56:
1305:. He found that if a finite field of characteristic 2 also has an automorphism whose square was the Frobenius map, then an analogue of Steinberg's construction gave the Suzuki groups. The fields with such an automorphism are those of order 2, and the corresponding groups are the Suzuki groups
2128:), but there are so many conventions that it is not possible to say exactly what groups these correspond to without it being specified explicitly. The source of the problem is that the simple group is not the orthogonal group O, nor the
1945:
There is a bewildering number of "accidental" isomorphisms between various small groups of Lie type (and alternating groups). For example, the groups SL(2, 4), PSL(2, 5), and the alternating group on 5 points are all isomorphic.
2026:) does not necessarily induce a surjective map of the corresponding groups with values in some (non algebraically closed) field. There are similar problems with the points of other algebraic groups with values in finite fields.
1174:, and field automorphisms induced by automorphisms of a finite field. Analogously to the unitary case, Steinberg constructed families of groups by taking fixed points of a product of a diagram and a field automorphism.
1535:), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the
1411:
are the ones whose structure is hardest to pin down explicitly. These groups also played a role in the discovery of the first modern sporadic group. They have involution centralizers of the form
1028:), which can be used to define the corresponding algebraic groups over the integers. In particular, he could take their points with values in any finite field. For the Lie algebras A
1965:
501:
476:
439:
2407:
956:. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series A
2791:, Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e Úd. corrigée, Exposé 162, vol. 15, Paris: Secrétariat math'ematique,
2710:
1572:
In general the finite group associated to an endomorphism of a simply connected simple algebraic group is the universal central extension of a simple group, so is
803:
2600:
2556:
1980:
There is no standard notation for the finite groups of Lie type, and the literature contains dozens of incompatible and confusing systems of notation for them.
2132:
PSO, but rather a subgroup of PSO, which accordingly does not have a classical notation. A particularly nasty trap is that some authors, such as the
2315:
2272:
1549:
1396:
858:
361:
2519:
2347:
311:
2129:
1580:. However some of the smallest groups in the families above are either not perfect or have a Schur multiplier larger than "expected".
796:
306:
1612:(2) Not perfect, but is isomorphic to the symmetric group on 6 points so its derived subgroup has index 2 and is simple of order 360.
1527:. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (
1539:, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their
1447:
The Suzuki groups are the only finite non-abelian simple groups with order not divisible by 3. They have order 2(2 + 1)(2 â 1).
1484:
1468:
2644:
722:
2705:
2416:
1278:
789:
1052:
this gave well known classical groups, but his construction also gave groups associated to the exceptional Lie algebras E
2331:
1950:
1025:
1495:â 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(
406:
220:
2268:
1252:
have no analogue over the reals, as the complex numbers have no automorphism of order 3. The symmetries of the D
1116:
Chevalley's construction did not give all of the known classical groups: it omitted the unitary groups and the non-
2822:
1960:. Some of the small alternating groups also have exceptional properties. The alternating groups usually have an
1961:
1000:
Chevalley groups can be thought of as Lie groups over finite fields. The theory was clarified by the theory of
604:
338:
215:
103:
2455:"A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface"
2683:
2228:) are the groups of points with values in the simple or the simply connected algebraic group. For example, A
1285:) found a new infinite series of groups that at first sight seemed unrelated to the known algebraic groups.
953:
1167:, which commute. The unitary group is the group of fixed points of the product of these two automorphisms.
1136:. This construction generalizes the usual construction of the unitary group from the general linear group.
2450:
2402:
2311:
2133:
1957:
1540:
1302:
1117:
842:
754:
544:
2817:
2812:
1561:
628:
1648:
Some cases where the group is perfect but has a Schur multiplier that is larger than expected include:
1968:. Alternating groups usually have a Schur multiplier of order 2, but the ones on 6 or 7 points have a
2719:
2339:
1516:
1504:
1317:(Strictly speaking, the group Suz(2) is not counted as a Suzuki group as it is not simple: it is the
926:
568:
556:
174:
108:
2278:
1370:
1164:
903:
143:
38:
484:
459:
422:
2745:
2491:
2438:
1159:
1021:
128:
100:
2594:
Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F
2550:
Ree, Rimhak (1960), "A family of simple groups associated with the simple Lie algebra of type (G
1531:). Although it was known since 19th century that other finite simple groups exist (for example,
1476:
2148:
the orthogonal group, but the corresponding simple group. The notation Ω, PΩ was introduced by
1556:
include all the finite simple groups other than the cyclic groups, the alternating groups, the
2771:
2737:
2663:
2619:
2575:
2515:
2454:
2430:
2383:
2343:
2149:
1969:
1464:
1353:
have extra automorphisms in characteristic 2 and 3. (Roughly speaking, in characteristic
1220:
1132:, the second of which was discovered at about the same time from a different point of view by
1124:
found a modification of
Chevalley's construction that gave these groups and two new families D
1108:
946:
915:
869:
857:
groups of Lie type does have a precise definition, and they make up most of the groups in the
699:
533:
270:
1455:
Finite groups of Lie type were among the first groups to be considered in mathematics, after
872:
may be viewed as the rational points of a reductive linear algebraic group over the field of
2761:
2727:
2653:
2609:
2565:
2483:
2420:
2373:
2156:†4 and thus the same notation may be used for a slightly different group, which agrees in
1577:
1512:
1005:
942:
934:
930:
898:
An initial approach to this question was the definition and detailed study of the so-called
684:
676:
668:
660:
652:
640:
580:
520:
510:
352:
294:
169:
138:
2796:
2757:
2693:
2675:
2631:
2587:
2529:
2395:
2357:
853:
does not have a widely accepted precise definition, but the important collection of finite
2792:
2753:
2689:
2671:
2627:
2583:
2525:
2511:
2391:
2353:
2212:
1536:
1508:
1460:
1318:
1017:
1001:
893:
839:
768:
761:
747:
704:
592:
515:
345:
259:
199:
79:
922:
investigated the orders of such groups, with a view to classifying cases of coincidence.
17:
2723:
1644:(3) Not perfect, but the derived group has index 3 and is the simple group of order 504.
2783:
2536:
1532:
1480:
1171:
1148:
1140:
949:
835:
775:
711:
401:
381:
318:
283:
204:
194:
179:
164:
118:
95:
2766:
2425:
1552:. Inspection of the list of finite simple groups shows that groups of Lie type over a
2806:
2495:
2191:, and people have different ideas on which should be included in the notation. The "A
1573:
1269:
938:
911:
831:
694:
616:
450:
323:
189:
2614:
2570:
2540:
1427:= 3, and by investigating groups with an involution centralizer of the similar form
1553:
1479:
in the 1830s. The systematic exploration of finite groups of Lie type started with
1456:
1380:
873:
864:
The name "groups of Lie type" is due to the close relationship with the (infinite)
854:
846:
821:
549:
248:
237:
184:
159:
154:
113:
84:
47:
1953:. Many of these special properties are related to certain sporadic simple groups.
2639:
1956:
Alternating groups sometimes behave as if they were groups of Lie type over the
1436:
817:
2711:
Proceedings of the
National Academy of Sciences of the United States of America
1637:(2) Not perfect, but the derived group has index 2 and is simple of order 6048.
1627:
1557:
1391:
is not simple, but it has a simple normal subgroup of index 3, isomorphic to A
1375:
1361:
in the Dynkin diagram when taking diagram automorphisms.) The smallest group F
1286:
919:
865:
716:
444:
2741:
2667:
2623:
2579:
2434:
2387:
2378:
1170:
In the same way, many
Chevalley groups have diagram automorphisms induced by
1273:
537:
28:
2775:
2658:
2732:
55:
2090:("orthogonal" groups) is particularly confusing. Some symbols used are O(
1257:
74:
1918:, so the Schur multiplier of the simple group has order 12 instead of 3.
1895:, so the Schur multiplier of the simple group has order 12 instead of 3.
1872:, so the Schur multiplier of the simple group has order 36 instead of 4.
1717:, so the Schur multiplier of the simple group has order 48 instead of 3.
1598:(3) = PSL(2, 3) Solvable of order 12 (the alternating group on 4 points)
2487:
2442:
2018:). The problem is that a surjective map of algebraic groups such as SL(
1941:, so the Schur multiplier of the simple group has order 4 instead of 1.
1849:, so the Schur multiplier of the simple group has order 2 instead of 1.
1834:, so the Schur multiplier of the simple group has order 2 instead of 1.
1819:, so the Schur multiplier of the simple group has order 3 instead of 1.
1804:, so the Schur multiplier of the simple group has order 2 instead of 1.
1789:, so the Schur multiplier of the simple group has order 4 instead of 1.
1766:, so the Schur multiplier of the simple group has order 6 instead of 2.
1751:, so the Schur multiplier of the simple group has order 2 instead of 1.
1732:, so the Schur multiplier of the simple group has order 2 instead of 1.
1694:, so the Schur multiplier of the simple group has order 2 instead of 1.
1679:, so the Schur multiplier of the simple group has order 6 instead of 2.
1664:, so the Schur multiplier of the simple group has order 2 instead of 1.
1139:
The unitary group arises as follows: the general linear group over the
416:
330:
1515:
realized that after an appropriate reformulation, many theorems about
2749:
2211:)" convention is far more common and is closer to the convention for
1626:(2) Not perfect, but the derived group has index 2 and is the simple
1301:
had an "extra" automorphism in characteristic 2 whose square was the
1020:(a sort of integral form but over finite fields) for all the complex
2471:
2297:
1591:(2) = SL(2, 2) Solvable of order 6 (the symmetric group on 3 points)
2183:). The problem is that there are two fields involved, one of order
2697:
2258:(4) may be any one of 4 different groups, depending on the author.
941:. There are several minor variations of these, given by taking
1234:
1101:
1519:
admit analogues for algebraic groups over an arbitrary field
1357:
one is allowed to ignore the arrow on bonds of multiplicity
1321:
of order 20.) Ree was able to find two new similar families
1157:(which corresponds to taking the transpose inverse), and a
2405:(1901b), "Theory of Linear Groups in An Arbitrary Field",
2364:
Chevalley, Claude (1955), "Sur certains groupes simples",
2173:) (and so on) for the group that other authors denote by A
1964:
of order 2, but the alternating group on 6 points has an
2298:
mathoverflow â Definition of âfinite group of Lie typeâ?
2201:)" convention is more logical and consistent, but the "A
2708:(1960), "A new type of simple groups of finite order",
2465:: 145â173, Reprinted in volume 5 of his collected works
2542:
Traité des substitutions et des équations algébriques
2459:
The
Quarterly Journal of Pure and Applied Mathematics
1619:(2) = Suz(2) Solvable of order 20 (a Frobenius group)
487:
462:
425:
495:
470:
433:
2408:Transactions of the American Mathematical Society
1523:, leading to construction of what are now called
2445:, Reprinted in volume II of his collected papers
2073:) (the symplectic group) or (confusingly) by Sp(
1949:For a complete list of these exceptions see the
1435:Ă PSL(2, 5) Janko found the sporadic group
884:are standard references for groups of Lie type.
1511:in the beginning of 20th century. In the 1950s
1369:is not simple, but it has a simple subgroup of
2238:) may mean either the special linear group SL(
2163:For the Steinberg groups, some authors write A
2044:) (the projective special linear group) or by
1016:concept was isolated. Chevalley constructed a
2601:Bulletin of the American Mathematical Society
2557:Bulletin of the American Mathematical Society
2246:) or the projective special linear group PSL(
1583:Cases where the group is not perfect include
797:
8:
2499:Leonard E. Dickson reported groups of type G
1992:) is not usually the same as the group PSL(
2785:Les "formes réelles" des groupes de type E
2218:Authors differ on whether groups such as A
2152:, though his definition is not simple for
2014:-valued points of the algebraic group PSL(
1548:The belief has now become a theorem â the
925:A classical group is, roughly speaking, a
804:
790:
242:
68:
33:
2765:
2731:
2657:
2613:
2569:
2424:
2377:
2307:
2305:
1507:. Other classical groups were studied by
1345:of simple groups by using the fact that F
1121:
1012:) on Lie algebras, by means of which the
1009:
877:
834:that are closely related to the group of
489:
488:
486:
464:
463:
461:
427:
426:
424:
2290:
1089:
1081:
360:
126:
36:
1925:(8) The Schur multiplier has an extra
1902:(4) The Schur multiplier has an extra
1879:(4) The Schur multiplier has an extra
1856:(9) The Schur multiplier has an extra
1841:(4) The Schur multiplier has an extra
1826:(4) The Schur multiplier has an extra
1811:(3) The Schur multiplier has an extra
1796:(2) The Schur multiplier has an extra
1773:(2) The Schur multiplier has an extra
1758:(3) The Schur multiplier has an extra
1743:(2) The Schur multiplier has an extra
1724:(2) The Schur multiplier has an extra
1701:(4) The Schur multiplier has an extra
1686:(2) The Schur multiplier has an extra
1671:(9) The Schur multiplier has an extra
1656:(4) The Schur multiplier has an extra
1550:classification of finite simple groups
1397:classification of finite simple groups
1282:
1172:automorphisms of their Dynkin diagrams
907:
881:
859:classification of finite simple groups
362:Classification of finite simple groups
2688:, Yale University, New Haven, Conn.,
7:
2640:"Variations on a theme of Chevalley"
2338:, Wiley Classics Library, New York:
1240:, from the order 3 automorphism of D
1226:, from the order 2 automorphism of E
1210:, from the order 2 automorphism of D
1191:, from the order 2 automorphism of A
1133:
2508:La géométrie des groupes classiques
2130:projective special orthogonal group
1966:outer automorphism group of order 4
1451:Relations with finite simple groups
1294:
1290:
992:of Chevalley and Steinberg groups.
2510:(3rd ed.), Berlin, New York:
1080:) had already been constructed by
25:
2426:10.1090/S0002-9947-1901-1500573-3
2084:The notation for groups of type D
1471:over prime finite fields, PSL(2,
1297:) knew that the algebraic group B
1469:projective special linear groups
54:
2615:10.1090/S0002-9904-1961-10527-2
2571:10.1090/S0002-9904-1960-10523-X
2472:"A new system of simple groups"
2366:The Tohoku Mathematical Journal
2187:, and its fixed field of order
2160:â„ 5 but not in lower dimension.
1485:projective special linear group
1379:(named after the mathematician
910:. These groups were studied by
2645:Pacific Journal of Mathematics
723:Infinite dimensional Lie group
1:
2417:American Mathematical Society
2275:of finite groups of Lie type)
2065:are sometimes denoted by Sp(2
2036:are sometimes denoted by PSL(
1026:universal enveloping algebras
2685:Lectures on Chevalley groups
2506:Dieudonné, Jean A. (1971) ,
1951:list of finite simple groups
496:{\displaystyle \mathbb {Z} }
471:{\displaystyle \mathbb {Z} }
434:{\displaystyle \mathbb {Z} }
1970:Schur multiplier of order 6
221:List of group theory topics
2839:
2682:Steinberg, Robert (1968),
2638:Steinberg, Robert (1959),
1267:
1256:diagram also give rise to
1099:
891:
26:
2545:, Paris: Gauthier-Villars
2336:Simple groups of Lie type
18:Finite groups of Lie type
1962:outer automorphism group
1568:Small groups of Lie type
1084:, and the ones of type E
954:projective linear groups
339:Elementary abelian group
216:Glossary of group theory
27:Not to be confused with
2470:Dickson, L. E. (1905),
2451:Dickson, Leonard Eugene
2415:(4), Providence, R.I.:
2403:Dickson, Leonard Eugene
1529:Tits simplicity theorem
1475:) being constructed by
1383:). The smallest group G
1147:given by reversing the
1118:split orthogonal groups
2782:Tits, Jacques (1958),
2659:10.2140/pjm.1959.9.875
2379:10.2748/tmj/1178245104
2269:DeligneâLusztig theory
2144:) for a group that is
1958:field with one element
1562:sporadic simple groups
1545:in the sense of Tits.
1303:Frobenius automorphism
952:, the latter yielding
902:over finite and other
843:linear algebraic group
755:Linear algebraic group
497:
472:
435:
2733:10.1073/pnas.46.6.868
2340:John Wiley & Sons
2273:representation theory
1984:The simple group PSL(
1517:semisimple Lie groups
498:
473:
436:
2059:The groups of type C
2029:The groups of type A
1505:finite simple groups
1483:'s theorem that the
1248:The groups of type D
1145:diagram automorphism
1072:. The ones of type G
1024:(or rather of their
485:
460:
423:
2724:1960PNAS...46..868S
2279:Modular Lie algebra
1165:complex conjugation
1022:simple Lie algebras
129:Group homomorphisms
39:Algebraic structure
2488:10.1007/BF01447497
1160:field automorphism
1076:(sometimes called
1004:, and the work of
830:usually refers to
820:, specifically in
605:Special orthogonal
493:
468:
431:
312:Lagrange's theorem
2521:978-0-387-05391-2
2368:, Second Series,
2349:978-0-471-50683-6
1467:groups, with the
1399:, the Ree groups
1264:SuzukiâRee groups
1202:orthogonal groups
943:derived subgroups
870:compact Lie group
851:group of Lie type
845:with values in a
827:group of Lie type
814:
813:
389:
388:
271:Alternating group
228:
227:
16:(Redirected from
2830:
2823:Algebraic groups
2799:
2778:
2769:
2735:
2701:
2696:, archived from
2678:
2661:
2634:
2617:
2590:
2573:
2546:
2532:
2498:
2466:
2446:
2428:
2398:
2381:
2360:
2332:Carter, Roger W.
2318:
2309:
2300:
2295:
2213:algebraic groups
1578:Schur multiplier
1576:and has trivial
1525:Chevalley groups
1513:Claude Chevalley
1491:) is simple for
1163:given by taking
1122:Steinberg (1959)
1096:Steinberg groups
1002:algebraic groups
996:Chevalley groups
900:classical groups
888:Classical groups
878:Dieudonné (1971)
806:
799:
792:
748:Algebraic groups
521:Hyperbolic group
511:Arithmetic group
502:
500:
499:
494:
492:
477:
475:
474:
469:
467:
440:
438:
437:
432:
430:
353:Schur multiplier
307:Cauchy's theorem
295:Quaternion group
243:
69:
58:
45:
34:
21:
2838:
2837:
2833:
2832:
2831:
2829:
2828:
2827:
2803:
2802:
2788:
2781:
2704:
2681:
2637:
2597:
2593:
2553:
2549:
2537:Jordan, Camille
2535:
2522:
2512:Springer-Verlag
2505:
2502:
2469:
2449:
2401:
2363:
2350:
2330:
2327:
2322:
2321:
2310:
2303:
2296:
2292:
2287:
2265:
2257:
2233:
2223:
2206:
2196:
2178:
2168:
2123:
2089:
2064:
2035:
2013:
2004:
1978:
1976:Notation issues
1924:
1901:
1878:
1855:
1840:
1825:
1810:
1795:
1772:
1757:
1742:
1738:
1723:
1700:
1685:
1670:
1655:
1643:
1636:
1625:
1618:
1611:
1604:
1597:
1590:
1570:
1537:sporadic groups
1509:Leonard Dickson
1477:Ăvariste Galois
1453:
1442:
1406:
1394:
1390:
1386:
1368:
1364:
1352:
1348:
1340:
1328:
1319:Frobenius group
1312:
1300:
1276:
1268:Main articles:
1266:
1255:
1251:
1243:
1238:
1233:the new series
1229:
1224:
1219:the new series
1215:
1209:
1196:
1190:
1156:
1141:complex numbers
1131:
1127:
1114:
1112:
1105:
1100:Main articles:
1098:
1087:
1075:
1071:
1067:
1063:
1059:
1055:
1051:
1045:
1039:
1033:
1018:Chevalley basis
1014:Chevalley group
998:
991:
985:
979:
973:
967:
961:
896:
894:Classical group
890:
836:rational points
810:
781:
780:
769:Abelian variety
762:Reductive group
750:
740:
739:
738:
737:
688:
680:
672:
664:
656:
629:Special unitary
540:
526:
525:
507:
506:
483:
482:
458:
457:
421:
420:
412:
411:
402:Discrete groups
391:
390:
346:Frobenius group
291:
278:
267:
260:Symmetric group
256:
240:
230:
229:
80:Normal subgroup
66:
46:
37:
32:
23:
22:
15:
12:
11:
5:
2836:
2834:
2826:
2825:
2820:
2815:
2805:
2804:
2801:
2800:
2786:
2779:
2718:(6): 868â870,
2706:Suzuki, Michio
2702:
2679:
2652:(3): 875â891,
2635:
2595:
2591:
2564:(6): 508â510,
2551:
2547:
2533:
2520:
2503:
2500:
2467:
2447:
2399:
2372:(1â2): 14â66,
2361:
2348:
2326:
2323:
2320:
2319:
2301:
2289:
2288:
2286:
2283:
2282:
2281:
2276:
2264:
2261:
2260:
2259:
2255:
2229:
2219:
2216:
2202:
2192:
2174:
2164:
2161:
2150:Jean Dieudonné
2119:
2085:
2082:
2060:
2057:
2030:
2027:
2009:
2000:
1977:
1974:
1943:
1942:
1922:
1919:
1899:
1896:
1876:
1873:
1853:
1850:
1838:
1835:
1823:
1820:
1808:
1805:
1793:
1790:
1770:
1767:
1755:
1752:
1740:
1736:
1733:
1721:
1718:
1698:
1695:
1683:
1680:
1668:
1665:
1653:
1646:
1645:
1641:
1638:
1634:
1631:
1623:
1620:
1616:
1613:
1609:
1606:
1602:
1599:
1595:
1592:
1588:
1569:
1566:
1533:Mathieu groups
1481:Camille Jordan
1452:
1449:
1440:
1409:
1408:
1404:
1392:
1388:
1384:
1373:2, called the
1366:
1362:
1350:
1346:
1343:
1342:
1338:
1331:
1330:
1326:
1315:
1314:
1310:
1298:
1265:
1262:
1253:
1249:
1246:
1245:
1241:
1236:
1231:
1227:
1222:
1217:
1211:
1205:
1198:
1192:
1186:
1183:unitary groups
1152:
1149:Dynkin diagram
1129:
1125:
1110:
1103:
1097:
1094:
1090:Dickson (1901)
1085:
1082:Dickson (1905)
1078:Dickson groups
1073:
1069:
1065:
1061:
1057:
1053:
1047:
1041:
1035:
1029:
997:
994:
987:
981:
975:
969:
963:
957:
927:special linear
916:Jean Dieudonné
892:Main article:
889:
886:
812:
811:
809:
808:
801:
794:
786:
783:
782:
779:
778:
776:Elliptic curve
772:
771:
765:
764:
758:
757:
751:
746:
745:
742:
741:
736:
735:
732:
729:
725:
721:
720:
719:
714:
712:Diffeomorphism
708:
707:
702:
697:
691:
690:
686:
682:
678:
674:
670:
666:
662:
658:
654:
649:
648:
637:
636:
625:
624:
613:
612:
601:
600:
589:
588:
577:
576:
569:Special linear
565:
564:
557:General linear
553:
552:
547:
541:
532:
531:
528:
527:
524:
523:
518:
513:
505:
504:
491:
479:
466:
453:
451:Modular groups
449:
448:
447:
442:
429:
413:
410:
409:
404:
398:
397:
396:
393:
392:
387:
386:
385:
384:
379:
374:
371:
365:
364:
358:
357:
356:
355:
349:
348:
342:
341:
336:
327:
326:
324:Hall's theorem
321:
319:Sylow theorems
315:
314:
309:
301:
300:
299:
298:
292:
287:
284:Dihedral group
280:
279:
274:
268:
263:
257:
252:
241:
236:
235:
232:
231:
226:
225:
224:
223:
218:
210:
209:
208:
207:
202:
197:
192:
187:
182:
177:
175:multiplicative
172:
167:
162:
157:
149:
148:
147:
146:
141:
133:
132:
124:
123:
122:
121:
119:Wreath product
116:
111:
106:
104:direct product
98:
96:Quotient group
90:
89:
88:
87:
82:
77:
67:
64:
63:
60:
59:
51:
50:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2835:
2824:
2821:
2819:
2816:
2814:
2811:
2810:
2808:
2798:
2794:
2790:
2789:
2780:
2777:
2773:
2768:
2763:
2759:
2755:
2751:
2747:
2743:
2739:
2734:
2729:
2725:
2721:
2717:
2713:
2712:
2707:
2703:
2700:on 2012-09-10
2699:
2695:
2691:
2687:
2686:
2680:
2677:
2673:
2669:
2665:
2660:
2655:
2651:
2647:
2646:
2641:
2636:
2633:
2629:
2625:
2621:
2616:
2611:
2607:
2603:
2602:
2592:
2589:
2585:
2581:
2577:
2572:
2567:
2563:
2559:
2558:
2548:
2544:
2543:
2538:
2534:
2531:
2527:
2523:
2517:
2513:
2509:
2504:
2497:
2493:
2489:
2485:
2481:
2477:
2473:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2440:
2436:
2432:
2427:
2422:
2418:
2414:
2410:
2409:
2404:
2400:
2397:
2393:
2389:
2385:
2380:
2375:
2371:
2367:
2362:
2359:
2355:
2351:
2345:
2341:
2337:
2333:
2329:
2328:
2324:
2317:
2313:
2308:
2306:
2302:
2299:
2294:
2291:
2284:
2280:
2277:
2274:
2270:
2267:
2266:
2262:
2253:
2249:
2245:
2241:
2237:
2232:
2227:
2222:
2217:
2214:
2210:
2205:
2200:
2195:
2190:
2186:
2182:
2177:
2172:
2167:
2162:
2159:
2155:
2151:
2147:
2143:
2139:
2135:
2131:
2127:
2122:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2088:
2083:
2080:
2076:
2072:
2068:
2063:
2058:
2055:
2051:
2047:
2043:
2039:
2033:
2028:
2025:
2021:
2017:
2012:
2008:
2003:
1999:
1995:
1991:
1987:
1983:
1982:
1981:
1975:
1973:
1971:
1967:
1963:
1959:
1954:
1952:
1947:
1940:
1936:
1932:
1928:
1920:
1917:
1913:
1909:
1905:
1897:
1894:
1890:
1886:
1882:
1874:
1871:
1867:
1863:
1859:
1851:
1848:
1844:
1836:
1833:
1829:
1821:
1818:
1814:
1806:
1803:
1799:
1791:
1788:
1784:
1780:
1776:
1768:
1765:
1761:
1753:
1750:
1746:
1734:
1731:
1727:
1719:
1716:
1712:
1708:
1704:
1696:
1693:
1689:
1681:
1678:
1674:
1666:
1663:
1659:
1651:
1650:
1649:
1639:
1632:
1629:
1621:
1614:
1607:
1600:
1593:
1586:
1585:
1584:
1581:
1579:
1575:
1567:
1565:
1563:
1560:, and the 26
1559:
1555:
1551:
1546:
1544:
1543:
1538:
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1450:
1448:
1445:
1443:
1439:
1434:
1430:
1426:
1422:
1418:
1414:
1402:
1401:
1400:
1398:
1387:(3) of type G
1382:
1378:
1377:
1372:
1365:(2) of type F
1360:
1356:
1336:
1335:
1334:
1324:
1323:
1322:
1320:
1313:(2) = Suz(2).
1308:
1307:
1306:
1304:
1296:
1292:
1288:
1284:
1280:
1275:
1271:
1270:Suzuki groups
1263:
1261:
1259:
1239:
1232:
1225:
1218:
1214:
1208:
1203:
1199:
1195:
1189:
1184:
1180:
1179:
1178:
1175:
1173:
1168:
1166:
1162:
1161:
1155:
1150:
1146:
1142:
1137:
1135:
1123:
1119:
1113:
1106:
1095:
1093:
1091:
1083:
1079:
1050:
1044:
1038:
1032:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
995:
993:
990:
984:
978:
972:
966:
960:
955:
951:
948:
944:
940:
939:unitary group
936:
932:
928:
923:
921:
917:
913:
912:L. E. Dickson
909:
908:Jordan (1870)
905:
901:
895:
887:
885:
883:
882:Carter (1989)
879:
875:
871:
867:
862:
860:
856:
852:
849:. The phrase
848:
844:
841:
837:
833:
832:finite groups
829:
828:
824:, the phrase
823:
819:
807:
802:
800:
795:
793:
788:
787:
785:
784:
777:
774:
773:
770:
767:
766:
763:
760:
759:
756:
753:
752:
749:
744:
743:
733:
730:
727:
726:
724:
718:
715:
713:
710:
709:
706:
703:
701:
698:
696:
693:
692:
689:
683:
681:
675:
673:
667:
665:
659:
657:
651:
650:
646:
642:
639:
638:
634:
630:
627:
626:
622:
618:
615:
614:
610:
606:
603:
602:
598:
594:
591:
590:
586:
582:
579:
578:
574:
570:
567:
566:
562:
558:
555:
554:
551:
548:
546:
543:
542:
539:
535:
530:
529:
522:
519:
517:
514:
512:
509:
508:
480:
455:
454:
452:
446:
443:
418:
415:
414:
408:
405:
403:
400:
399:
395:
394:
383:
380:
378:
375:
372:
369:
368:
367:
366:
363:
359:
354:
351:
350:
347:
344:
343:
340:
337:
335:
333:
329:
328:
325:
322:
320:
317:
316:
313:
310:
308:
305:
304:
303:
302:
296:
293:
290:
285:
282:
281:
277:
272:
269:
266:
261:
258:
255:
250:
247:
246:
245:
244:
239:
238:Finite groups
234:
233:
222:
219:
217:
214:
213:
212:
211:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
152:
151:
150:
145:
142:
140:
137:
136:
135:
134:
131:
130:
125:
120:
117:
115:
112:
110:
107:
105:
102:
99:
97:
94:
93:
92:
91:
86:
83:
81:
78:
76:
73:
72:
71:
70:
65:Basic notions
62:
61:
57:
53:
52:
49:
44:
40:
35:
30:
19:
2818:Lie algebras
2813:Group theory
2784:
2715:
2709:
2698:the original
2684:
2649:
2643:
2605:
2599:
2561:
2555:
2541:
2507:
2479:
2475:
2462:
2458:
2412:
2406:
2369:
2365:
2335:
2293:
2251:
2247:
2243:
2239:
2235:
2230:
2225:
2220:
2208:
2203:
2198:
2193:
2188:
2184:
2180:
2175:
2170:
2165:
2157:
2153:
2145:
2141:
2137:
2125:
2120:
2115:
2111:
2107:
2103:
2099:
2095:
2091:
2086:
2078:
2074:
2070:
2066:
2061:
2053:
2049:
2045:
2041:
2037:
2031:
2023:
2019:
2015:
2010:
2006:
2001:
1997:
1993:
1989:
1985:
1979:
1955:
1948:
1944:
1938:
1934:
1930:
1926:
1915:
1911:
1907:
1903:
1892:
1888:
1884:
1880:
1869:
1865:
1861:
1857:
1846:
1842:
1831:
1827:
1816:
1812:
1801:
1797:
1786:
1782:
1778:
1774:
1763:
1759:
1748:
1744:
1729:
1725:
1714:
1710:
1706:
1702:
1691:
1687:
1676:
1672:
1661:
1657:
1647:
1605:(4) Solvable
1582:
1571:
1554:finite field
1547:
1541:
1528:
1524:
1520:
1500:
1496:
1492:
1488:
1472:
1454:
1446:
1437:
1432:
1428:
1424:
1420:
1416:
1412:
1410:
1395:(8). In the
1381:Jacques Tits
1374:
1358:
1354:
1344:
1332:
1316:
1277:
1247:
1212:
1206:
1201:
1193:
1187:
1182:
1177:These gave:
1176:
1169:
1158:
1153:
1144:
1138:
1115:
1077:
1048:
1042:
1036:
1030:
1013:
999:
988:
982:
976:
970:
964:
958:
924:
899:
897:
874:real numbers
863:
850:
847:finite field
826:
825:
822:group theory
815:
644:
632:
620:
608:
596:
584:
572:
560:
376:
331:
288:
275:
264:
253:
249:Cyclic group
127:
114:Free product
85:Group action
48:Group theory
43:Group theory
42:
2608:: 115â116,
2482:: 137â150,
2419:: 363â394,
1465:alternating
1134:Tits (1958)
818:mathematics
534:Topological
373:alternating
2807:Categories
2476:Math. Ann.
2325:References
1628:Tits group
1558:Tits group
1376:Tits group
935:symplectic
931:orthogonal
920:Emil Artin
868:, since a
866:Lie groups
641:Symplectic
581:Orthogonal
538:Lie groups
445:Free group
170:continuous
109:Direct sum
2742:0027-8424
2668:0030-8730
2624:0002-9904
2580:0002-9904
2496:179178145
2435:0002-9947
2388:0040-8735
2334:(1989) ,
1461:symmetric
1419:Ă PSL(2,
1274:Ree group
1006:Chevalley
950:quotients
840:reductive
705:Conformal
593:Euclidean
200:nilpotent
29:Lie group
2776:16590684
2539:(1870),
2453:(1901),
2263:See also
2136:, use O(
2022:) â PSL(
1542:geometry
1258:triality
1200:further
700:Poincaré
545:Solenoid
417:Integers
407:Lattices
382:sporadic
377:Lie type
205:solvable
195:dihedral
180:additive
165:infinite
75:Subgroup
2797:0106247
2758:0120283
2720:Bibcode
2694:0466335
2676:0109191
2632:0125155
2588:0125155
2530:0310083
2443:1986251
2396:0073602
2358:0407163
2254:). So A
2110:), PSO(
1739:(2) = C
1574:perfect
1487:PSL(2,
1289: (
1281: (
1068:, and G
1008: (
947:central
695:Lorentz
617:Unitary
516:Lattice
456:PSL(2,
190:abelian
101:(Semi-)
2795:
2774:
2767:222949
2764:
2756:
2748:
2740:
2692:
2674:
2666:
2630:
2622:
2586:
2578:
2528:
2518:
2494:
2441:
2433:
2394:
2386:
2356:
2346:
1457:cyclic
1423:) for
1279:Suzuki
1143:has a
904:fields
855:simple
550:Circle
481:SL(2,
370:cyclic
334:-group
185:cyclic
160:finite
155:simple
139:kernel
2750:70960
2746:JSTOR
2492:S2CID
2439:JSTOR
2316:p. xi
2312:ATLAS
2285:Notes
2134:ATLAS
2005:) of
1503:) of
1371:index
1349:and G
937:, or
838:of a
734:Sp(â)
731:SU(â)
144:image
2772:PMID
2738:ISSN
2664:ISSN
2620:ISSN
2598:)",
2576:ISSN
2554:)",
2516:ISBN
2431:ISSN
2384:ISSN
2344:ISBN
2250:+1,
2242:+1,
2118:), Ω
1463:and
1333:and
1295:1961
1291:1960
1283:1960
1272:and
1181:the
1107:and
1010:1955
914:and
880:and
728:O(â)
717:Loop
536:and
2762:PMC
2728:doi
2654:doi
2610:doi
2566:doi
2484:doi
2421:doi
2374:doi
2146:not
2098:),
1407:(3)
1341:(3)
1329:(2)
1287:Ree
1128:, E
1088:by
1064:, F
1060:, E
1056:, E
1046:, D
1040:, C
1034:, B
986:, D
974:, D
968:, C
962:, B
945:or
906:by
816:In
643:Sp(
631:SU(
607:SO(
571:SL(
559:GL(
2809::
2793:MR
2770:,
2760:,
2754:MR
2752:,
2744:,
2736:,
2726:,
2716:46
2714:,
2690:MR
2672:MR
2670:,
2662:,
2648:,
2642:,
2628:MR
2626:,
2618:,
2606:67
2604:,
2584:MR
2582:,
2574:,
2562:66
2560:,
2526:MR
2524:,
2514:,
2490:,
2480:60
2478:,
2474:,
2463:33
2461:,
2457:,
2437:,
2429:,
2411:,
2392:MR
2390:,
2382:,
2354:MR
2352:,
2342:,
2314:,
2304:^
2140:,
2114:,
2106:,
2094:,
2081:).
2077:,
2069:,
2056:).
2052:,
2040:,
2034:â1
1996:,
1988:,
1972:.
1937:/2
1933:Ă
1929:/2
1914:/2
1910:Ă
1906:/2
1891:/2
1887:Ă
1883:/2
1868:/3
1864:Ă
1860:/3
1845:/2
1830:/2
1815:/3
1800:/2
1785:/2
1781:Ă
1777:/2
1762:/3
1747:/2
1728:/2
1713:/4
1709:Ă
1705:/4
1690:/2
1675:/3
1660:/2
1564:.
1499:,
1459:,
1444:.
1431:/2
1415:/2
1293:,
1260:.
1120:.
1092:.
980:,A
933:,
929:,
918:.
876:.
861:.
619:U(
595:E(
583:O(
41:â
2787:6
2730::
2722::
2656::
2650:9
2612::
2596:4
2568::
2552:2
2501:2
2486::
2423::
2413:2
2376::
2370:7
2271:(
2256:2
2252:q
2248:n
2244:q
2240:n
2236:q
2234:(
2231:n
2226:q
2224:(
2221:n
2215:.
2209:q
2207:(
2204:n
2199:q
2197:(
2194:n
2189:q
2185:q
2181:q
2179:(
2176:n
2171:q
2169:(
2166:n
2158:n
2154:n
2142:q
2138:n
2126:q
2124:(
2121:n
2116:q
2112:n
2108:q
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31:.
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