41:
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346:
296:
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781:
291:
707:
774:
940:
1346:
In 2017 John F. R. Duncan, Michael H. Mertens, and Ken Ono proved theorems that establish an analogue of
391:
205:
888:
123:
1350:
for the O'Nan group. Their results "reveal a role for the O'Nan pariah group as a provider of hidden
1690:
1454:
1179:
1139:
1110:
995:
589:
323:
200:
88:
469:
444:
407:
1351:
896:
739:
529:
1362:." The O'Nan moonshine results "also represent the intersection of moonshine theory with the
821:
613:
1653:
1625:
1589:
1582:
1546:
1463:
1375:
825:
553:
541:
159:
93:
8:
1632:
Yoshiara, Satoshi (1985), "The maximal subgroups of the sporadic simple group of O'Nan",
1347:
884:
128:
23:
1467:
1594:
1559:
1436:
1411:
113:
85:
1366:, which, since its inception in the 1960s, has become a driving force for research in
1641:
1613:
1608:
1573:
1554:
1534:
1441:
1363:
1317:
1291:
979:
957:
900:
518:
361:
255:
1033:
representations over the field with 7 elements, exchanged by an outer automorphism.
684:
1673:
1603:
1568:
1526:
1517:
O'Nan, Michael E. (1976), "Some evidence for the existence of a new simple group",
1479:
1471:
1431:
1423:
1053:
991:
876:
802:
669:
661:
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645:
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625:
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337:
279:
154:
1649:
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1578:
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872:
753:
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64:
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Journal of the
Faculty of Science. University of Tokyo. Section IA. Mathematics
1530:
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386:
366:
303:
268:
189:
179:
164:
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103:
80:
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1617:
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1367:
1011:
679:
601:
435:
308:
174:
1445:
1015:
806:
534:
233:
222:
169:
144:
139:
98:
69:
32:
1484:
1007:
1475:
914:). The following simple groups have Sylow 2-subgroups of Alperin type:
701:
429:
1030:
522:
1371:
892:
59:
1672:
401:
315:
903:
to a Sylow 2-Subgroup of a group of type (Z/2Z ×Z/2Z ×Z/2Z).PSL
40:
1385:
An informal description of these developments was written by
1410:
Duncan, John F. R.; Mertens, Michael H.; Ono, Ken (2017),
1014:. Thus it is one of the 6 sporadic groups called the
472:
447:
410:
1674:"Atlas of Finite Group Representations: O'Nan group"
1592:(1985), "The maximal subgroups of the O'Nan group",
480:
455:
418:
1497:"Moonshine Link Discovered for Pariah Symmetries"
1682:
1409:
1379:
1555:"A new construction of the O'Nan simple group"
1519:Proceedings of the London Mathematical Society
782:
1452:Griess, R. L. (1982), "The Friendly Giant",
1335:two classes, fused by an outer automorphism
1309:two classes, fused by an outer automorphism
1259:two classes, fused by an outer automorphism
831: 460,815,505,920 = 2
789:
775:
1607:
1572:
1516:
1491:
1483:
1435:
1390:
880:
474:
449:
412:
1631:
1045:
1029:showed that its triple cover has two 45-
949:(q), if q is congruent to 3 or 5 mod 8,
928:(q), if q is congruent to 3 or 5 mod 8,
1683:
1588:
1451:
1041:
999:
347:Classification of finite simple groups
975:= 2 and the extension does not split.
1552:
1036:
1026:
13:
1341:
1021:
14:
1702:
1661:
953:and the extension does not split.
39:
1138:the subgroup fixed by an outer
1380:Duncan, Mertens & Ono 2017
1178:the centralizer of an (inner)
708:Infinite dimensional Lie group
1:
986:= 2 and the extension splits.
1609:10.1016/0021-8693(85)90059-6
1574:10.1016/0021-8693(88)90141-X
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
7:
1329:
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1303:
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1206:
1201:
1172:
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1103:
1098:
1048:independently found the 13
932:and the extension does not
206:List of group theory topics
10:
1707:
1428:10.1038/s41467-017-00660-y
1422:(1), Article number: 670,
1402:
863:
1109:two classes, fused by an
968:and the extension splits.
1531:10.1112/plms/s3-32.3.421
1455:Inventiones Mathematicae
996:outer automorphism group
324:Elementary abelian group
201:Glossary of group theory
1553:Ryba, A. J. E. (1988),
1668:MathWorld: O'Nan Group
740:Linear algebraic group
482:
457:
420:
1495:(22 September 2017).
1416:Nature Communications
1332:= 2·3·7·11·19·31
1281:= 3·5·7·11·19·31
1064:Maximal subgroups of
994:has order 3, and its
971:For the O'Nan group,
822:sporadic simple group
483:
458:
421:
16:Sporadic simple group
1376:mathematical physics
1130:= 2·3·5·7·11·19
470:
445:
408:
1468:1982InMat..69....1G
1387:Erica Klarreich
1348:monstrous moonshine
1232:= 2·7·11·19·31
1209:= 2·7·11·19·31
1175:= 3·7·11·19·31
1068:
114:Group homomorphisms
24:Algebraic structure
1595:Journal of Algebra
1560:Journal of Algebra
1476:10.1007/BF01389186
1412:"Pariah moonshine"
1306:= 2·3·7·19·31
1256:= 2·3·7·11·19
1111:outer automorphism
1106:= 2·3·5·11·31
1063:
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
1590:Wilson, Robert A.
1382:, article 670).
1364:Langlands program
1339:
1338:
1054:maximal subgroups
1050:conjugacy classes
1037:Maximal subgroups
1002::94) showed that
980:Higman-Sims group
958:alternating group
877:Michael O'Nan
875:and was found by
871:is one of the 26
799:
798:
374:
373:
256:Alternating group
213:
212:
1698:
1677:
1656:
1628:
1611:
1585:
1576:
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1521:, Third Series,
1513:
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1509:
1493:Klarreich, Erica
1488:
1487:
1448:
1439:
1199:
1069:
1062:
992:Schur multiplier
883:) in a study of
858:
818:O'Nan–Sims group
803:abstract algebra
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
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338:Schur multiplier
292:Cauchy's theorem
280:Quaternion group
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1691:Sporadic groups
1681:
1680:
1664:
1659:
1507:
1505:
1502:Quanta Magazine
1405:
1396:Quanta Magazine
1360:elliptic curves
1356:quadratic forms
1344:
1342:O'Nan moonshine
1331:
1326:
1321:
1305:
1301:= 2·3·5·11
1300:
1295:
1280:
1275:
1270:
1255:
1251:= 2·3·5·31
1250:
1245:
1231:
1226:
1222:
1208:
1203:
1197:
1193:
1174:
1169:
1165:
1161:
1157:
1135:= 2·3·7·31
1134:
1129:
1124:
1105:
1101:= 2·3·7·19
1100:
1095:
1046:Yoshiara (1985)
1039:
1024:
1022:Representations
963:
948:
941:Steinberg group
927:
920:Chevalley group
913:
906:
899:type", meaning
873:sporadic groups
866:
856:
801:In the area of
795:
766:
765:
754:Abelian variety
747:Reductive group
735:
725:
724:
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614:Special unitary
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331:Frobenius group
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65:Normal subgroup
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5:
1704:
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1663:
1662:External links
1660:
1658:
1657:
1640:(1): 105–141,
1629:
1602:(2): 467–473,
1586:
1567:(1): 173–197,
1550:
1525:(3): 421–479,
1514:
1489:
1449:
1406:
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1327:= 2·3·5·7
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1170:= 2·3·5·7
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998:has order 2. (
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223:Finite groups
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55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
21:
20:
1637:
1633:
1599:
1593:
1564:
1558:
1522:
1518:
1506:. Retrieved
1500:
1459:
1453:
1419:
1415:
1394:
1384:
1345:
1276:= 2·3·7
1227:= 2·3·5
1204:= 2·3·5
1183:
1143:
1065:
1060:as follows:
1057:
1040:
1025:
1006:cannot be a
1003:
989:
983:
972:
965:
950:
943:
929:
922:
908:
868:
867:
852:
848:
844:
843: 7
840:
839: 5
836:
835: 3
832:
817:
813:
810:
807:group theory
800:
629:
617:
605:
593:
581:
569:
557:
545:
316:
273:
260:
249:
238:
234:Cyclic group
112:
99:Free product
70:Group action
33:Group theory
28:Group theory
27:
1462:(1): 1007,
1330:182,863,296
1031:dimensional
1027:Ryba (1988)
1008:subquotient
1000:Griess 1982
855: 31 ≈ 5
811:O'Nan group
519:Topological
358:alternating
1304:58,183,776
1279:42,858,585
1254:30,968,784
1230:17,778,376
1207:17,778,376
1180:involution
1140:involution
901:isomorphic
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
1646:0040-8980
1618:0021-8693
1539:0024-6115
1508:23 August
1173:2,857,239
1133:2,624,832
1099:3,753,792
1084:Comments
1075:Structure
805:known as
690:Conformal
578:Euclidean
185:nilpotent
1685:Category
1446:28935903
1372:geometry
1352:symmetry
1194:(3:4 × A
978:For the
956:For the
939:For the
918:For the
893:subgroup
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
1654:0783183
1626:0812997
1583:0921973
1547:0401905
1464:Bibcode
1437:5608900
1403:Sources
1389: (
1168:161,280
1128:175,560
1104:122,760
1016:pariahs
1010:of the
897:Alperin
887:with a
879: (
864:History
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
1652:
1644:
1624:
1616:
1581:
1545:
1537:
1444:
1434:
1274:10,752
1249:14,880
1225:25,920
1202:25,920
885:groups
853:·
849:·
845:·
841:·
837:·
833:·
809:, the
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
1393:) in
1325:2,520
1314:12,13
1299:7,920
1288:10,11
1219:3:2.D
1162:(4):2
1096:(7):2
1081:Index
1078:Order
966:n = 1
951:n = 1
934:split
930:n = 1
889:Sylow
826:order
820:is a
719:Sp(∞)
716:SU(∞)
129:image
1642:ISSN
1614:ISSN
1535:ISSN
1510:2020
1442:PMID
1391:2017
1378:." (
1374:and
1358:and
1246:(31)
1044:and
990:The
895:of "
881:1976
713:O(∞)
702:Loop
521:and
1604:doi
1569:doi
1565:112
1527:doi
1480:hdl
1472:doi
1432:PMC
1424:doi
1354:to
1271:(2)
1239:7,8
1184:O'N
1182:in
1146::2
1144:O'N
1142:in
1089:1,2
1072:No.
1066:O'N
1058:O'N
1056:of
1052:of
1004:O'N
869:O'N
859:10.
824:of
816:or
814:O'N
628:Sp(
616:SU(
592:SO(
556:SL(
544:GL(
1687::
1650:MR
1648:,
1638:32
1636:,
1622:MR
1620:,
1612:,
1600:97
1598:,
1579:MR
1577:,
1563:,
1557:,
1543:MR
1541:,
1533:,
1523:32
1499:.
1478:,
1470:,
1460:69
1458:,
1440:,
1430:,
1418:,
1414:,
1399:.
1370:,
1294:11
1267:4L
1221:10
1198:)2
1018:.
982:,
964:,
891:2-
604:U(
580:E(
568:O(
26:→
1676:.
1606::
1571::
1529::
1512:.
1482::
1474::
1466::
1426::
1420:8
1320:7
1318:A
1292:M
1269:3
1264:9
1244:2
1242:L
1216:6
1196:6
1190:5
1164:1
1160:3
1158:L
1156:2
1154:4
1151:4
1123:1
1121:J
1117:3
1094:3
1092:L
984:n
973:n
962:8
960:A
947:4
944:D
936:.
926:2
923:G
912:2
909:F
907:(
905:3
857:×
790:e
783:t
776:v
672:8
670:E
664:7
662:E
656:6
654:E
648:4
646:F
640:2
638:G
632:)
630:n
620:)
618:n
608:)
606:n
596:)
594:n
584:)
582:n
572:)
570:n
560:)
558:n
548:)
546:n
488:)
475:Z
463:)
450:Z
426:)
413:Z
404:(
317:p
282:Q
274:n
271:D
261:n
258:A
250:n
247:S
239:n
236:Z
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