551:
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
730:
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
853:-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
663:
321:
546:
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
382:) 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
219:
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
394:) 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
864:
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 (
562:
230:
412:
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
735:
is a finite group, then the generalized
Fitting subgroup contains its own centralizer. This means that in some sense the generalized Fitting subgroup controls
543:
contains its own centralizer. The generalized
Fitting subgroup is the smallest subgroup that contains the Fitting subgroup and all normal semisimple subgroups.
831:
977:
898:
1014:
834:, this allows one to guess over which field a simple group should be defined. Note that a few groups are of characteristic
1045:
206:
925:
96:
437:
1040:
658:{\displaystyle \operatorname {Fit} ^{*}(G)=\bigcap \{HC_{G}(H/K):H/K{\text{ a chief factor of }}G\}.}
135:
111:, the Fitting subgroup is defined to be the subgroup generated by the nilpotent normal subgroups of
775:). In particular there are only a finite number of groups with given generalized Fitting subgroup.
142:
is again a normal nilpotent subgroup. It may also be explicitly constructed as the product of the
1009:, Grundlehren der Mathematischen Wissenschaften, vol. 243, Berlin-New York: Springer-Verlag,
907:
886:
873:
346:
209:
869:
316:{\displaystyle \operatorname {Fit} (G)=\bigcap \{C_{G}(H/K):H/K{\text{ a chief factor of }}G\}.}
165:
is a finite non-trivial solvable group then the
Fitting subgroup is always non-trivial, i.e. if
1010:
991:
973:
942:
894:
823:
349:
934:
32:
17:
1024:
987:
954:
794:
and exert a great deal of control over the structure of the group (allowing what is called
1020:
1002:
983:
969:
961:
950:
360:
202:
131:
61:
58:
857:-local subgroup has a known component, it is often possible to identify the whole group (
795:
194:
80:
71:. Intuitively, it represents the smallest subgroup which "controls" the structure of
1034:
357:
138:
which says that the product of a finite collection of normal nilpotent subgroups of
476:
399:
216:
143:
119:
54:
47:
36:
405:
The layer is also the maximal normal semisimple subgroup, where a group is called
327:
198:
28:
910:; Seitz, Gary M. (1976), "On groups with a standard component of known type",
946:
872:. It is especially effective in the exceptional cases where components or
995:
845:
If a simple group is not of Lie type over a field of given characteristic
197:. Since the Fitting subgroup of a finite solvable group contains its own
507:) is a product of non-abelian simple groups then the derived subgroup of
64:
527:
contains the
Fitting subgroup and all normal semisimple subgroups, then
938:
409:
if it is a perfect central extension of a product of simple groups.
115:. For infinite groups, the Fitting subgroup is not always nilpotent.
923:
Bender, Helmut (1970), "On groups with abelian Sylow 2-subgroups",
201:, this gives a method of understanding finite solvable groups as
420:
that contains its own centralizer and the Fitting group. If
134:
of the Fitting subgroup of a finite group is guaranteed by
193:
is not itself nilpotent, giving rise to the concept of
386:
with this structure. The generalized Fitting subgroup
565:
233:
118:
The remainder of this article deals exclusively with
94:, which is generated by the Fitting subgroup and the
657:
315:
87:is not solvable, a similar role is played by the
553:
107:For an arbitrary (not necessarily finite) group
858:
511:is a normal semisimple subgroup mapping onto
8:
787:-subgroups of a finite group are called the
751:) is contained in the automorphism group of
649:
594:
307:
255:
455:, which is the same as the intersection of
641:
633:
616:
604:
570:
564:
299:
291:
274:
262:
232:
177:)≠1. Similarly the Fitting subgroup of
826:defined over a field of characteristic
479:as it is characteristically simple. If
221:
865:
832:classification of finite simple groups
491:) is a product of cyclic groups then
7:
798:). A finite group is said to be of
495:must be in the Fitting subgroup. If
556:, Chapter X, Theorem 5.4, p. 126):
25:
337:The generalized Fitting subgroup
968:(in German), Berlin, New York:
690:) if and only if there is some
893:, Cambridge University Press,
868:) and has come to be known as
783:The normalizers of nontrivial
624:
610:
585:
579:
282:
268:
246:
240:
1:
822:-local subgroup, because any
643: a chief factor of
301: a chief factor of
1005:; Blackburn, Norman (1982),
554:Huppert & Blackburn 1982
475:) is a product of simple or
224:, Kap.VI, Satz 5.4, p.686):
215:In a nilpotent group, every
169:≠1 is finite solvable, then
89:generalized Fitting subgroup
31:, especially in the area of
18:Generalized Fitting subgroup
859:Aschbacher & Seitz 1976
830:has this property. In the
1062:
759:), and the centralizer of
743:modulo the centralizer of
926:Mathematische Zeitschrift
436:. If not, pick a minimal
363:of a simple group.) The
189:) will be nontrivial if
57:, is the unique largest
838:type for more than one
438:characteristic subgroup
205:of nilpotent groups by
150:over all of the primes
659:
539:) must be trivial, so
428:we want to prove that
424:is the centralizer of
352:subgroup. (A group is
326:The generalization to
317:
154:dividing the order of
660:
318:
212:of nilpotent groups.
1046:Functional subgroups
876:are not applicable.
698:such that for every
563:
231:
126:The Fitting subgroup
1007:Finite groups. III.
908:Aschbacher, Michael
891:Finite Group Theory
887:Aschbacher, Michael
874:signalizer functors
333:groups is similar.
210:automorphism groups
939:10.1007/BF01109839
767:) is contained in
655:
313:
979:978-3-540-03825-2
900:978-0-521-78675-1
824:group of Lie type
818:-group for every
644:
451:is the center of
361:central extension
302:
136:Fitting's theorem
16:(Redirected from
1053:
1027:
1003:Huppert, Bertram
998:
966:Endliche Gruppen
957:
919:
903:
792:-local subgroups
668:Here an element
664:
662:
661:
656:
645:
642:
637:
620:
609:
608:
575:
574:
432:is contained in
345:of a group is a
322:
320:
319:
314:
303:
300:
295:
278:
267:
266:
41:Fitting subgroup
21:
1061:
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1001:
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970:Springer-Verlag
960:
922:
906:
901:
885:
882:
870:Bender's method
800:characteristic
781:
724:
681:
600:
566:
561:
560:
549:quasi-nilpotent
339:
258:
229:
228:
128:
23:
22:
15:
12:
11:
5:
1059:
1057:
1049:
1048:
1043:
1033:
1032:
1029:
1028:
1015:
999:
978:
958:
920:
912:Osaka J. Math.
904:
899:
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796:local analysis
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777:
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195:Fitting length
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53:, named after
24:
14:
13:
10:
9:
6:
4:
3:
2:
1058:
1047:
1044:
1042:
1041:Finite groups
1039:
1038:
1036:
1026:
1022:
1018:
1016:3-540-10633-2
1012:
1008:
1004:
1000:
997:
993:
989:
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477:cyclic groups
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419:
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400:simple groups
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393:
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385:
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366:
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355:
351:
348:
344:
336:
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332:
330:
310:
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296:
292:
288:
285:
279:
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271:
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252:
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234:
227:
226:
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223:
218:
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211:
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200:
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176:
172:
168:
164:
159:
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149:
145:
141:
137:
133:
125:
123:
121:
120:finite groups
116:
114:
110:
105:
103:
99:
98:
93:
90:
86:
82:
78:
74:
70:
66:
63:
60:
56:
52:
49:
45:
42:
38:
34:
30:
19:
1006:
965:
930:
924:
918:(3): 439–482
915:
911:
890:
863:
854:
850:
846:
844:
839:
835:
827:
819:
815:
811:
807:
801:
799:
789:
788:
784:
782:
779:Applications
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
732:
727:
725:
715:
711:
707:
703:
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695:
691:
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683:
678:
673:
669:
667:
548:
545:
540:
536:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
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472:
468:
464:
460:
456:
452:
448:
444:
440:
433:
429:
425:
421:
417:
413:
411:
406:
404:
398:-groups and
395:
391:
387:
383:
379:
375:
371:
367:
364:
353:
342:
340:
328:
325:
222:Huppert 1967
217:chief factor
214:
190:
186:
182:
178:
174:
170:
166:
162:
160:
155:
151:
147:
139:
129:
117:
112:
108:
106:
101:
95:
91:
88:
84:
76:
72:
68:
55:Hans Fitting
50:
48:finite group
43:
40:
37:group theory
26:
962:Huppert, B.
933:: 164–176,
866:Bender 1970
849:, then the
356:if it is a
354:quasisimple
350:quasisimple
199:centralizer
29:mathematics
1035:Categories
880:References
739:, because
722:Properties
407:semisimple
331:-nilpotent
203:extensions
132:nilpotency
97:components
947:0025-5874
592:⋂
577:
572:∗
523:). So if
347:subnormal
343:component
253:⋂
238:
62:nilpotent
35:known as
964:(1967),
889:(2000),
463:. Then
447:, where
207:faithful
81:solvable
65:subgroup
1025:0650245
988:0224703
955:0288180
814:) is a
358:perfect
144:p-cores
83:. When
33:algebra
1023:
1013:
996:527050
994:
986:
976:
953:
945:
897:
672:is in
445:C/Z(H)
441:M/Z(H)
59:normal
39:, the
374:) or
365:layer
75:when
46:of a
1011:ISBN
992:OCLC
974:ISBN
943:ISSN
895:ISBN
804:type
714:mod
459:and
449:Z(H)
130:The
935:doi
931:117
861:).
806:if
726:If
702:in
694:in
568:Fit
443:of
416:of
235:Fit
161:If
146:of
104:.
100:of
79:is
67:of
27:In
1037::
1021:MR
1019:,
990:,
984:MR
982:,
972:,
951:MR
949:,
941:,
929:,
916:13
914:,
842:.
718:.
710:≡
706:,
402:.
341:A
158:.
122:.
937::
855:p
851:p
847:p
840:p
836:p
828:p
820:p
816:p
812:G
810:(
808:F
802:p
790:p
785:p
773:G
771:(
769:F
765:G
763:(
761:F
757:G
755:(
753:F
749:G
747:(
745:F
741:G
737:G
733:G
728:G
716:K
712:x
708:x
704:H
700:x
696:H
692:h
688:K
686:/
684:H
682:(
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676:C
674:H
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653:.
650:}
647:G
639:K
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628::
625:)
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618:/
614:H
611:(
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589:=
586:)
583:G
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487:(
485:Z
483:/
481:M
473:H
471:(
469:Z
467:/
465:M
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453:H
434:H
430:C
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418:G
414:H
396:p
392:G
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388:F
384:G
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376:L
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311:.
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286::
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272:H
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256:{
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247:)
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241:(
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148:G
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69:G
51:G
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20:)
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