45:
1294:. 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
2117:), 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
1934:
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.
2015:) 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.
1163:, 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.
1524:), 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
1400:
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
1017:), 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
1954:
490:
465:
428:
2396:
945:. 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
2780:, Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e Úd. corrigée, Exposé 162, vol. 15, Paris: Secrétariat math'ematique,
2699:
1561:
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
792:
2589:
2545:
1969:
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.
2121:
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
2304:
2261:
1538:
1385:
847:
350:
2508:
2336:
300:
2118:
1569:. However some of the smallest groups in the families above are either not perfect or have a Schur multiplier larger than "expected".
785:
295:
1601:(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.
1516:. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (
1528:, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their
1436:
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).
1473:
1457:
2633:
711:
2694:
2405:
1267:
778:
1041:
this gave well known classical groups, but his construction also gave groups associated to the exceptional Lie algebras E
2320:
1939:
1014:
1484:â 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(
395:
209:
2257:
1241:
have no analogue over the reals, as the complex numbers have no automorphism of order 3. The symmetries of the D
1105:
Chevalley's construction did not give all of the known classical groups: it omitted the unitary groups and the non-
2811:
1949:. Some of the small alternating groups also have exceptional properties. The alternating groups usually have an
1950:
989:
Chevalley groups can be thought of as Lie groups over finite fields. The theory was clarified by the theory of
593:
327:
204:
92:
2444:"A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface"
2672:
2217:) are the groups of points with values in the simple or the simply connected algebraic group. For example, A
1274:) found a new infinite series of groups that at first sight seemed unrelated to the known algebraic groups.
942:
1156:, which commute. The unitary group is the group of fixed points of the product of these two automorphisms.
1125:. This construction generalizes the usual construction of the unitary group from the general linear group.
2439:
2391:
2300:
2122:
1946:
1529:
1291:
1106:
831:
743:
533:
2806:
2801:
1550:
617:
1637:
Some cases where the group is perfect but has a Schur multiplier that is larger than expected include:
1957:. Alternating groups usually have a Schur multiplier of order 2, but the ones on 6 or 7 points have a
2708:
2328:
1505:
1493:
1306:(Strictly speaking, the group Suz(2) is not counted as a Suzuki group as it is not simple: it is the
915:
557:
545:
163:
97:
2267:
1359:
1153:
892:
132:
27:
473:
448:
411:
2734:
2480:
2427:
1148:
1010:
117:
89:
2583:
Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F
2539:
Ree, Rimhak (1960), "A family of simple groups associated with the simple Lie algebra of type (G
1520:). Although it was known since 19th century that other finite simple groups exist (for example,
1465:
2137:
the orthogonal group, but the corresponding simple group. The notation Ω, PΩ was introduced by
1545:
include all the finite simple groups other than the cyclic groups, the alternating groups, the
2760:
2726:
2652:
2608:
2564:
2504:
2443:
2419:
2372:
2332:
2138:
1958:
1453:
1342:
have extra automorphisms in characteristic 2 and 3. (Roughly speaking, in characteristic
1209:
1121:, the second of which was discovered at about the same time from a different point of view by
1113:
found a modification of
Chevalley's construction that gave these groups and two new families D
1097:
935:
904:
858:
846:
groups of Lie type does have a precise definition, and they make up most of the groups in the
688:
522:
259:
1444:
Finite groups of Lie type were among the first groups to be considered in mathematics, after
861:
may be viewed as the rational points of a reductive linear algebraic group over the field of
2750:
2716:
2642:
2598:
2554:
2472:
2409:
2362:
2145:†4 and thus the same notation may be used for a slightly different group, which agrees in
1566:
1501:
994:
931:
923:
919:
887:
An initial approach to this question was the definition and detailed study of the so-called
673:
665:
657:
649:
641:
629:
569:
509:
499:
341:
283:
158:
127:
2785:
2746:
2682:
2664:
2620:
2576:
2518:
2384:
2346:
842:
does not have a widely accepted precise definition, but the important collection of finite
2781:
2742:
2678:
2660:
2616:
2572:
2514:
2500:
2380:
2342:
2201:
1525:
1497:
1449:
1307:
1006:
990:
882:
828:
757:
750:
736:
693:
581:
504:
334:
248:
188:
68:
911:
investigated the orders of such groups, with a view to classifying cases of coincidence.
2712:
1633:(3) Not perfect, but the derived group has index 3 and is the simple group of order 504.
2772:
2525:
1521:
1469:
1160:
1137:
1129:
938:
824:
764:
700:
390:
370:
307:
272:
193:
183:
168:
153:
107:
84:
2755:
2414:
1541:. Inspection of the list of finite simple groups shows that groups of Lie type over a
2795:
2484:
2180:, and people have different ideas on which should be included in the notation. The "A
1562:
1258:
927:
900:
820:
683:
605:
439:
312:
178:
2603:
2559:
2529:
1416:= 3, and by investigating groups with an involution centralizer of the similar form
1542:
1468:
in the 1830s. The systematic exploration of finite groups of Lie type started with
1445:
1369:
862:
853:
The name "groups of Lie type" is due to the close relationship with the (infinite)
843:
835:
810:
538:
237:
226:
173:
148:
143:
102:
73:
36:
1942:. Many of these special properties are related to certain sporadic simple groups.
2628:
1945:
Alternating groups sometimes behave as if they were groups of Lie type over the
1425:
806:
2700:
Proceedings of the
National Academy of Sciences of the United States of America
1626:(2) Not perfect, but the derived group has index 2 and is simple of order 6048.
1616:
1546:
1380:
is not simple, but it has a simple normal subgroup of index 3, isomorphic to A
1364:
1350:
in the Dynkin diagram when taking diagram automorphisms.) The smallest group F
1275:
908:
854:
705:
433:
2730:
2656:
2612:
2568:
2423:
2376:
2367:
1159:
In the same way, many
Chevalley groups have diagram automorphisms induced by
1262:
526:
17:
2764:
2647:
2721:
44:
2079:("orthogonal" groups) is particularly confusing. Some symbols used are O(
1246:
63:
1907:, so the Schur multiplier of the simple group has order 12 instead of 3.
1884:, so the Schur multiplier of the simple group has order 12 instead of 3.
1861:, so the Schur multiplier of the simple group has order 36 instead of 4.
1706:, so the Schur multiplier of the simple group has order 48 instead of 3.
1587:(3) = PSL(2, 3) Solvable of order 12 (the alternating group on 4 points)
2476:
2431:
2007:). The problem is that a surjective map of algebraic groups such as SL(
1930:, so the Schur multiplier of the simple group has order 4 instead of 1.
1838:, so the Schur multiplier of the simple group has order 2 instead of 1.
1823:, so the Schur multiplier of the simple group has order 2 instead of 1.
1808:, so the Schur multiplier of the simple group has order 3 instead of 1.
1793:, so the Schur multiplier of the simple group has order 2 instead of 1.
1778:, so the Schur multiplier of the simple group has order 4 instead of 1.
1755:, so the Schur multiplier of the simple group has order 6 instead of 2.
1740:, so the Schur multiplier of the simple group has order 2 instead of 1.
1721:, so the Schur multiplier of the simple group has order 2 instead of 1.
1683:, so the Schur multiplier of the simple group has order 2 instead of 1.
1668:, so the Schur multiplier of the simple group has order 6 instead of 2.
1653:, so the Schur multiplier of the simple group has order 2 instead of 1.
1128:
The unitary group arises as follows: the general linear group over the
405:
319:
1504:
realized that after an appropriate reformulation, many theorems about
2738:
2200:)" convention is far more common and is closer to the convention for
1615:(2) Not perfect, but the derived group has index 2 and is the simple
1290:
had an "extra" automorphism in characteristic 2 whose square was the
1009:(a sort of integral form but over finite fields) for all the complex
2460:
2286:
1580:(2) = SL(2, 2) Solvable of order 6 (the symmetric group on 3 points)
2172:). The problem is that there are two fields involved, one of order
2686:
2247:(4) may be any one of 4 different groups, depending on the author.
930:. There are several minor variations of these, given by taking
1223:
1090:
1508:
admit analogues for algebraic groups over an arbitrary field
1346:
one is allowed to ignore the arrow on bonds of multiplicity
1310:
of order 20.) Ree was able to find two new similar families
1146:(which corresponds to taking the transpose inverse), and a
2394:(1901b), "Theory of Linear Groups in An Arbitrary Field",
2353:
Chevalley, Claude (1955), "Sur certains groupes simples",
2162:) (and so on) for the group that other authors denote by A
1953:
of order 2, but the alternating group on 6 points has an
2287:
mathoverflow â Definition of âfinite group of Lie typeâ?
2190:)" convention is more logical and consistent, but the "A
2697:(1960), "A new type of simple groups of finite order",
2454:: 145â173, Reprinted in volume 5 of his collected works
2531:
Traité des substitutions et des équations algébriques
2448:
The
Quarterly Journal of Pure and Applied Mathematics
1608:(2) = Suz(2) Solvable of order 20 (a Frobenius group)
476:
451:
414:
484:
459:
422:
2397:Transactions of the American Mathematical Society
1512:, leading to construction of what are now called
2434:, Reprinted in volume II of his collected papers
2062:) (the symplectic group) or (confusingly) by Sp(
1938:For a complete list of these exceptions see the
1424:Ă PSL(2, 5) Janko found the sporadic group
873:are standard references for groups of Lie type.
1500:in the beginning of 20th century. In the 1950s
1358:is not simple, but it has a simple subgroup of
2227:) may mean either the special linear group SL(
2152:For the Steinberg groups, some authors write A
2033:) (the projective special linear group) or by
1005:concept was isolated. Chevalley constructed a
2590:Bulletin of the American Mathematical Society
2546:Bulletin of the American Mathematical Society
2235:) or the projective special linear group PSL(
1572:Cases where the group is not perfect include
786:
8:
2488:Leonard E. Dickson reported groups of type G
1981:) is not usually the same as the group PSL(
2774:Les "formes réelles" des groupes de type E
2207:Authors differ on whether groups such as A
2141:, though his definition is not simple for
2003:-valued points of the algebraic group PSL(
1537:The belief has now become a theorem â the
914:A classical group is, roughly speaking, a
793:
779:
231:
57:
22:
2754:
2720:
2646:
2602:
2558:
2413:
2366:
2296:
2294:
1496:. Other classical groups were studied by
1334:of simple groups by using the fact that F
1110:
1001:) on Lie algebras, by means of which the
998:
866:
823:that are closely related to the group of
478:
477:
475:
453:
452:
450:
416:
415:
413:
2279:
1078:
1070:
349:
115:
25:
1914:(8) The Schur multiplier has an extra
1891:(4) The Schur multiplier has an extra
1868:(4) The Schur multiplier has an extra
1845:(9) The Schur multiplier has an extra
1830:(4) The Schur multiplier has an extra
1815:(4) The Schur multiplier has an extra
1800:(3) The Schur multiplier has an extra
1785:(2) The Schur multiplier has an extra
1762:(2) The Schur multiplier has an extra
1747:(3) The Schur multiplier has an extra
1732:(2) The Schur multiplier has an extra
1713:(2) The Schur multiplier has an extra
1690:(4) The Schur multiplier has an extra
1675:(2) The Schur multiplier has an extra
1660:(9) The Schur multiplier has an extra
1645:(4) The Schur multiplier has an extra
1539:classification of finite simple groups
1386:classification of finite simple groups
1271:
1161:automorphisms of their Dynkin diagrams
896:
870:
848:classification of finite simple groups
351:Classification of finite simple groups
2677:, Yale University, New Haven, Conn.,
7:
2629:"Variations on a theme of Chevalley"
2327:, Wiley Classics Library, New York:
1229:, from the order 3 automorphism of D
1215:, from the order 2 automorphism of E
1199:, from the order 2 automorphism of D
1180:, from the order 2 automorphism of A
1122:
2497:La géométrie des groupes classiques
2119:projective special orthogonal group
1955:outer automorphism group of order 4
1440:Relations with finite simple groups
1283:
1279:
981:of Chevalley and Steinberg groups.
2499:(3rd ed.), Berlin, New York:
1069:) had already been constructed by
14:
2415:10.1090/S0002-9947-1901-1500573-3
2073:The notation for groups of type D
1460:over prime finite fields, PSL(2,
1286:) knew that the algebraic group B
1458:projective special linear groups
43:
2604:10.1090/S0002-9904-1961-10527-2
2560:10.1090/S0002-9904-1960-10523-X
2461:"A new system of simple groups"
2355:The Tohoku Mathematical Journal
2176:, and its fixed field of order
2149:â„ 5 but not in lower dimension.
1474:projective special linear group
1368:(named after the mathematician
899:. These groups were studied by
2634:Pacific Journal of Mathematics
712:Infinite dimensional Lie group
1:
2406:American Mathematical Society
2264:of finite groups of Lie type)
2054:are sometimes denoted by Sp(2
2025:are sometimes denoted by PSL(
1015:universal enveloping algebras
2674:Lectures on Chevalley groups
2495:Dieudonné, Jean A. (1971) ,
1940:list of finite simple groups
485:{\displaystyle \mathbb {Z} }
460:{\displaystyle \mathbb {Z} }
423:{\displaystyle \mathbb {Z} }
1959:Schur multiplier of order 6
210:List of group theory topics
2828:
2671:Steinberg, Robert (1968),
2627:Steinberg, Robert (1959),
1256:
1245:diagram also give rise to
1088:
880:
15:
2534:, Paris: Gauthier-Villars
2325:Simple groups of Lie type
1951:outer automorphism group
1557:Small groups of Lie type
1073:, and the ones of type E
943:projective linear groups
328:Elementary abelian group
205:Glossary of group theory
16:Not to be confused with
2459:Dickson, L. E. (1905),
2440:Dickson, Leonard Eugene
2404:(4), Providence, R.I.:
2392:Dickson, Leonard Eugene
1518:Tits simplicity theorem
1464:) being constructed by
1372:). The smallest group G
1136:given by reversing the
1107:split orthogonal groups
2771:Tits, Jacques (1958),
2648:10.2140/pjm.1959.9.875
2368:10.2748/tmj/1178245104
2258:DeligneâLusztig theory
2133:) for a group that is
1947:field with one element
1551:sporadic simple groups
1534:in the sense of Tits.
1292:Frobenius automorphism
941:, the latter yielding
891:over finite and other
832:linear algebraic group
744:Linear algebraic group
486:
461:
424:
2722:10.1073/pnas.46.6.868
2329:John Wiley & Sons
2262:representation theory
1973:The simple group PSL(
1506:semisimple Lie groups
487:
462:
425:
2048:The groups of type C
2018:The groups of type A
1494:finite simple groups
1472:'s theorem that the
1237:The groups of type D
1134:diagram automorphism
1061:. The ones of type G
1013:(or rather of their
474:
449:
412:
2713:1960PNAS...46..868S
2268:Modular Lie algebra
1154:complex conjugation
1011:simple Lie algebras
118:Group homomorphisms
28:Algebraic structure
2477:10.1007/BF01447497
1149:field automorphism
1065:(sometimes called
993:, and the work of
819:usually refers to
809:, specifically in
594:Special orthogonal
482:
457:
420:
301:Lagrange's theorem
2510:978-0-387-05391-2
2357:, Second Series,
2338:978-0-471-50683-6
1456:groups, with the
1388:, the Ree groups
1253:SuzukiâRee groups
1191:orthogonal groups
932:derived subgroups
859:compact Lie group
840:group of Lie type
834:with values in a
816:group of Lie type
803:
802:
378:
377:
260:Alternating group
217:
216:
2819:
2812:Algebraic groups
2788:
2767:
2758:
2724:
2690:
2685:, archived from
2667:
2650:
2623:
2606:
2579:
2562:
2535:
2521:
2487:
2455:
2435:
2417:
2387:
2370:
2349:
2321:Carter, Roger W.
2307:
2298:
2289:
2284:
2202:algebraic groups
1567:Schur multiplier
1565:and has trivial
1514:Chevalley groups
1502:Claude Chevalley
1480:) is simple for
1152:given by taking
1111:Steinberg (1959)
1085:Steinberg groups
991:algebraic groups
985:Chevalley groups
889:classical groups
877:Classical groups
867:Dieudonné (1971)
795:
788:
781:
737:Algebraic groups
510:Hyperbolic group
500:Arithmetic group
491:
489:
488:
483:
481:
466:
464:
463:
458:
456:
429:
427:
426:
421:
419:
342:Schur multiplier
296:Cauchy's theorem
284:Quaternion group
232:
58:
47:
34:
23:
2827:
2826:
2822:
2821:
2820:
2818:
2817:
2816:
2792:
2791:
2777:
2770:
2693:
2670:
2626:
2586:
2582:
2542:
2538:
2526:Jordan, Camille
2524:
2511:
2501:Springer-Verlag
2494:
2491:
2458:
2438:
2390:
2352:
2339:
2319:
2316:
2311:
2310:
2299:
2292:
2285:
2281:
2276:
2254:
2246:
2222:
2212:
2195:
2185:
2167:
2157:
2112:
2078:
2053:
2024:
2002:
1993:
1967:
1965:Notation issues
1913:
1890:
1867:
1844:
1829:
1814:
1799:
1784:
1761:
1746:
1731:
1727:
1712:
1689:
1674:
1659:
1644:
1632:
1625:
1614:
1607:
1600:
1593:
1586:
1579:
1559:
1526:sporadic groups
1498:Leonard Dickson
1466:Ăvariste Galois
1442:
1431:
1395:
1383:
1379:
1375:
1357:
1353:
1341:
1337:
1329:
1317:
1308:Frobenius group
1301:
1289:
1265:
1257:Main articles:
1255:
1244:
1240:
1232:
1227:
1222:the new series
1218:
1213:
1208:the new series
1204:
1198:
1185:
1179:
1145:
1130:complex numbers
1120:
1116:
1103:
1101:
1094:
1089:Main articles:
1087:
1076:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1034:
1028:
1022:
1007:Chevalley basis
1003:Chevalley group
987:
980:
974:
968:
962:
956:
950:
885:
883:Classical group
879:
825:rational points
799:
770:
769:
758:Abelian variety
751:Reductive group
739:
729:
728:
727:
726:
677:
669:
661:
653:
645:
618:Special unitary
529:
515:
514:
496:
495:
472:
471:
447:
446:
410:
409:
401:
400:
391:Discrete groups
380:
379:
335:Frobenius group
280:
267:
256:
249:Symmetric group
245:
229:
219:
218:
69:Normal subgroup
55:
35:
26:
21:
12:
11:
5:
2825:
2823:
2815:
2814:
2809:
2804:
2794:
2793:
2790:
2789:
2775:
2768:
2707:(6): 868â870,
2695:Suzuki, Michio
2691:
2668:
2641:(3): 875â891,
2624:
2584:
2580:
2553:(6): 508â510,
2540:
2536:
2522:
2509:
2492:
2489:
2456:
2436:
2388:
2361:(1â2): 14â66,
2350:
2337:
2315:
2312:
2309:
2308:
2290:
2278:
2277:
2275:
2272:
2271:
2270:
2265:
2253:
2250:
2249:
2248:
2244:
2218:
2208:
2205:
2191:
2181:
2163:
2153:
2150:
2139:Jean Dieudonné
2108:
2074:
2071:
2049:
2046:
2019:
2016:
1998:
1989:
1966:
1963:
1932:
1931:
1911:
1908:
1888:
1885:
1865:
1862:
1842:
1839:
1827:
1824:
1812:
1809:
1797:
1794:
1782:
1779:
1759:
1756:
1744:
1741:
1729:
1725:
1722:
1710:
1707:
1687:
1684:
1672:
1669:
1657:
1654:
1642:
1635:
1634:
1630:
1627:
1623:
1620:
1612:
1609:
1605:
1602:
1598:
1595:
1591:
1588:
1584:
1581:
1577:
1558:
1555:
1522:Mathieu groups
1470:Camille Jordan
1441:
1438:
1429:
1398:
1397:
1393:
1381:
1377:
1373:
1362:2, called the
1355:
1351:
1339:
1335:
1332:
1331:
1327:
1320:
1319:
1315:
1304:
1303:
1299:
1287:
1254:
1251:
1242:
1238:
1235:
1234:
1230:
1225:
1220:
1216:
1211:
1206:
1200:
1194:
1187:
1181:
1175:
1172:unitary groups
1141:
1138:Dynkin diagram
1118:
1114:
1099:
1092:
1086:
1083:
1079:Dickson (1901)
1074:
1071:Dickson (1905)
1067:Dickson groups
1062:
1058:
1054:
1050:
1046:
1042:
1036:
1030:
1024:
1018:
986:
983:
976:
970:
964:
958:
952:
946:
916:special linear
905:Jean Dieudonné
881:Main article:
878:
875:
801:
800:
798:
797:
790:
783:
775:
772:
771:
768:
767:
765:Elliptic curve
761:
760:
754:
753:
747:
746:
740:
735:
734:
731:
730:
725:
724:
721:
718:
714:
710:
709:
708:
703:
701:Diffeomorphism
697:
696:
691:
686:
680:
679:
675:
671:
667:
663:
659:
655:
651:
647:
643:
638:
637:
626:
625:
614:
613:
602:
601:
590:
589:
578:
577:
566:
565:
558:Special linear
554:
553:
546:General linear
542:
541:
536:
530:
521:
520:
517:
516:
513:
512:
507:
502:
494:
493:
480:
468:
455:
442:
440:Modular groups
438:
437:
436:
431:
418:
402:
399:
398:
393:
387:
386:
385:
382:
381:
376:
375:
374:
373:
368:
363:
360:
354:
353:
347:
346:
345:
344:
338:
337:
331:
330:
325:
316:
315:
313:Hall's theorem
310:
308:Sylow theorems
304:
303:
298:
290:
289:
288:
287:
281:
276:
273:Dihedral group
269:
268:
263:
257:
252:
246:
241:
230:
225:
224:
221:
220:
215:
214:
213:
212:
207:
199:
198:
197:
196:
191:
186:
181:
176:
171:
166:
164:multiplicative
161:
156:
151:
146:
138:
137:
136:
135:
130:
122:
121:
113:
112:
111:
110:
108:Wreath product
105:
100:
95:
93:direct product
87:
85:Quotient group
79:
78:
77:
76:
71:
66:
56:
53:
52:
49:
48:
40:
39:
13:
10:
9:
6:
4:
3:
2:
2824:
2813:
2810:
2808:
2805:
2803:
2800:
2799:
2797:
2787:
2783:
2779:
2778:
2769:
2766:
2762:
2757:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2723:
2718:
2714:
2710:
2706:
2702:
2701:
2696:
2692:
2689:on 2012-09-10
2688:
2684:
2680:
2676:
2675:
2669:
2666:
2662:
2658:
2654:
2649:
2644:
2640:
2636:
2635:
2630:
2625:
2622:
2618:
2614:
2610:
2605:
2600:
2596:
2592:
2591:
2581:
2578:
2574:
2570:
2566:
2561:
2556:
2552:
2548:
2547:
2537:
2533:
2532:
2527:
2523:
2520:
2516:
2512:
2506:
2502:
2498:
2493:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2429:
2425:
2421:
2416:
2411:
2407:
2403:
2399:
2398:
2393:
2389:
2386:
2382:
2378:
2374:
2369:
2364:
2360:
2356:
2351:
2348:
2344:
2340:
2334:
2330:
2326:
2322:
2318:
2317:
2313:
2306:
2302:
2297:
2295:
2291:
2288:
2283:
2280:
2273:
2269:
2266:
2263:
2259:
2256:
2255:
2251:
2242:
2238:
2234:
2230:
2226:
2221:
2216:
2211:
2206:
2203:
2199:
2194:
2189:
2184:
2179:
2175:
2171:
2166:
2161:
2156:
2151:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2111:
2106:
2102:
2098:
2094:
2090:
2086:
2082:
2077:
2072:
2069:
2065:
2061:
2057:
2052:
2047:
2044:
2040:
2036:
2032:
2028:
2022:
2017:
2014:
2010:
2006:
2001:
1997:
1992:
1988:
1984:
1980:
1976:
1972:
1971:
1970:
1964:
1962:
1960:
1956:
1952:
1948:
1943:
1941:
1936:
1929:
1925:
1921:
1917:
1909:
1906:
1902:
1898:
1894:
1886:
1883:
1879:
1875:
1871:
1863:
1860:
1856:
1852:
1848:
1840:
1837:
1833:
1825:
1822:
1818:
1810:
1807:
1803:
1795:
1792:
1788:
1780:
1777:
1773:
1769:
1765:
1757:
1754:
1750:
1742:
1739:
1735:
1723:
1720:
1716:
1708:
1705:
1701:
1697:
1693:
1685:
1682:
1678:
1670:
1667:
1663:
1655:
1652:
1648:
1640:
1639:
1638:
1628:
1621:
1618:
1610:
1603:
1596:
1589:
1582:
1575:
1574:
1573:
1570:
1568:
1564:
1556:
1554:
1552:
1549:, and the 26
1548:
1544:
1540:
1535:
1533:
1532:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1447:
1439:
1437:
1434:
1432:
1428:
1423:
1419:
1415:
1411:
1407:
1403:
1391:
1390:
1389:
1387:
1376:(3) of type G
1371:
1367:
1366:
1361:
1354:(2) of type F
1349:
1345:
1325:
1324:
1323:
1313:
1312:
1311:
1309:
1302:(2) = Suz(2).
1297:
1296:
1295:
1293:
1285:
1281:
1277:
1273:
1269:
1264:
1260:
1259:Suzuki groups
1252:
1250:
1248:
1228:
1221:
1214:
1207:
1203:
1197:
1192:
1188:
1184:
1178:
1173:
1169:
1168:
1167:
1164:
1162:
1157:
1155:
1151:
1150:
1144:
1139:
1135:
1131:
1126:
1124:
1112:
1108:
1102:
1095:
1084:
1082:
1080:
1072:
1068:
1039:
1033:
1027:
1021:
1016:
1012:
1008:
1004:
1000:
996:
992:
984:
982:
979:
973:
967:
961:
955:
949:
944:
940:
937:
933:
929:
928:unitary group
925:
921:
917:
912:
910:
906:
902:
901:L. E. Dickson
898:
897:Jordan (1870)
894:
890:
884:
876:
874:
872:
871:Carter (1989)
868:
864:
860:
856:
851:
849:
845:
841:
838:. The phrase
837:
833:
830:
826:
822:
821:finite groups
818:
817:
813:, the phrase
812:
808:
796:
791:
789:
784:
782:
777:
776:
774:
773:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
742:
741:
738:
733:
732:
722:
719:
716:
715:
713:
707:
704:
702:
699:
698:
695:
692:
690:
687:
685:
682:
681:
678:
672:
670:
664:
662:
656:
654:
648:
646:
640:
639:
635:
631:
628:
627:
623:
619:
616:
615:
611:
607:
604:
603:
599:
595:
592:
591:
587:
583:
580:
579:
575:
571:
568:
567:
563:
559:
556:
555:
551:
547:
544:
543:
540:
537:
535:
532:
531:
528:
524:
519:
518:
511:
508:
506:
503:
501:
498:
497:
469:
444:
443:
441:
435:
432:
407:
404:
403:
397:
394:
392:
389:
388:
384:
383:
372:
369:
367:
364:
361:
358:
357:
356:
355:
352:
348:
343:
340:
339:
336:
333:
332:
329:
326:
324:
322:
318:
317:
314:
311:
309:
306:
305:
302:
299:
297:
294:
293:
292:
291:
285:
282:
279:
274:
271:
270:
266:
261:
258:
255:
250:
247:
244:
239:
236:
235:
234:
233:
228:
227:Finite groups
223:
222:
211:
208:
206:
203:
202:
201:
200:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
145:
142:
141:
140:
139:
134:
131:
129:
126:
125:
124:
123:
120:
119:
114:
109:
106:
104:
101:
99:
96:
94:
91:
88:
86:
83:
82:
81:
80:
75:
72:
70:
67:
65:
62:
61:
60:
59:
54:Basic notions
51:
50:
46:
42:
41:
38:
33:
29:
24:
19:
2807:Lie algebras
2802:Group theory
2773:
2704:
2698:
2687:the original
2673:
2638:
2632:
2594:
2588:
2550:
2544:
2530:
2496:
2468:
2464:
2451:
2447:
2401:
2395:
2358:
2354:
2324:
2282:
2240:
2236:
2232:
2228:
2224:
2219:
2214:
2209:
2197:
2192:
2187:
2182:
2177:
2173:
2169:
2164:
2159:
2154:
2146:
2142:
2134:
2130:
2126:
2114:
2109:
2104:
2100:
2096:
2092:
2088:
2084:
2080:
2075:
2067:
2063:
2059:
2055:
2050:
2042:
2038:
2034:
2030:
2026:
2020:
2012:
2008:
2004:
1999:
1995:
1990:
1986:
1982:
1978:
1974:
1968:
1944:
1937:
1933:
1927:
1923:
1919:
1915:
1904:
1900:
1896:
1892:
1881:
1877:
1873:
1869:
1858:
1854:
1850:
1846:
1835:
1831:
1820:
1816:
1805:
1801:
1790:
1786:
1775:
1771:
1767:
1763:
1752:
1748:
1737:
1733:
1718:
1714:
1703:
1699:
1695:
1691:
1680:
1676:
1665:
1661:
1650:
1646:
1636:
1594:(4) Solvable
1571:
1560:
1543:finite field
1536:
1530:
1517:
1513:
1509:
1489:
1485:
1481:
1477:
1461:
1443:
1435:
1426:
1421:
1417:
1413:
1409:
1405:
1401:
1399:
1384:(8). In the
1370:Jacques Tits
1363:
1347:
1343:
1333:
1321:
1305:
1266:
1236:
1201:
1195:
1190:
1182:
1176:
1171:
1166:These gave:
1165:
1158:
1147:
1142:
1133:
1127:
1104:
1066:
1037:
1031:
1025:
1019:
1002:
988:
977:
971:
965:
959:
953:
947:
913:
888:
886:
863:real numbers
852:
839:
836:finite field
815:
814:
811:group theory
804:
633:
621:
609:
597:
585:
573:
561:
549:
365:
320:
277:
264:
253:
242:
238:Cyclic group
116:
103:Free product
74:Group action
37:Group theory
32:Group theory
31:
2597:: 115â116,
2471:: 137â150,
2408:: 363â394,
1454:alternating
1123:Tits (1958)
807:mathematics
523:Topological
362:alternating
2796:Categories
2465:Math. Ann.
2314:References
1617:Tits group
1547:Tits group
1365:Tits group
924:symplectic
920:orthogonal
909:Emil Artin
857:, since a
855:Lie groups
630:Symplectic
570:Orthogonal
527:Lie groups
434:Free group
159:continuous
98:Direct sum
2731:0027-8424
2657:0030-8730
2613:0002-9904
2569:0002-9904
2485:179178145
2424:0002-9947
2377:0040-8735
2323:(1989) ,
1450:symmetric
1408:Ă PSL(2,
1263:Ree group
995:Chevalley
939:quotients
829:reductive
694:Conformal
582:Euclidean
189:nilpotent
18:Lie group
2765:16590684
2528:(1870),
2442:(1901),
2252:See also
2125:, use O(
2011:) â PSL(
1531:geometry
1247:triality
1189:further
689:Poincaré
534:Solenoid
406:Integers
396:Lattices
371:sporadic
366:Lie type
194:solvable
184:dihedral
169:additive
154:infinite
64:Subgroup
2786:0106247
2747:0120283
2709:Bibcode
2683:0466335
2665:0109191
2621:0125155
2577:0125155
2519:0310083
2432:1986251
2385:0073602
2347:0407163
2243:). So A
2099:), PSO(
1728:(2) = C
1563:perfect
1476:PSL(2,
1278: (
1270: (
1057:, and G
997: (
936:central
684:Lorentz
606:Unitary
505:Lattice
445:PSL(2,
179:abelian
90:(Semi-)
2784:
2763:
2756:222949
2753:
2745:
2737:
2729:
2681:
2663:
2655:
2619:
2611:
2575:
2567:
2517:
2507:
2483:
2430:
2422:
2383:
2375:
2345:
2335:
1446:cyclic
1412:) for
1268:Suzuki
1132:has a
893:fields
844:simple
539:Circle
470:SL(2,
359:cyclic
323:-group
174:cyclic
149:finite
144:simple
128:kernel
2739:70960
2735:JSTOR
2481:S2CID
2428:JSTOR
2305:p. xi
2301:ATLAS
2274:Notes
2123:ATLAS
1994:) of
1492:) of
1360:index
1338:and G
926:, or
827:of a
723:Sp(â)
720:SU(â)
133:image
2761:PMID
2727:ISSN
2653:ISSN
2609:ISSN
2587:)",
2565:ISSN
2543:)",
2505:ISBN
2420:ISSN
2373:ISSN
2333:ISBN
2239:+1,
2231:+1,
2107:), Ω
1452:and
1322:and
1284:1961
1280:1960
1272:1960
1261:and
1170:the
1096:and
999:1955
903:and
869:and
717:O(â)
706:Loop
525:and
2751:PMC
2717:doi
2643:doi
2599:doi
2555:doi
2473:doi
2410:doi
2363:doi
2135:not
2087:),
1396:(3)
1330:(3)
1318:(2)
1276:Ree
1117:, E
1077:by
1053:, F
1049:, E
1045:, E
1035:, D
1029:, C
1023:, B
975:, D
963:, D
957:, C
951:, B
934:or
895:by
805:In
632:Sp(
620:SU(
596:SO(
560:SL(
548:GL(
2798::
2782:MR
2759:,
2749:,
2743:MR
2741:,
2733:,
2725:,
2715:,
2705:46
2703:,
2679:MR
2661:MR
2659:,
2651:,
2637:,
2631:,
2617:MR
2615:,
2607:,
2595:67
2593:,
2573:MR
2571:,
2563:,
2551:66
2549:,
2515:MR
2513:,
2503:,
2479:,
2469:60
2467:,
2463:,
2452:33
2450:,
2446:,
2426:,
2418:,
2400:,
2381:MR
2379:,
2371:,
2343:MR
2341:,
2331:,
2303:,
2293:^
2129:,
2103:,
2095:,
2083:,
2070:).
2066:,
2058:,
2045:).
2041:,
2029:,
2023:â1
1985:,
1977:,
1961:.
1926:/2
1922:Ă
1918:/2
1903:/2
1899:Ă
1895:/2
1880:/2
1876:Ă
1872:/2
1857:/3
1853:Ă
1849:/3
1834:/2
1819:/2
1804:/3
1789:/2
1774:/2
1770:Ă
1766:/2
1751:/3
1736:/2
1717:/2
1702:/4
1698:Ă
1694:/4
1679:/2
1664:/3
1649:/2
1553:.
1488:,
1448:,
1433:.
1420:/2
1404:/2
1282:,
1249:.
1109:.
1081:.
969:,A
922:,
918:,
907:.
865:.
850:.
608:U(
584:E(
572:O(
30:â
2776:6
2719::
2711::
2645::
2639:9
2601::
2585:4
2557::
2541:2
2490:2
2475::
2412::
2402:2
2365::
2359:7
2260:(
2245:2
2241:q
2237:n
2233:q
2229:n
2225:q
2223:(
2220:n
2215:q
2213:(
2210:n
2204:.
2198:q
2196:(
2193:n
2188:q
2186:(
2183:n
2178:q
2174:q
2170:q
2168:(
2165:n
2160:q
2158:(
2155:n
2147:n
2143:n
2131:q
2127:n
2115:q
2113:(
2110:n
2105:q
2101:n
2097:q
2093:n
2091:(
2089:O
2085:q
2081:n
2076:n
2068:q
2064:n
2060:q
2056:n
2051:n
2043:q
2039:n
2037:(
2035:L
2031:q
2027:n
2021:n
2013:n
2009:n
2005:n
2000:q
1996:F
1991:q
1987:F
1983:n
1979:q
1975:n
1928:Z
1924:Z
1920:Z
1916:Z
1912:2
1910:B
1905:Z
1901:Z
1897:Z
1893:Z
1889:6
1887:E
1882:Z
1878:Z
1874:Z
1870:Z
1866:5
1864:A
1859:Z
1855:Z
1851:Z
1847:Z
1843:3
1841:A
1836:Z
1832:Z
1828:3
1826:A
1821:Z
1817:Z
1813:2
1811:G
1806:Z
1802:Z
1798:2
1796:G
1791:Z
1787:Z
1783:4
1781:F
1776:Z
1772:Z
1768:Z
1764:Z
1760:4
1758:D
1753:Z
1749:Z
1745:3
1743:B
1738:Z
1734:Z
1730:3
1726:3
1724:B
1719:Z
1715:Z
1711:3
1709:A
1704:Z
1700:Z
1696:Z
1692:Z
1688:2
1686:A
1681:Z
1677:Z
1673:2
1671:A
1666:Z
1662:Z
1658:1
1656:A
1651:Z
1647:Z
1643:1
1641:A
1631:2
1629:G
1624:2
1622:G
1619:.
1613:4
1611:F
1606:2
1604:B
1599:2
1597:B
1592:2
1590:A
1585:1
1583:A
1578:1
1576:A
1510:k
1490:q
1486:n
1482:q
1478:q
1462:p
1430:1
1427:J
1422:Z
1418:Z
1414:q
1410:q
1406:Z
1402:Z
1394:2
1392:G
1382:1
1378:2
1374:2
1356:4
1352:4
1348:p
1344:p
1340:2
1336:4
1328:2
1326:G
1316:4
1314:F
1300:2
1298:B
1288:2
1243:4
1239:4
1233:.
1231:4
1226:4
1224:D
1219:;
1217:6
1212:6
1210:E
1205:;
1202:n
1196:n
1193:D
1186:;
1183:n
1177:n
1174:A
1143:n
1140:A
1119:6
1115:4
1100:6
1098:E
1093:4
1091:D
1075:6
1063:2
1059:2
1055:4
1051:8
1047:7
1043:6
1038:n
1032:n
1026:n
1020:n
978:n
972:n
966:n
960:n
954:n
948:n
794:e
787:t
780:v
676:8
674:E
668:7
666:E
660:6
658:E
652:4
650:F
644:2
642:G
636:)
634:n
624:)
622:n
612:)
610:n
600:)
598:n
588:)
586:n
576:)
574:n
564:)
562:n
552:)
550:n
492:)
479:Z
467:)
454:Z
430:)
417:Z
408:(
321:p
286:Q
278:n
275:D
265:n
262:A
254:n
251:S
243:n
240:Z
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
