48:
1232:
981:
is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many
Hauptmoduln from simple combinations of dimensions of sporadic groups. For
1041:
1023:
493:
468:
431:
1227:{\displaystyle T_{3C}(\tau )={\Big (}j(3\tau ){\Big )}^{1/3}={\frac {1}{q}}\,+\,248q^{2}\,+\,4124q^{5}\,+\,34752q^{8}\,+\,213126q^{11}\,+\,1057504q^{14}+\cdots \,}
795:
19:
This article is about the sporadic simple group found by John G. Thompson. For the three unusual infinite groups F, T and V found by
Richard Thompson, see
1027:
353:
303:
788:
298:
714:
781:
904:(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the
896:. It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the
398:
212:
1824:
966:
596:
330:
207:
95:
924:
is a product of the
Thompson group and a group of order 3, as a result of which the Thompson group acts on a
925:
746:
536:
820:
620:
954:× Th, so Th centralizes 3 involutions alongside the 3-cycle. These involutions are centralized by the
824:
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166:
100:
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978:
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30:
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529:
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928:
over the field with 3 elements. This vertex operator algebra contains the E
1675:
Linton, Stephen A. (1989), "The maximal subgroups of the
Thompson group",
47:
908:(which unlike the Thompson group is a subgroup of the compact Lie group E
66:
408:
322:
950:
The full normalizer of a 3C element in the
Monster group is S
892:
of a certain lattice in the 248-dimensional Lie algebra of E
1031:
920:
The centralizer of an element of order 3 of type 3C in the
1260:
found the 16 conjugacy classes of maximal subgroups of
1810:
1044:
992:
977:
Conway and Norton suggested in their 1979 paper that
479:
454:
417:
1711:
Smith, P. E. (1976), "A simple subgroup of M? and E
1226:
1017:
876:is one of the 26 sporadic groups and was found by
487:
462:
425:
1095:
1072:
1717:The Bulletin of the London Mathematical Society
1490:normalizer of a subgroup of order 3 (class 3C)
1446:normalizer of a subgroup of order 3 (class 3B)
1406:normalizer of a subgroup of order 3 (class 3A)
958:, which therefore contains Th as a subgroup.
789:
8:
1677:Journal of the London Mathematical Society
796:
782:
234:
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25:
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478:
456:
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453:
419:
418:
416:
1266:
986:, the relevant McKay-Thompson series is
969:of the Thompson group are both trivial.
881:
830: 90,745,943,887,872,000
808:In the area of modern algebra known as
352:
118:
28:
1257:
354:Classification of finite simple groups
885:
7:
1521:normalizer of a subgroup of order 5
1754:(1976), "A conjugacy theorem for E
14:
46:
973:Generalized monstrous moonshine
1089:
1080:
1064:
1058:
1012:
1006:
888:. They constructed it as the
715:Infinite dimensional Lie group
1:
1018:{\displaystyle T_{3C}(\tau )}
1775:10.1016/0021-8693(76)90235-0
488:{\displaystyle \mathbb {Z} }
463:{\displaystyle \mathbb {Z} }
426:{\displaystyle \mathbb {Z} }
213:List of group theory topics
1841:
1362:centralizer of involution
939:, giving the embedding of
18:
1805:MathWorld: Thompson group
967:outer automorphism group
331:Elementary abelian group
208:Glossary of group theory
1689:10.1112/jlms/s2-39.1.79
926:vertex operator algebra
1228:
1019:
747:Linear algebraic group
489:
464:
427:
1268:Maximal subgroups of
1229:
1020:
884:) and constructed by
821:sporadic simple group
490:
465:
428:
16:Sporadic simple group
1729:10.1112/blms/8.2.161
1042:
990:
878:John G. Thompson
477:
452:
415:
1272:
979:monstrous moonshine
121:Group homomorphisms
31:Algebraic structure
1761:Journal of Algebra
1475:(3 × 3 : 2 ·
1267:
1224:
1015:
956:Baby monster group
890:automorphism group
597:Special orthogonal
485:
460:
423:
304:Lagrange's theorem
1752:Thompson, John G.
1679:, Second Series,
1666:
1665:
1326:= 2·3·5·7·31
1253:Maximal subgroups
1125:
932:Lie algebra over
806:
805:
381:
380:
263:Alternating group
220:
219:
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963:Schur multiplier
863:
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740:Algebraic groups
513:Hyperbolic group
503:Arithmetic group
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299:Cauchy's theorem
287:Quaternion group
235:
61:
50:
37:
26:
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1825:Sporadic groups
1815:
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1801:
1757:
1750:
1714:
1710:
1674:
1671:
1659:
1655:
1639:
1624:
1620:
1604:
1599:
1584:= 2·3·5·19
1583:
1578:
1562:
1557:
1551:7 : (3 × 2
1540:
1535:
1517:
1513:
1504:
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1464:
1460:
1442:
1438:
1429:
1426:
1425:
1424:
1420:
1417:
1416:
1415:
1403:= 2·3·7·13
1402:
1397:
1381:= 2·3·7·19
1380:
1375:
1358:
1354:
1346:
1343:
1342:
1341:
1331:Dempwolff group
1325:
1320:
1304:= 2·3·7·13
1303:
1298:
1255:
1207:
1189:
1171:
1153:
1135:
1092:
1045:
1040:
1039:
1026:
993:
988:
987:
975:
953:
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938:
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916:Representations
911:
906:Dempwolff group
903:
898:Chevalley group
895:
871:
861:
802:
773:
772:
761:Abelian variety
754:Reductive group
742:
732:
731:
730:
729:
680:
672:
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621:Special unitary
532:
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394:Discrete groups
383:
382:
338:Frobenius group
283:
270:
259:
252:Symmetric group
248:
232:
222:
221:
72:Normal subgroup
58:
38:
29:
24:
21:Thompson groups
17:
12:
11:
5:
1838:
1836:
1828:
1827:
1817:
1816:
1813:
1812:
1807:
1800:
1799:External links
1797:
1796:
1795:
1768:(2): 525–530,
1755:
1748:
1723:(2): 161–165,
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1359:= 2·3·5·7
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814:Thompson group
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768:Elliptic curve
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704:Diffeomorphism
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549:General linear
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311:Sylow theorems
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276:Dihedral group
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167:multiplicative
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96:direct product
90:
88:Quotient group
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1646:
1645:
1642:
1640:= 3·5·31
1637:
1634:
1631:
1630:
1627:
1622:
1617:
1614:
1611:
1610:
1607:
1605:= 2·3·13
1602:
1596:
1593:
1590:
1589:
1586:
1581:
1579:(19) : 2
1575:
1572:
1569:
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1258:Linton (1989)
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922:Monster group
915:
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853: 19
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230:Finite groups
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57:Basic notions
54:
53:
49:
45:
44:
41:
36:
32:
27:
22:
1765:
1759:
1720:
1716:
1683:(1): 79–88,
1680:
1676:
1660:= 2·3·5
1650:
1635:31 : 15
1625:= 2·3·5
1615:
1594:
1573:
1563:= 2·3·7
1552:
1541:= 2·3·5
1530:
1518:= 2·3·5
1508:
1487:= 2·3·5
1476:
1455:
1433:
1392:
1376:(8) : 6
1370:
1349:
1324:319,979,520
1315:
1302:634,023,936
1299:(2) : 3
1293:
1269:
1264:as follows:
1261:
1256:
1242:
1238:
1236:
983:
976:
960:
949:
940:
933:
919:
886:Smith (1976)
873:
872:
854:
850:
846:
845: 7
842:
841: 5
838:
837: 3
834:
816:
813:
810:group theory
807:
636:
624:
612:
600:
588:
576:
564:
552:
323:
280:
267:
256:
245:
241:Cyclic group
119:
106:Free product
77:Group action
40:Group theory
35:Group theory
34:
1401:25,474,176
1379:33,094,656
1357:92,897,280
1247:j-invariant
526:Topological
365:alternating
1669:References
1482:) : 2
1465:= 2·3
1443:= 2·3
633:Symplectic
573:Orthogonal
530:Lie groups
437:Free group
162:continuous
101:Direct sum
1784:0021-8693
1737:0024-6093
1697:0024-6107
1529:5 :
1507: : 4
1432: : 2
1285:Comments
1279:Structure
1245:) is the
1221:⋯
1087:τ
1062:τ
1010:τ
833:= 2
697:Conformal
585:Euclidean
192:nilpotent
1819:Category
1485:349,920
1463:944,784
1441:944,784
965:and the
692:Poincaré
537:Solenoid
409:Integers
399:Lattices
374:sporadic
369:Lie type
197:solvable
187:dihedral
172:additive
157:infinite
67:Subgroup
1792:0399193
1745:0409630
1705:0989921
1539:12,000
1516:12,000
1205:1057504
1032:A007245
1030::
880: (
869:History
857: 31
687:Lorentz
609:Unitary
508:Lattice
448:PSL(2,
182:abelian
93:(Semi-)
1790:
1782:
1743:
1735:
1715:(3)",
1703:
1695:
1603:5,616
1582:6,840
1561:7,056
1414:(3 × 3
1187:213126
943:into E
855:·
851:·
847:·
843:·
839:·
835:·
812:, the
542:Circle
473:SL(2,
362:cyclic
326:-group
177:cyclic
152:finite
147:simple
131:kernel
1423:) · 3
1391:(3 x
1282:Order
1169:34752
947:(3).
825:order
819:is a
726:Sp(∞)
723:SU(∞)
136:image
1780:ISSN
1733:ISSN
1693:ISSN
1658:120
1638:465
1623:720
1329:the
1237:and
1151:4124
1028:OEIS
961:The
882:1976
720:O(∞)
709:Loop
528:and
1770:doi
1758:",
1725:doi
1685:doi
1600:(3)
1536:(5)
1321:(2)
1276:No.
1133:248
1035:),
912:).
864:10.
860:≈ 9
823:of
635:Sp(
623:SU(
599:SO(
563:SL(
551:GL(
1821::
1788:MR
1786:,
1778:,
1766:38
1764:,
1741:MR
1739:,
1731:,
1719:,
1701:MR
1699:,
1691:,
1681:39
1647:16
1632:15
1619:10
1612:14
1591:13
1570:12
1548:11
1531:GL
1526:10
1270:Th
1262:Th
1249:.
1213:14
1195:11
984:Th
941:Th
874:Th
817:Th
611:U(
587:E(
575:O(
33:→
1772::
1756:8
1727::
1721:8
1713:8
1687::
1654:5
1651:S
1616:M
1598:3
1595:L
1577:2
1574:L
1558:)
1556:4
1553:S
1534:2
1512:4
1509:S
1503:+
1498:5
1495:9
1480:6
1477:A
1472:8
1459:4
1456:S
1451:7
1437:4
1434:S
1428:+
1419:+
1411:6
1396:2
1393:G
1388:5
1374:3
1371:U
1367:4
1353:9
1350:A
1345:+
1340:2
1337:3
1319:5
1316:L
1314:2
1311:2
1297:4
1294:D
1290:1
1243:τ
1241:(
1239:j
1218:+
1209:q
1201:+
1191:q
1183:+
1177:8
1173:q
1165:+
1159:5
1155:q
1147:+
1141:2
1137:q
1129:+
1123:q
1120:1
1115:=
1110:3
1106:/
1102:1
1096:)
1090:)
1084:3
1081:(
1078:j
1073:(
1068:=
1065:)
1059:(
1054:C
1051:3
1047:T
1025:(
1013:)
1007:(
1002:C
999:3
995:T
952:3
945:8
937:3
934:F
930:8
910:8
902:8
900:E
894:8
862:×
797:e
790:t
783:v
679:8
677:E
671:7
669:E
663:6
661:E
655:4
653:F
647:2
645:G
639:)
637:n
627:)
625:n
615:)
613:n
603:)
601:n
591:)
589:n
579:)
577:n
567:)
565:n
555:)
553:n
495:)
482:Z
470:)
457:Z
433:)
420:Z
411:(
324:p
289:Q
281:n
278:D
268:n
265:A
257:n
254:S
246:n
243:Z
23:.
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