22:
43:
94:
66:
73:
113:
80:
47:
62:
127:
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800:
32:
138:
87:
51:
36:
175:
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602:
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the transitivity requirement in the previous definition is only necessary in the case when |
131:
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is an example of a block system. The corresponding equivalence relation is the smallest
542:} together with the complement of the union of these sets is a block system containing
938:
286:
149:
145:
21:
610:
137:"Block system" redirects here. For the railway signalling system, see
849:
is primitive if and only if either the group action is trivial (then
346:
301:
is a singleton set then these two partitions are identical). A
15:
282:
such that the induced action on the block system is trivial.
828:
are in one-to-one correspondence with the subgroups of
644:The stabilizer of a block contains the stabilizer
317:if it has no other block systems. For a non-empty
530:satisfies the given condition then the system {
345:. A block can be characterized as a non-empty
341:Each element of some block system is called a
293:is non-empty then the partition into one set
8:
50:. Unsourced material may be challenged and
874:(then the stabilizers of all elements of
824:is transitive then the blocks containing
653:of each of its elements. Conversely, if
507:, and by the same method it follows that
114:Learn how and when to remove this message
787:It follows that the blocks containing
526:. In the other direction, if the set
297:itself is a block system as well (if
63:"Block" permutation group theory
7:
48:adding citations to reliable sources
694:is a block contained in the orbit
14:
333:and the group action is trivial.
20:
1:
878:are the maximal subgroups of
189:. In terms of the associated
128:modular representation theory
464:-invariance it follows that
251:induces a natural action of
961:
337:Characterization of blocks
278:-invariant equivalence on
136:
125:
816:. In particular, if the
801:one-to-one correspondence
412:is a block, and for some
289:is a block system and if
255:on any block system for
126:Not to be confused with
585:} is a block system on
201:-invariance means that
139:Signalling block system
803:with the subgroups of
857:}) or the stabilizer
593:Stabilizers of blocks
569:acts transitively on
305:(and thus non-empty)
191:equivalence relation
44:improve this article
930:Congruence relation
841:. In this case the
557:is a block for any
428:≠ ∅. Then for some
285:The partition into
945:Permutation groups
549:In particular, if
356:such that for all
791:and contained in
678:, then the orbit
665:is a subgroup of
124:
123:
116:
98:
952:
868:maximal subgroup
601:is a block, the
553:is a block then
488:. The condition
243:. The action of
132:Aschbacher block
119:
112:
108:
105:
99:
97:
56:
24:
16:
960:
959:
955:
954:
953:
951:
950:
949:
935:
934:
926:
915:
902:
893:
865:
840:
815:
779:
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741:
702:and containing
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652:
623:
595:
339:
142:
135:
120:
109:
103:
100:
57:
55:
41:
25:
12:
11:
5:
958:
956:
948:
947:
937:
936:
933:
932:
925:
922:
911:
898:
889:
861:
836:
811:
771:
746:
737:
673:
648:
642:
641:
619:
594:
591:
573:then the set {
403:
402:
384:
338:
335:
313:is said to be
287:singleton sets
221:
220:
122:
121:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
957:
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923:
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919:
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901:
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873:
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864:
860:
856:
852:
848:
844:
839:
835:
831:
827:
823:
819:
814:
810:
806:
802:
798:
794:
790:
785:
783:
778:
774:
770:
766:
762:
758:
754:
749:
745:
740:
736:
732:
728:
725:and subgroup
724:
720:
716:
712:
707:
705:
701:
697:
693:
689:
685:
681:
676:
672:
668:
664:
660:
656:
651:
647:
639:
635:
631:
627:
622:
618:
615:
614:
613:
612:
608:
604:
600:
592:
590:
588:
584:
580:
576:
572:
568:
564:
560:
556:
552:
547:
545:
541:
537:
533:
529:
525:
521:
517:
513:
510:
506:
503:
499:
496:also implies
495:
491:
487:
483:
479:
475:
471:
467:
463:
460:and from the
459:
455:
451:
447:
443:
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431:
427:
423:
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411:
407:
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351:
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328:
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308:
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296:
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288:
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140:
133:
129:
118:
115:
107:
96:
93:
89:
86:
82:
79:
75:
72:
68:
65: –
64:
60:
59:Find sources:
53:
49:
45:
39:
38:
34:
29:This article
27:
23:
18:
17:
917:
912:
908:
904:
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842:
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781:
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730:
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543:
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429:
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409:
408:Assume that
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154:block system
153:
150:group theory
143:
110:
101:
91:
84:
77:
70:
58:
42:Please help
30:
832:containing
807:containing
733:containing
721:containing
669:containing
518:, and thus
262:The set of
146:mathematics
603:stabilizer
401:entirely).
303:transitive
187:-invariant
74:newspapers
883:conjugate
565:, and if
364:, either
315:primitive
176:partition
104:June 2019
31:does not
939:Category
924:See also
894:because
717:, block
709:For any
611:subgroup
231:and all
223:for all
212:implies
182:that is
156:for the
799:are in
609:is the
480:and so
472:. Thus
452:, then
266:of the
88:scholar
52:removed
37:sources
690:under
444:. Let
406:Proof:
397:moves
379:fixes
347:subset
264:orbits
158:action
90:
83:
76:
69:
61:
866:is a
845:-set
820:-set
742:it's
436:it's
420:it's
393:= ∅ (
343:block
321:-set
309:-set
270:-set
174:is a
167:on a
162:group
160:of a
95:JSTOR
81:books
767:and
661:and
624:= {
383:) or
152:, a
148:and
67:news
35:any
33:cite
920:).
885:to
870:of
853:= {
686:of
605:of
597:If
352:of
247:on
193:on
178:of
169:set
144:In
130:or
46:by
941::
916:⋅
907:⋅
903:=
900:gx
784:.
780:=
759:∩
755:=
729:⊆
713:∈
706:.
657:∈
640:}.
636:=
634:gB
632:|
628:∈
589:.
581:∈
577:|
575:gB
561:∈
555:gB
546:.
538:∈
534:|
532:gB
524:gB
522:⊆
514:⊆
500:~
492:~
490:gx
484:⊆
482:gB
478:gy
476:~
470:gy
468:~
466:gx
456:~
448:∈
440:~
438:gx
432:∈
424:∩
422:gB
416:∈
389:∩
387:gB
371:=
369:gB
360:∈
329:|=
259:.
239:∈
235:,
227:∈
218:gy
216:~
214:gx
208:~
197:,
918:g
913:x
909:G
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896:G
891:x
887:G
880:G
876:X
872:G
863:x
859:G
855:x
851:X
847:X
843:G
838:x
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830:G
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805:G
797:x
795:.
793:G
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782:H
777:x
775:.
773:H
769:G
765:x
763:.
761:G
757:B
753:x
751:.
748:B
744:G
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727:H
723:x
719:B
715:X
711:x
704:x
700:x
698:.
696:G
692:H
688:x
684:x
682:.
680:H
675:x
671:G
667:G
663:H
659:X
655:x
650:x
646:G
638:B
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626:g
621:B
617:G
607:B
599:B
587:X
583:G
579:g
571:X
567:G
563:G
559:g
551:B
544:B
540:G
536:g
528:B
520:B
516:B
512:B
509:g
505:x
502:g
498:x
494:x
486:B
474:y
462:G
458:y
454:x
450:B
446:y
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430:x
426:B
418:G
414:g
410:B
399:B
395:g
391:B
381:B
377:g
375:(
373:B
362:G
358:g
354:X
350:B
331:2
327:X
323:X
319:G
311:X
307:G
299:X
295:X
291:X
280:X
276:G
272:X
268:G
257:X
253:G
249:X
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237:y
233:x
229:G
225:g
210:y
206:x
199:G
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172:X
165:G
141:.
134:.
117:)
111:(
106:)
102:(
92:·
85:·
78:·
71:·
54:.
40:.
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