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The spider relation can only be defined directly if the diagram has at least 2 nodes in all 3 directions. However, it is possible to define the spider relation for a larger group, then consider the subgroup generated by fewer
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as a subgroup of larger Y-group. However, if we simply adjoin the spider relation to the
Coxeter group, we obtain the double cover
165:
424:
Many subgroups of the (bi)monster can be defined by adjoining the spider relation to smaller
Coxeter diagrams, most notably the
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155:
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125:
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52:
1027:
703:
1480:, London Math. Soc. Lecture Note Ser., vol. 165, Cambridge: Cambridge University Press, pp. 24–45,
1100:
279:
1142:
502:, respectively. Other groups, which would be infinite without the spider relation, are summarized below:
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1181:
1228:
886:
775:
592:
526:
349:
282:
of the bimonster could be given by adding a certain extra relation to the presentation defined by the
936:
828:
645:
25:
1456:, Lecture Notes in Pure and Applied Mathematics, vol. 111, New York: Dekker, pp. 27–50,
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Hyperbolic reflection groups, completely replicable functions, the
Monster and the Bimonster
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29:
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33:
467:
are finite even without adjoining additional relations. They are the
Coxeter groups
1644:
1389:
1364:
Basak, Tathagata (2007), "The complex
Lorentzian Leech lattice and the Bimonster",
115:
1485:
17:
1608:
275:
1613:
1472:; Pritchard, A. D. (1992), "Hyperbolic reflections for the Bimonster and 3Fi
234:
Actually, the 3 outermost nodes are redundant. This is because the subgroup
228:
1520:
1429:
1325:
1380:
412:, the group generated is the bimonster. This was proved in 1990 by
1564:
Soicher, Leonard H. (1989), "From the
Monster to the Bimonster",
1541:, Ph.D. thesis, Princeton University, Department of Mathematics,
405:. Once this relation is added, and the diagram is extended to
1504:; Simons, Christopher S. (2001), "26 implies the Bimonster",
1413:
Ivanov, A. A. (1999), "Y-groups via
Transitive Extension",
289:
diagram. More specifically, the affine E6 Coxeter group is
1445:; Soicher, Leonard H. (1988), "The Bimonster, the group
1652:
1231:
1184:
1145:
1103:
1030:
986:
939:
889:
831:
778:
706:
648:
595:
529:
352:
295:
55:
416:; the proof was simplified in 1999 by A. A. Ivanov.
1478:
Groups, Combinatorics & Geometry (Durham, 1990)
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1078:
1005:
961:
914:
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262:: it automatically generates the 3 extra nodes of
248:Coxeter group. It generates the remaining node of
89:
1299:As mentioned before, the 3 outermost nodes of
1285:
1283:
1281:
1672:
8:
346:, which can be reduced to the finite group
339:{\displaystyle \mathbb {Z} ^{6}:O_{5}(3):2}
90:{\displaystyle Bi=M\wr \mathbb {Z} _{2}.\,}
1679:
1665:
1218:This is the group obtained when realizing
1079:{\displaystyle 2\times 2^{2}.^{2}E_{6}(2)}
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1379:
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1239:
1230:
1186:
1185:
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302:
298:
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294:
86:
77:
73:
72:
54:
1454:Computers in Algebra (Chicago, IL, 1985)
1452:, and the projective plane of order 3",
1313:is sufficient to generate the bimonster.
754:{\displaystyle 2^{7}:(O_{7}(2)\times 2)}
504:
100:The Bimonster is also a quotient of the
1211:
401:by adding a single relation called the
1121:{\displaystyle 2\times 2.\mathbb {B} }
255:. This pattern extends all the way to
7:
1633:
1631:
1160:{\displaystyle 2\times \mathbb {M} }
1537:Simons, Christopher Smyth (1997),
14:
1199:{\displaystyle \mathbb {M} \wr 2}
1635:
1622:Note: incorrectly named here as
1268:{\displaystyle 2.O_{8}^{+}(3):2}
915:{\displaystyle 2\times 2Fi_{22}}
807:{\displaystyle O_{9}(2)\times 2}
624:{\displaystyle O_{7}(3)\times 2}
571:{\displaystyle 3^{5}:O_{5}(3):2}
394:{\displaystyle 3^{5}:O_{5}(3):2}
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962:{\displaystyle 2\times Fi_{23}}
865:{\displaystyle O_{10}^{-}(2):2}
1390:10.1016/j.jalgebra.2006.05.033
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682:{\displaystyle O_{8}^{+}(3):2}
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321:
1:
1651:. You can help Knowledge by
1581:10.1016/0021-8693(89)90064-1
1486:10.1017/CBO9780511629259.006
1719:
1630:
1006:{\displaystyle 3.Fi_{24}}
1647:-related article is a
1521:10.1006/jabr.2000.8494
1430:10.1006/jabr.1999.7882
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104:corresponding to the
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1703:Group theory stubs
1606:Weisstein, Eric W.
1567:Journal of Algebra
1507:Journal of Algebra
1416:Journal of Algebra
1367:Journal of Algebra
1306:are redundant, so
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1443:Norton, Simon P.
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1330:simple Lie group
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1502:Conway, John H.
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1022:
993:
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827:
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494:
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414:Simon P. Norton
411:
403:spider relation
366:
353:
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311:
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118:with 16 nodes:
113:
71:
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50:
45:
12:
11:
5:
1716:
1714:
1706:
1705:
1700:
1690:
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1599:
1598:External links
1596:
1595:
1594:
1574:(2): 275–280,
1561:
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1514:(2): 805–814,
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426:Fischer groups
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420:Other Y-groups
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106:Dynkin diagram
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30:wreath product
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508:Y-group name
507:
506:
503:
498:
491:
484:
477:
470:
463:
456:
449:
442:
435:
432:. The groups
431:
427:
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121:
120:
119:
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114:, a Y-shaped
110:
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102:Coxeter group
83:
78:
68:
65:
62:
59:
56:
49:
48:
47:
42:
38:
35:
34:monster group
31:
27:
23:
19:
1698:Group theory
1653:expanding it
1645:group theory
1642:
1627:
1621:
1612:
1571:
1565:
1538:
1511:
1505:
1477:
1453:
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1420:
1414:
1381:math/0508228
1374:(1): 32–56,
1371:
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283:
280:presentation
271:
263:
256:
249:
235:
233:
226:
108:
99:
40:
36:
28:that is the
21:
15:
1609:"Bimonster"
276:conjectured
18:mathematics
1692:Categories
1358:References
1346:Affine E_6
1614:MathWorld
1406:125231322
1191:≀
1150:×
1108:×
1035:×
944:×
894:×
843:−
799:×
743:×
616:×
69:≀
22:bimonster
1326:Triality
1320:See also
428:and the
1590:0992763
1557:2696217
1530:1805481
1494:1200248
1462:1060755
1398:2301231
278:that a
241:is the
32:of the
1588:
1555:
1545:
1528:
1492:
1460:
1404:
1396:
1290:nodes.
495:, and
460:, and
20:, the
1643:This
1402:S2CID
1376:arXiv
472:i+j+1
116:graph
39:with
26:group
24:is a
1649:stub
1543:ISBN
1576:doi
1572:121
1516:doi
1512:235
1482:doi
1476:",
1450:555
1425:doi
1421:218
1386:doi
1372:309
1352:222
1341:111
1311:444
1304:555
1223:224
1175:444
1136:344
1094:334
1021:333
977:244
930:234
880:233
822:144
769:134
697:133
639:224
586:223
520:222
479:i+j
465:124
458:123
451:122
444:ij1
437:ij0
410:444
287:444
267:555
260:444
253:125
239:124
112:555
16:In
1694::
1611:.
1586:MR
1584:,
1570:,
1553:MR
1551:,
1526:MR
1524:,
1510:,
1490:MR
1488:,
1474:24
1458:MR
1441:;
1419:,
1400:,
1394:MR
1392:,
1384:,
1370:,
1336:,
1328:-
1280:^
1233:2.
1111:2.
999:24
988:3.
955:23
908:22
838:10
488:,
481:,
474:,
453:,
446:,
439:,
269:.
46::
1680:e
1673:t
1666:v
1655:.
1624:)
1620:(
1617:.
1593:.
1578::
1560:.
1533:.
1518::
1497:.
1484::
1465:.
1447:Y
1434:.
1427::
1409:.
1388::
1378::
1349:Y
1338:Y
1334:4
1332:D
1308:Y
1301:Y
1275:.
1263:2
1260::
1257:)
1254:3
1251:(
1246:+
1241:8
1237:O
1220:Y
1194:2
1187:M
1172:Y
1154:M
1147:2
1133:Y
1115:B
1105:2
1091:Y
1074:)
1071:2
1068:(
1063:6
1059:E
1053:2
1049:.
1043:2
1039:2
1032:2
1018:Y
995:i
991:F
974:Y
951:i
947:F
941:2
927:Y
904:i
900:F
897:2
891:2
877:Y
860:2
857::
854:)
851:2
848:(
834:O
819:Y
802:2
796:)
793:2
790:(
785:9
781:O
766:Y
749:)
746:2
740:)
737:2
734:(
729:7
725:O
721:(
718::
713:7
709:2
694:Y
677:2
674::
671:)
668:3
665:(
660:+
655:8
651:O
636:Y
619:2
613:)
610:3
607:(
602:7
598:O
583:Y
566:2
563::
560:)
557:3
554:(
549:5
545:O
541::
536:5
532:3
517:Y
500:8
497:E
493:7
490:E
486:6
483:E
476:D
469:A
462:Y
455:Y
448:Y
441:Y
434:Y
407:Y
389:2
386::
383:)
380:3
377:(
372:5
368:O
364::
359:5
355:3
334:2
331::
328:)
325:3
322:(
317:5
313:O
309::
304:6
299:Z
284:Y
264:Y
257:Y
250:Y
245:8
243:E
236:Y
109:Y
84:.
79:2
74:Z
66:M
63:=
60:i
57:B
44:2
41:Z
37:M
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