36:
1126:
889:
1346:
761:
294:
634:
1533:
539:
475:
576:
578:. This causes only a small amount of confusion, because one can usually distinguish between the core of a group and the core of a subgroup within a group. The
1121:{\displaystyle O_{\pi _{1},\pi _{2},\dots ,\pi _{n+1}}(G)/O_{\pi _{1},\pi _{2},\dots ,\pi _{n}}(G)=O_{\pi _{n+1}}(G/O_{\pi _{1},\pi _{2},\dots ,\pi _{n}}(G))}
542:
209:
57:
1226:
1680:
1650:
1627:
1601:
79:
1672:
1642:
639:
1668:
1435:′-core is the intersection of the kernels of the ordinary (complex) irreducible representations that lie in the principal
1412:
50:
44:
1593:
1400:
169:
61:
173:
358:
332:
1708:
351:
1463:-core is the intersection of the centralizers of the abelian chief factors whose order is divisible by
1660:
393:
328:
1502:
109:
1585:
592:
503:
1676:
1646:
1623:
1597:
444:
324:
320:
1447:-core is the intersection of the kernels of the irreducible representations in the principal
1686:
1544:
478:
347:
336:
1690:
552:
397:
157:
105:
1612:
1496:
1165:
430:
350:. Equivalently, it is a subgroup that occurs as the isotropy subgroup of a transitive,
365:
case generalizes to finding the normal core in case of subgroups of arbitrary groups.
1702:
1424:
1215:
362:
1559:′-core of its solvable radical in order to better mimic properties of the 2′-core.
1416:
482:
409:
389:
316:
93:
299:
Under this more general definition, the normal core is the core with respect to
97:
1619:
378:
17:
1491:
A related subgroup in concept and notation is the solvable radical. The
1475:-constrained group, an irreducible module over a field of characteristic
117:
377:"P-core" redirects here. For the computer central processing units, see
423:
1483:′-core of the group is contained in the kernel of the representation.
185:
289:{\displaystyle \mathrm {Core} _{S}(H):=\bigcap _{s\in S}{s^{-1}Hs}.}
1467:(all of which are irreducible representations over a field of size
1423:-core of a finite group is the intersection of the kernels of the
307:. The normal core of any normal subgroup is the subgroup itself.
1341:{\displaystyle C_{G}(O_{p',p}(G)/O_{p'}(G))\subseteq O_{p',p}(G)}
29:
477:, and in particular appears in one of the definitions of the
1535:. There is some variance in the literature in defining the
335:, the normal core of any isotropy subgroup is precisely the
782:-core can also be defined as the unique largest subnormal
541:. In the area of finite insoluble groups, including the
756:{\displaystyle O_{p',p}(G)/O_{p'}(G)=O_{p}(G/O_{p'}(G))}
1547:'s N-group papers, but not his later work) define the
422:
of a finite group is defined to be its largest normal
1505:
1229:
1194:
if and only if it is equal to some term of its upper
892:
642:
595:
555:
506:
447:
212:
1543:. A few authors in only a few papers (for instance
1611:
1527:
1340:
1120:
755:
628:
570:
533:
469:
288:
27:Any of certain special normal subgroups of a group
327:of any point acts as the identity on its entire
1479:lies in the principal block if and only if the
1471:lying in the principal block). For a finite,
315:Normal cores are important in the context of
8:
1637:Huppert, Bertram; Blackburn, Norman (1982).
1510:
1504:
1415:, which studies the actions of groups on
1312:
1282:
1273:
1247:
1234:
1228:
1098:
1079:
1066:
1061:
1052:
1032:
1027:
1003:
984:
971:
966:
957:
934:
915:
902:
897:
891:
730:
721:
709:
682:
673:
647:
641:
600:
594:
554:
545:, the 2′-core is often called simply the
511:
505:
452:
446:
267:
262:
250:
228:
214:
211:
195:is the intersection of the conjugates of
80:Learn how and when to remove this message
1451:-block over any field of characteristic
43:This article includes a list of general
1568:
1399:Just as normal cores are important for
790:′-core as the unique largest subnormal
346:is a subgroup whose normal core is the
1179:if and only if it is equal to its own
802:-core as the unique largest subnormal
543:classification of finite simple groups
1610:Doerk, Klaus; Hawkes, Trevor (1992).
1187:-core. A finite group is said to be
7:
392:, though some aspects generalize to
112:. The two most common types are the
1379:-nilpotent if and only if it has a
771:-core is the unique largest normal
1511:
1363:-soluble. Every soluble group is
853:of primes, one defines subgroups O
492:is the largest normal subgroup of
429:. It is the normal core of every
224:
221:
218:
215:
49:it lacks sufficient corresponding
25:
1455:. Also, for a finite group, the
1427:over any field of characteristic
379:Intel Core § 12th generation
1665:A Course in the Theory of Groups
1499:normal subgroup, and is denoted
1439:-block. For a finite group, the
1172:. A finite group is said to be
34:
1671:. Vol. 80 (2nd ed.).
323:, where the normal core of the
180:). More generally, the core of
1528:{\displaystyle O_{\infty }(G)}
1522:
1516:
1335:
1329:
1302:
1299:
1293:
1270:
1264:
1240:
1115:
1112:
1106:
1046:
1017:
1011:
954:
948:
750:
747:
741:
715:
699:
693:
670:
664:
623:
617:
565:
559:
528:
522:
464:
458:
331:. Thus, in case the action is
240:
234:
1:
1669:Graduate Texts in Mathematics
1551:′-core of an insoluble group
1495:is defined to be the largest
1413:modular representation theory
1135:-series is formed by taking
1425:irreducible representations
1205:is the length of its upper
629:{\displaystyle O_{p',p}(G)}
1725:
1594:Cambridge University Press
1431:. For a finite group, the
1375:-constrained. A group is
763:. For a finite group, the
496:whose order is coprime to
376:
104:is any of certain special
1411:′-cores are important in
1351:Every nilpotent group is
1209:-series. A finite group
534:{\displaystyle O_{p'}(G)}
470:{\displaystyle O_{p}(G)}
359:hidden subgroup problem
64:more precise citations.
1529:
1355:-nilpotent, and every
1342:
1122:
757:
630:
572:
535:
471:
290:
168:(or equivalently, the
1661:Robinson, Derek J. S.
1614:Finite Soluble Groups
1530:
1343:
1123:
806:-nilpotent subgroup.
775:-nilpotent subgroup.
758:
631:
573:
536:
472:
394:locally finite groups
357:The solution for the
291:
164:that is contained in
1575:Robinson (1996) p.16
1503:
1387:, which is just its
1367:-soluble, and every
1359:-nilpotent group is
1227:
890:
794:′-subgroup; and the
640:
593:
571:{\displaystyle O(G)}
553:
504:
445:
210:
191: ⊆
1590:Finite Group Theory
1586:Aschbacher, Michael
433:of the group. The
1525:
1371:-soluble group is
1338:
1118:
753:
626:
568:
531:
485:. Similarly, the
467:
344:core-free subgroup
286:
261:
184:with respect to a
1620:Walter de Gruyter
1487:Solvable radicals
441:is often denoted
325:isotropy subgroup
246:
90:
89:
82:
16:(Redirected from
1716:
1694:
1656:
1639:Finite Groups II
1633:
1617:
1606:
1576:
1573:
1545:John G. Thompson
1534:
1532:
1531:
1526:
1515:
1514:
1493:solvable radical
1347:
1345:
1344:
1339:
1328:
1327:
1320:
1292:
1291:
1290:
1277:
1263:
1262:
1255:
1239:
1238:
1164:there is also a
1127:
1125:
1124:
1119:
1105:
1104:
1103:
1102:
1084:
1083:
1071:
1070:
1056:
1045:
1044:
1043:
1042:
1010:
1009:
1008:
1007:
989:
988:
976:
975:
961:
947:
946:
945:
944:
920:
919:
907:
906:
821:-core begin the
762:
760:
759:
754:
740:
739:
738:
725:
714:
713:
692:
691:
690:
677:
663:
662:
655:
635:
633:
632:
627:
616:
615:
608:
577:
575:
574:
569:
540:
538:
537:
532:
521:
520:
519:
500:and is denoted
479:Fitting subgroup
476:
474:
473:
468:
457:
456:
431:Sylow p-subgroup
398:profinite groups
384:In this section
348:trivial subgroup
295:
293:
292:
287:
282:
275:
274:
260:
233:
232:
227:
106:normal subgroups
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
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1673:Springer-Verlag
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1643:Springer Verlag
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1159:
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1094:
1075:
1062:
1057:
1028:
1023:
999:
980:
967:
962:
930:
911:
898:
893:
888:
887:
878:
877:
867:
860:
852:
842:
835:
786:-subgroup; the
731:
726:
705:
683:
678:
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591:
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551:
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512:
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448:
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406:
382:
375:
339:of the action.
313:
263:
213:
208:
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158:normal subgroup
156:is the largest
150:normal interior
138:
133:
131:The normal core
86:
75:
69:
66:
56:Please help to
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1272:
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1266:
1261:
1258:
1254:
1251:
1246:
1242:
1237:
1233:
1213:is said to be
1154:
1139:
1129:
1128:
1117:
1114:
1111:
1108:
1101:
1097:
1093:
1090:
1087:
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1078:
1074:
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752:
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729:
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698:
695:
689:
686:
681:
676:
672:
669:
666:
661:
658:
654:
651:
646:
636:is defined by
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524:
518:
515:
510:
466:
463:
460:
455:
451:
405:
402:
388:will denote a
374:
367:
354:group action.
312:
309:
303: =
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296:
285:
281:
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273:
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266:
259:
256:
253:
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245:
242:
239:
236:
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226:
223:
220:
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152:of a subgroup
137:
134:
132:
129:
96:, a branch of
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1721:
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1682:0-387-94461-3
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1652:0-387-10632-4
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1629:3-11-012892-6
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1603:0-521-78675-4
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1458:
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1418:
1417:vector spaces
1414:
1410:
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1402:
1401:group actions
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1267:
1259:
1256:
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1249:
1244:
1235:
1231:
1222:
1218:
1217:
1216:p-constrained
1212:
1208:
1204:
1202:
1198:-series; its
1197:
1193:
1191:
1186:
1182:
1178:
1176:
1171:
1169:
1163:
1158:
1153:
1149:
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1134:
1109:
1099:
1095:
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1085:
1080:
1076:
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1063:
1058:
1053:
1049:
1039:
1036:
1033:
1029:
1024:
1020:
1014:
1004:
1000:
996:
993:
990:
985:
981:
977:
972:
968:
963:
958:
951:
941:
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935:
931:
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924:
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903:
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875:
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864:
857:
850:
846:
839:
832:
828:
826:
820:
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797:
793:
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776:
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766:
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735:
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667:
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368:
366:
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318:
317:group actions
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99:
95:
84:
81:
73:
70:December 2023
63:
59:
53:
52:
46:
41:
32:
31:
19:
1709:Group theory
1664:
1638:
1613:
1589:
1571:
1556:
1552:
1548:
1540:
1536:
1492:
1490:
1480:
1476:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1420:
1408:
1404:
1398:
1395:Significance
1388:
1382:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1352:
1350:
1220:
1219:for a prime
1214:
1210:
1206:
1200:
1199:
1195:
1189:
1188:
1184:
1180:
1174:
1173:
1167:
1161:
1156:
1151:
1147:
1141:
1136:
1132:
1130:
880:
873:
869:
862:
855:
848:
844:
837:
830:
829:. For sets
824:
822:
818:
814:
810:
808:
803:
799:
795:
791:
787:
783:
779:
777:
772:
768:
764:
584:
580:
579:
549:and denoted
546:
497:
493:
487:
486:
483:finite group
438:
434:
424:
417:
416:
412:
407:
390:finite group
385:
383:
370:
356:
343:
341:
314:
311:Significance
304:
300:
298:
200:
196:
192:
188:
181:
177:
170:intersection
165:
161:
153:
149:
145:
141:
140:For a group
139:
127:of a group.
122:
121:
113:
101:
94:group theory
91:
76:
67:
48:
1407:-cores and
1385:-complement
146:normal core
114:normal core
98:mathematics
62:introducing
18:Normal core
1691:0836.20001
1563:References
1539:′-core of
1177:-nilpotent
1131:The upper
589:, denoted
404:Definition
333:transitive
174:conjugates
136:Definition
45:references
1512:∞
1403:on sets,
1306:⊆
1096:π
1089:…
1077:π
1064:π
1030:π
1001:π
994:…
982:π
969:π
932:π
925:…
913:π
900:π
437:-core of
427:-subgroup
269:−
255:∈
248:⋂
1703:Category
1663:(1996).
1588:(2000),
1497:solvable
1391:′-core.
1318:′
1288:′
1253:′
1192:-soluble
736:′
688:′
653:′
606:′
517:′
352:faithful
120:and the
118:subgroup
1555:as the
1381:normal
1203:-length
1170:-series
868:, ...,
843:, ...,
827:-series
396:and to
363:abelian
361:in the
203:, i.e.
172:of the
58:improve
1689:
1679:
1649:
1626:
1600:
1419:. The
1166:lower
1150:′ and
883:) by:
823:upper
813:′ and
490:′-core
415:, the
408:For a
337:kernel
199:under
186:subset
144:, the
47:, but
587:-core
481:of a
420:-core
410:prime
373:-core
329:orbit
125:-core
116:of a
110:group
108:of a
1677:ISBN
1647:ISBN
1624:ISBN
1598:ISBN
809:The
778:The
547:core
369:The
321:sets
102:core
100:, a
1687:Zbl
1223:if
319:on
176:of
160:of
148:or
92:In
1705::
1685:.
1675:.
1667:.
1645:.
1641:.
1622:.
1618:.
1596:,
1592:,
1459:′,
1443:′,
1348:.
1183:′,
1162:p;
1160:=
1146:=
1144:−1
876:+1
861:,
851:+1
836:,
817:′,
798:′,
767:′,
583:′,
400:.
342:A
244::=
1693:.
1655:.
1632:.
1557:p
1553:G
1549:p
1541:G
1537:p
1523:)
1520:G
1517:(
1508:O
1481:p
1477:p
1473:p
1469:p
1465:p
1461:p
1457:p
1453:p
1449:p
1445:p
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