126:
228:
868:
824:
779:
531:
359:
743:
280:
254:
396:
306:
695:
472:
443:
166:
146:
60:
37:
895:
903:
1080:
949:
1054:
966:
Hering, Christoph (1985), "Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II",
1106:
69:
1009:
919:
171:
831:
405:
is a group such that there exists a group action that's 2-transitive and faithful. Similarly we can define
792:
841:
797:
755:
477:
907:
1049:, Grundlehren der Mathematischen Wissenschaften, vol. 243, Berlin-New York: Springer-Verlag,
311:
704:
1076:
1050:
1026:
985:
945:
259:
233:
1018:
975:
883:
835:
375:
285:
1090:
1064:
1038:
997:
959:
834:. In other words, the n-dimensional projective transforms act transitively on the space of
673:
1086:
1060:
1034:
1004:
993:
955:
941:
891:
879:
698:
448:
419:
934:
887:
667:
Every group is trivially 1-transitive, by its action on itself by left-multiplication.
151:
131:
45:
22:
1100:
980:
656:
1075:, Pure and Applied Mathematics, vol. 289, Boca Raton: Chapman & Hall/CRC,
63:
785:
1030:
989:
749:
1022:
940:, Graduate Texts in Mathematics, vol. 163, Berlin, New York:
1007:(1957), "Zweifach transitive, auflösbare Permutationsgruppen",
128:. That is, assuming (without a real loss of generality) that
474:, since the induced action on the distinct set of pairs is
1071:
Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007),
544:
permutation groups can be defined for any natural number
844:
800:
791:
acts sharply (n+2)-transitively on the n-dimensional
758:
707:
676:
480:
451:
422:
378:
314:
288:
262:
236:
174:
154:
134:
72:
48:
25:
933:
862:
818:
773:
737:
689:
525:
466:
437:
390:
353:
300:
274:
248:
222:
160:
140:
120:
66:transitively on the set of distinct ordered pairs
54:
31:
121:{\displaystyle \{(x,y)\in S\times S:x\neq y\}}
8:
1045:Huppert, Bertram; Blackburn, Norman (1982),
898:. The insoluble groups were classified by (
732:
708:
115:
73:
745:, then the action is sharply n-transitive.
886:is 2-transitive, but not conversely. The
979:
854:
850:
847:
846:
843:
810:
806:
803:
802:
799:
765:
761:
760:
757:
706:
681:
675:
479:
450:
421:
377:
313:
287:
261:
235:
173:
153:
133:
71:
47:
24:
932:Dixon, John D.; Mortimer, Brian (1996),
223:{\displaystyle (x,y),(w,z)\in S\times S}
896:list of transitive finite linear groups
890:2-transitive groups were classified by
830:is because the (n+2) points must be in
904:classification of finite simple groups
899:
874:Classifications of 2-transitive groups
1073:Handbook of finite translation planes
536:The definition works in general with
7:
548:. Specifically, a permutation group
14:
863:{\displaystyle \mathbb {RP} ^{n}}
819:{\displaystyle \mathbb {RP} ^{n}}
774:{\displaystyle \mathbb {R} ^{n}}
595:with the property that all the
878:Every 2-transitive group is a
526:{\displaystyle g(x,y)=(gx,gy)}
520:
502:
496:
484:
348:
336:
330:
318:
205:
193:
187:
175:
88:
76:
1:
563:if, given two sets of points
981:10.1016/0021-8693(85)90179-6
882:, but not conversely. Every
354:{\displaystyle g(x,y)=(w,z)}
784:The group of n-dimensional
748:The group of n-dimensional
738:{\displaystyle \{1,...,n\}}
621:, there is a group element
1123:
1010:Mathematische Zeitschrift
920:Multiply transitive group
894:and are described in the
168:, for each pair of pairs
659:are important examples.
832:general linear position
752:acts 2-transitively on
275:{\displaystyle w\neq z}
249:{\displaystyle x\neq y}
16:Concept in group theory
864:
820:
775:
750:homothety-translations
739:
691:
527:
468:
439:
392:
391:{\displaystyle g\in G}
355:
302:
301:{\displaystyle g\in G}
276:
250:
224:
162:
142:
122:
56:
33:
865:
821:
793:real projective space
786:projective transforms
776:
740:
692:
690:{\displaystyle S_{n}}
528:
469:
440:
393:
356:
303:
277:
251:
225:
163:
143:
123:
57:
34:
908:almost simple groups
842:
798:
756:
705:
674:
478:
467:{\displaystyle gy=z}
449:
438:{\displaystyle gx=w}
420:
376:
364:The group action is
312:
286:
260:
234:
172:
152:
148:acts on the left of
132:
70:
46:
23:
1047:Finite groups. III.
542:multiply transitive
40:acts 2-transitively
1107:Permutation groups
1023:10.1007/BF01160336
968:Journal of Algebra
936:Permutation groups
860:
816:
771:
735:
687:
540:replacing 2. Such
523:
464:
435:
403:2-transitive group
388:
351:
298:
272:
246:
220:
158:
138:
118:
52:
29:
1082:978-1-58488-605-1
951:978-0-387-94599-6
836:projective frames
411:-transitive group
282:, there exists a
161:{\displaystyle S}
141:{\displaystyle G}
55:{\displaystyle S}
32:{\displaystyle G}
1114:
1093:
1067:
1041:
1005:Huppert, Bertram
1000:
983:
962:
939:
884:Zassenhaus group
869:
867:
866:
861:
859:
858:
853:
825:
823:
822:
817:
815:
814:
809:
780:
778:
777:
772:
770:
769:
764:
744:
742:
741:
736:
696:
694:
693:
688:
686:
685:
532:
530:
529:
524:
473:
471:
470:
465:
444:
442:
441:
436:
397:
395:
394:
389:
360:
358:
357:
352:
307:
305:
304:
299:
281:
279:
278:
273:
255:
253:
252:
247:
229:
227:
226:
221:
167:
165:
164:
159:
147:
145:
144:
139:
127:
125:
124:
119:
61:
59:
58:
53:
38:
36:
35:
30:
1122:
1121:
1117:
1116:
1115:
1113:
1112:
1111:
1097:
1096:
1083:
1070:
1057:
1044:
1003:
965:
952:
942:Springer-Verlag
931:
928:
916:
892:Bertram Huppert
880:primitive group
876:
845:
840:
839:
801:
796:
795:
759:
754:
753:
703:
702:
699:symmetric group
677:
672:
671:
665:
646:
637:
616:
603:
594:
585:
578:
569:
476:
475:
447:
446:
418:
417:
374:
373:
310:
309:
284:
283:
258:
257:
232:
231:
170:
169:
150:
149:
130:
129:
68:
67:
44:
43:
21:
20:
17:
12:
11:
5:
1120:
1118:
1110:
1109:
1099:
1098:
1095:
1094:
1081:
1068:
1055:
1042:
1001:
974:(1): 151–164,
963:
950:
927:
924:
923:
922:
915:
912:
875:
872:
857:
852:
849:
813:
808:
805:
768:
763:
734:
731:
728:
725:
722:
719:
716:
713:
710:
684:
680:
664:
661:
657:Mathieu groups
651:between 1 and
642:
633:
612:
599:
590:
583:
574:
567:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
486:
483:
463:
460:
457:
454:
434:
431:
428:
425:
416:Equivalently,
387:
384:
381:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
297:
294:
291:
271:
268:
265:
245:
242:
239:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
177:
157:
137:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
84:
81:
78:
75:
51:
28:
15:
13:
10:
9:
6:
4:
3:
2:
1119:
1108:
1105:
1104:
1102:
1092:
1088:
1084:
1078:
1074:
1069:
1066:
1062:
1058:
1056:3-540-10633-2
1052:
1048:
1043:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1011:
1006:
1002:
999:
995:
991:
987:
982:
977:
973:
969:
964:
961:
957:
953:
947:
943:
938:
937:
930:
929:
925:
921:
918:
917:
913:
911:
909:
905:
901:
897:
893:
889:
885:
881:
873:
871:
855:
837:
833:
829:
811:
794:
790:
787:
782:
766:
751:
746:
729:
726:
723:
720:
717:
714:
711:
700:
682:
678:
668:
662:
660:
658:
654:
650:
645:
641:
636:
632:
628:
624:
620:
615:
611:
607:
602:
598:
593:
589:
582:
577:
573:
566:
562:
560:
555:
551:
547:
543:
539:
534:
517:
514:
511:
508:
505:
499:
493:
490:
487:
481:
461:
458:
455:
452:
432:
429:
426:
423:
414:
412:
408:
404:
399:
385:
382:
379:
371:
367:
362:
345:
342:
339:
333:
327:
324:
321:
315:
295:
292:
289:
269:
266:
263:
243:
240:
237:
217:
214:
211:
208:
202:
199:
196:
190:
184:
181:
178:
155:
135:
112:
109:
106:
103:
100:
97:
94:
91:
85:
82:
79:
65:
49:
41:
26:
1072:
1046:
1014:
1008:
971:
967:
935:
906:and are all
902:) using the
877:
827:
788:
783:
747:
669:
666:
652:
648:
643:
639:
634:
630:
626:
622:
618:
613:
609:
608:and all the
605:
600:
596:
591:
587:
580:
575:
571:
564:
558:
557:
553:
549:
545:
541:
537:
535:
415:
410:
406:
402:
400:
369:
365:
363:
39:
18:
1017:: 126–150,
900:Hering 1985
629:which maps
561:-transitive
398:is unique.
370:-transitive
926:References
701:acting on
556:points is
552:acting on
308:such that
1031:0025-5874
990:0021-8693
647:for each
383:∈
293:∈
267:≠
241:≠
215:×
209:∈
110:≠
98:×
92:∈
42:on a set
1101:Category
914:See also
888:solvable
663:Examples
619:distinct
606:distinct
372:if such
19:A group
1091:2290291
1065:0650245
1039:0094386
998:0780488
960:1409812
697:be the
407:sharply
366:sharply
1089:
1079:
1063:
1053:
1037:
1029:
996:
988:
958:
948:
828:almost
826:. The
789:almost
655:. The
586:, ...
570:, ...
62:if it
230:with
1077:ISBN
1051:ISBN
1027:ISSN
986:ISSN
946:ISBN
670:Let
617:are
604:are
579:and
445:and
256:and
64:acts
1019:doi
976:doi
838:of
638:to
625:in
361:.
1103::
1087:MR
1085:,
1061:MR
1059:,
1035:MR
1033:,
1025:,
1015:68
1013:,
994:MR
992:,
984:,
972:93
970:,
956:MR
954:,
944:,
910:.
870:.
781:.
533:.
413:.
401:A
1021::
978::
856:n
851:P
848:R
812:n
807:P
804:R
767:n
762:R
733:}
730:n
727:,
724:.
721:.
718:.
715:,
712:1
709:{
683:n
679:S
653:k
649:i
644:i
640:b
635:i
631:a
627:G
623:g
614:i
610:b
601:i
597:a
592:k
588:b
584:1
581:b
576:k
572:a
568:1
565:a
559:k
554:n
550:G
546:k
538:k
521:)
518:y
515:g
512:,
509:x
506:g
503:(
500:=
497:)
494:y
491:,
488:x
485:(
482:g
462:z
459:=
456:y
453:g
433:w
430:=
427:x
424:g
409:2
386:G
380:g
368:2
349:)
346:z
343:,
340:w
337:(
334:=
331:)
328:y
325:,
322:x
319:(
316:g
296:G
290:g
270:z
264:w
244:y
238:x
218:S
212:S
206:)
203:z
200:,
197:w
194:(
191:,
188:)
185:y
182:,
179:x
176:(
156:S
136:G
116:}
113:y
107:x
104::
101:S
95:S
89:)
86:y
83:,
80:x
77:(
74:{
50:S
27:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.