1014:
800:
99:
acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the
1291:
1242:
1438:
1366:
1173:
1128:
1083:
1496:
Galois used a different terminology, because most of the terminology in this statement was introduced afterwards, partly for clarifying the concepts introduced by Galois.
928:
724:
1193:
1037:
381:
277:
937:
320:
1400:
1328:
884:
850:
830:
680:
733:
1459:
95:
While primitive permutation groups are transitive, not all transitive permutation groups are primitive. The simplest example is the
1454:
1485:
1561:
627:
There are a large number of primitive groups of degree 16. As
Carmichael notes, all of these groups, except for the
1464:
1566:
279:
In fact, every transitive group of prime degree is primitive (since the number of elements of a partition fixed by
383:
1247:
1198:
452:
1405:
1333:
105:
155:
In the same letter in which he introduced the term "primitive", Galois stated the following theorem:
1133:
1088:
1042:
389:
An equivalent definition of primitivity relies on the fact that every transitive action of a group
1511:
1504:
889:
685:
65:
136:
1009:{\displaystyle \sigma ={\begin{pmatrix}1&2&3&4\\2&3&4&1\end{pmatrix}}.}
1373:
632:
459:
338:
57:
35:
1178:
1022:
348:
244:
426:
290:
96:
1378:
1306:
862:
835:
808:
658:
1537:
1301:
857:
653:
628:
334:
386:
results from this and the fact that there are polynomials with a symmetric Galois group.
164:
1555:
101:
1486:
http://www.galois.ihp.fr/ressources/vie-et-oeuvre-de-galois/lettres/lettre-testament
640:
636:
207:
191:
187:
176:
144:
104:
acting on a 2-element set. This is because for a non-transitive action, either the
41:
795:{\displaystyle \eta ={\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}}.}
237:
is an odd prime number, then the order of a solvable transitive group of degree
31:
72:-action preserves are the trivial partitions into either a single set or into |
17:
1521:
1542:
1531:
Ginn, Boston, 1937. Reprinted by Dover
Publications, New York, 1956.
439:, and imprimitive otherwise (that is, if there is a proper subgroup
451:
is a proper subgroup). These imprimitive actions are examples of
406:
1467:, a classification of finite primitive groups into various types
540:
393:
is isomorphic to an action arising from the canonical action of
458:
The numbers of primitive groups of small degree were stated by
322:
is the cardinality of the affine group of an affine space with
1518:
619:
1508:
The primitive permutation groups of degree less than 2500
27:
Permutation group that preserves no non-trivial partition
1529:
Introduction to the Theory of Groups of Finite Order.
952:
748:
417:. A group action is primitive if it is isomorphic to
1408:
1381:
1336:
1309:
1250:
1201:
1181:
1136:
1091:
1045:
1025:
940:
892:
865:
838:
811:
736:
688:
661:
351:
345:
are not solvable, since their order are greater than
293:
247:
139:
in his last letter, in which he used the French term
1297:
Every transitive group of prime degree is primitive
1432:
1394:
1360:
1322:
1285:
1236:
1187:
1167:
1122:
1077:
1031:
1008:
922:
878:
844:
824:
794:
718:
674:
375:
314:
271:
233:A corollary of this result of Galois is that, if
116:, or the group action is trivial, in which case
639:on the 4-dimensional space over the 2-element
222:acts is finite, its cardinality is called the
8:
1427:
1409:
1355:
1337:
1162:
1150:
1117:
1105:
917:
893:
713:
695:
1522:Data Library "Primitive Permutation Groups"
1407:
1386:
1380:
1335:
1314:
1308:
1277:
1261:
1249:
1228:
1212:
1200:
1180:
1141:
1135:
1096:
1090:
1066:
1053:
1044:
1024:
947:
939:
891:
870:
864:
837:
816:
810:
743:
735:
687:
666:
660:
350:
292:
246:
112:form a nontrivial partition preserved by
464:
1477:
1039:is not primitive, since the partition
333:is a prime number greater than 3, the
84:does preserve a nontrivial partition,
7:
1286:{\displaystyle \sigma (X_{2})=X_{1}}
1237:{\displaystyle \sigma (X_{1})=X_{2}}
135:This terminology was introduced by
1460:Jordan's theorem (symmetric group)
25:
1514:292 (2005), no. 1, 154–183.
1455:Block (permutation group theory)
76:| singleton sets. Otherwise, if
1433:{\displaystyle \{1,\ldots ,n\}}
1361:{\displaystyle \{1,\ldots ,n\}}
1267:
1254:
1218:
1205:
1072:
1046:
367:
355:
309:
297:
263:
251:
1:
1168:{\displaystyle X_{2}=\{2,4\}}
1123:{\displaystyle X_{1}=\{1,3\}}
1078:{\displaystyle (X_{1},X_{2})}
1440:is primitive for every
635:group, are subgroups of the
923:{\displaystyle \{1,2,3,4\}}
832:and the group generated by
719:{\displaystyle X=\{1,2,3\}}
1583:
186:may be identified with an
44:on a non-empty finite set
120:nontrivial partitions of
1538:"Primitive Group Action"
128:| ≥ 3) are preserved by
1527:Carmichael, Robert D.,
1368:is primitive for every
1188:{\displaystyle \sigma }
1032:{\displaystyle \sigma }
1019:The group generated by
453:induced representations
376:{\displaystyle p(p-1).}
272:{\displaystyle p(p-1).}
167:acting on a finite set
1505:Roney-Dougal, Colva M.
1434:
1396:
1362:
1324:
1287:
1238:
1189:
1169:
1124:
1079:
1033:
1010:
924:
880:
846:
826:
796:
720:
676:
377:
316:
315:{\displaystyle p(p-1)}
273:
212:
143:for an equation whose
1484:Galois' last letter:
1435:
1397:
1395:{\displaystyle A_{n}}
1363:
1325:
1323:{\displaystyle S_{n}}
1288:
1239:
1190:
1170:
1125:
1080:
1034:
1011:
925:
881:
879:{\displaystyle S_{4}}
847:
845:{\displaystyle \eta }
827:
825:{\displaystyle S_{3}}
797:
721:
677:
675:{\displaystyle S_{3}}
378:
317:
283:must be a divisor of
274:
206:as a subgroup of the
157:
1406:
1379:
1334:
1307:
1248:
1199:
1179:
1134:
1089:
1043:
1023:
938:
890:
863:
836:
809:
734:
686:
659:
384:Abel–Ruffini theorem
349:
329:It follows that, if
291:
245:
171:, then the order of
1465:O'Nan–Scott theorem
1175:is preserved under
930:and the permutation
726:and the permutation
124:(which exists for |
1562:Permutation groups
1512:Journal of Algebra
1444: > 2.
1430:
1402:acting on the set
1392:
1358:
1330:acting on the set
1320:
1283:
1234:
1185:
1165:
1120:
1075:
1029:
1006:
997:
920:
886:acting on the set
876:
842:
822:
792:
783:
716:
682:acting on the set
672:
373:
312:
269:
141:Ă©quation primitive
80:is transitive and
1567:Integer sequences
1374:alternating group
856:Now consider the
625:
624:
460:Robert Carmichael
339:alternating group
36:permutation group
16:(Redirected from
1574:
1548:
1547:
1497:
1494:
1488:
1482:
1439:
1437:
1436:
1431:
1401:
1399:
1398:
1393:
1391:
1390:
1367:
1365:
1364:
1359:
1329:
1327:
1326:
1321:
1319:
1318:
1292:
1290:
1289:
1284:
1282:
1281:
1266:
1265:
1243:
1241:
1240:
1235:
1233:
1232:
1217:
1216:
1194:
1192:
1191:
1186:
1174:
1172:
1171:
1166:
1146:
1145:
1129:
1127:
1126:
1121:
1101:
1100:
1084:
1082:
1081:
1076:
1071:
1070:
1058:
1057:
1038:
1036:
1035:
1030:
1015:
1013:
1012:
1007:
1002:
1001:
929:
927:
926:
921:
885:
883:
882:
877:
875:
874:
851:
849:
848:
843:
831:
829:
828:
823:
821:
820:
801:
799:
798:
793:
788:
787:
725:
723:
722:
717:
681:
679:
678:
673:
671:
670:
465:
382:
380:
379:
374:
344:
332:
325:
321:
319:
318:
313:
286:
282:
278:
276:
275:
270:
241:is a divisor of
240:
236:
175:is a power of a
97:Klein four-group
21:
1582:
1581:
1577:
1576:
1575:
1573:
1572:
1571:
1552:
1551:
1535:
1534:
1501:
1500:
1495:
1491:
1483:
1479:
1474:
1451:
1404:
1403:
1382:
1377:
1376:
1332:
1331:
1310:
1305:
1304:
1302:symmetric group
1273:
1257:
1246:
1245:
1224:
1208:
1197:
1196:
1177:
1176:
1137:
1132:
1131:
1092:
1087:
1086:
1062:
1049:
1041:
1040:
1021:
1020:
996:
995:
990:
985:
980:
974:
973:
968:
963:
958:
948:
936:
935:
888:
887:
866:
861:
860:
858:symmetric group
852:are primitive.
834:
833:
812:
807:
806:
782:
781:
776:
771:
765:
764:
759:
754:
744:
732:
731:
684:
683:
662:
657:
656:
654:symmetric group
649:
347:
346:
342:
335:symmetric group
330:
323:
289:
288:
284:
280:
243:
242:
238:
234:
163:is a primitive
153:
137:Évariste Galois
28:
23:
22:
18:Primitive group
15:
12:
11:
5:
1580:
1578:
1570:
1569:
1564:
1554:
1553:
1550:
1549:
1536:Todd Rowland.
1532:
1525:
1515:
1499:
1498:
1489:
1476:
1475:
1473:
1470:
1469:
1468:
1462:
1457:
1450:
1447:
1446:
1445:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1389:
1385:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1317:
1313:
1298:
1280:
1276:
1272:
1269:
1264:
1260:
1256:
1253:
1231:
1227:
1223:
1220:
1215:
1211:
1207:
1204:
1184:
1164:
1161:
1158:
1155:
1152:
1149:
1144:
1140:
1119:
1116:
1113:
1110:
1107:
1104:
1099:
1095:
1074:
1069:
1065:
1061:
1056:
1052:
1048:
1028:
1017:
1016:
1005:
1000:
994:
991:
989:
986:
984:
981:
979:
976:
975:
972:
969:
967:
964:
962:
959:
957:
954:
953:
951:
946:
943:
932:
931:
919:
916:
913:
910:
907:
904:
901:
898:
895:
873:
869:
841:
819:
815:
803:
802:
791:
786:
780:
777:
775:
772:
770:
767:
766:
763:
760:
758:
755:
753:
750:
749:
747:
742:
739:
728:
727:
715:
712:
709:
706:
703:
700:
697:
694:
691:
669:
665:
648:
645:
623:
622:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
544:
543:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
413:a subgroup of
372:
369:
366:
363:
360:
357:
354:
311:
308:
305:
302:
299:
296:
268:
265:
262:
259:
256:
253:
250:
198:elements, and
165:solvable group
152:
149:
147:is primitive.
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1579:
1568:
1565:
1563:
1560:
1559:
1557:
1545:
1544:
1539:
1533:
1530:
1526:
1523:
1520:
1516:
1513:
1509:
1506:
1503:
1502:
1493:
1490:
1487:
1481:
1478:
1471:
1466:
1463:
1461:
1458:
1456:
1453:
1452:
1448:
1443:
1424:
1421:
1418:
1415:
1412:
1387:
1383:
1375:
1371:
1352:
1349:
1346:
1343:
1340:
1315:
1311:
1303:
1299:
1296:
1295:
1294:
1278:
1274:
1270:
1262:
1258:
1251:
1229:
1225:
1221:
1213:
1209:
1202:
1182:
1159:
1156:
1153:
1147:
1142:
1138:
1114:
1111:
1108:
1102:
1097:
1093:
1067:
1063:
1059:
1054:
1050:
1026:
1003:
998:
992:
987:
982:
977:
970:
965:
960:
955:
949:
944:
941:
934:
933:
914:
911:
908:
905:
902:
899:
896:
871:
867:
859:
855:
854:
853:
839:
817:
813:
789:
784:
778:
773:
768:
761:
756:
751:
745:
740:
737:
730:
729:
710:
707:
704:
701:
698:
692:
689:
667:
663:
655:
652:Consider the
651:
650:
646:
644:
642:
638:
634:
630:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
549:
546:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
466:
463:
461:
456:
454:
450:
446:
442:
438:
434:
431:
429:
424:
420:
416:
412:
408:
404:
400:
396:
392:
387:
385:
370:
364:
361:
358:
352:
340:
336:
327:
306:
303:
300:
294:
266:
260:
257:
254:
248:
231:
229:
225:
221:
217:
211:
209:
205:
201:
197:
193:
189:
185:
181:
178:
174:
170:
166:
162:
156:
150:
148:
146:
142:
138:
133:
131:
127:
123:
119:
115:
111:
107:
103:
102:trivial group
98:
93:
91:
87:
83:
79:
75:
71:
67:
64:and the only
63:
59:
55:
51:
47:
43:
40:
37:
33:
19:
1541:
1528:
1507:
1492:
1480:
1441:
1369:
1018:
804:
641:finite field
637:affine group
626:
457:
448:
444:
440:
436:
432:
427:
422:
418:
414:
410:
402:
398:
394:
390:
388:
328:
232:
227:
223:
219:
215:
213:
208:affine group
203:
199:
195:
192:finite field
188:affine space
183:
179:
177:prime number
172:
168:
160:
158:
154:
145:Galois group
140:
134:
129:
125:
121:
117:
113:
109:
94:
89:
85:
81:
77:
73:
69:
61:
58:transitively
53:
49:
45:
38:
29:
633:alternating
397:on the set
214:If the set
182:. Further,
90:imprimitive
32:mathematics
1556:Categories
1472:References
341:of degree
326:elements.
151:Properties
88:is called
66:partitions
48:is called
1543:MathWorld
1419:…
1347:…
1252:σ
1203:σ
1183:σ
1027:σ
942:σ
840:η
738:η
629:symmetric
462:in 1937:
447:of which
362:−
304:−
258:−
218:on which
190:over the
50:primitive
1449:See also
1372:and the
647:Examples
430:subgroup
337:and the
202:acts on
1195:, i.e.
620:A000019
428:maximal
287:), and
1085:where
547:Number
468:Degree
425:for a
407:cosets
224:degree
106:orbits
42:acting
805:Both
194:with
56:acts
1517:The
1300:The
1244:and
1130:and
631:and
541:OEIS
409:for
68:the
34:, a
1519:GAP
455:.
443:of
435:of
405:of
226:of
159:If
118:all
108:of
92:.
60:on
52:if
30:In
1558::
1540:.
1510:,
1293:.
643:.
595:10
592:22
571:11
537:24
534:23
531:22
528:21
525:20
522:19
519:18
516:17
513:16
510:15
507:14
504:13
501:12
498:11
495:10
230:.
132:.
1546:.
1524:.
1442:n
1428:}
1425:n
1422:,
1416:,
1413:1
1410:{
1388:n
1384:A
1370:n
1356:}
1353:n
1350:,
1344:,
1341:1
1338:{
1316:n
1312:S
1279:1
1275:X
1271:=
1268:)
1263:2
1259:X
1255:(
1230:2
1226:X
1222:=
1219:)
1214:1
1210:X
1206:(
1163:}
1160:4
1157:,
1154:2
1151:{
1148:=
1143:2
1139:X
1118:}
1115:3
1112:,
1109:1
1106:{
1103:=
1098:1
1094:X
1073:)
1068:2
1064:X
1060:,
1055:1
1051:X
1047:(
1004:.
999:)
993:1
988:4
983:3
978:2
971:4
966:3
961:2
956:1
950:(
945:=
918:}
915:4
912:,
909:3
906:,
903:2
900:,
897:1
894:{
872:4
868:S
818:3
814:S
790:.
785:)
779:1
774:3
769:2
762:3
757:2
752:1
746:(
741:=
714:}
711:3
708:,
705:2
702:,
699:1
696:{
693:=
690:X
668:3
664:S
616:5
613:7
610:4
607:9
604:4
601:8
598:4
589:6
586:4
583:9
580:6
577:8
574:9
568:7
565:7
562:4
559:5
556:2
553:2
550:1
492:9
489:8
486:7
483:6
480:5
477:4
474:3
471:2
449:H
445:G
441:K
437:G
433:H
423:H
421:/
419:G
415:G
411:H
403:H
401:/
399:G
395:G
391:G
371:.
368:)
365:1
359:p
356:(
353:p
343:p
331:p
324:p
310:)
307:1
301:p
298:(
295:p
285:p
281:G
267:.
264:)
261:1
255:p
252:(
249:p
239:p
235:p
228:G
220:G
216:X
210:.
204:X
200:G
196:p
184:X
180:p
173:X
169:X
161:G
130:G
126:X
122:X
114:G
110:G
86:G
82:G
78:G
74:X
70:G
62:X
54:G
46:X
39:G
20:)
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