454:M
450:H
446:C
442:H
423:H
419:C
415:H
411:C
407:G
403:H
385:p
381:G
379:(
377:F
373:G
369:G
367:(
365:L
361:G
359:(
357:E
318:p
300:.
297:}
294:G
286:K
282:/
278:H
275::
272:)
269:K
265:/
261:H
258:(
253:G
249:C
245:{
239:=
236:)
233:G
230:(
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168:G
164:G
162:(
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152:G
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141:p
137:G
129:G
102:G
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58:G
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33:F
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