52:
1902:
giving a uniform construction of the classical groups and the groups of exceptional type in a 1955 paper. This omitted certain known groups (the projective unitary groups), which were obtained by "twisting" the
Chevalley construction. The remaining groups of Lie type were produced by Steinberg, Tits,
1777:
There are two threads in the history of finite simple groups â the discovery and construction of specific simple groups and families, which took place from the work of Galois in the 1820s to the construction of the
Monster in 1981; and proof that this list was complete, which began in the 19th
1932:
These groups (the groups of Lie type, together with the cyclic groups, alternating groups, and the five exceptional
Mathieu groups) were believed to be a complete list, but after a lull of almost a century since the work of Mathieu, in 1964 the first
1949:". The Monster is the largest sporadic simple group having order of 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. The Monster has a faithful 196,883-dimensional representation in the 196,884-dimensional
2477:
1120:
1778:
century, most significantly took place 1955 through 1983 (when victory was initially declared), but was only generally agreed to be finished in 2004. By 2018, its publication was envisioned as a series of 12
933:
2256:, MonografĂas de la Real Academia de Ciencias Exactas, FĂsicas, QuĂmicas y Naturales de Zaragoza, vol. 26, Real Academia de Ciencias Exactas, FĂsicas, QuĂmicas y Naturales de Zaragoza
1361:
1806:
on five or more points are simple (and hence not solvable), which he proved in 1831, was the reason that one could not solve the quintic in radicals. Galois also constructed the
1621:
1181:
1521:
There is as yet no known classification for general (infinite) simple groups, and no such classification is expected. One reason for this is the existence of continuum-many
2359:
497:
472:
435:
1288:
1403:
1684:
1449:
2327:
1654:
1315:
1249:
1218:
2246:
1290:, i.e. the group of even finitely supported permutations of the integers, is simple. This group can be written as the increasing union of the finite simple groups
2120:
1507:
1487:
1423:
1140:
1052:
1032:
1012:
992:
964:
2153:
1848:
in 1861 and 1873, were also simple. Since these five groups were constructed by methods which did not yield infinitely many possibilities, they were called "
2397:
2132:
832:
and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding
1968:
Soon after the construction of the
Monster in 1981, a proof, totaling more than 10,000 pages, was supplied that group theorists had successfully
799:
1972:, with victory declared in 1983 by Daniel Gorenstein. This was premature â some gaps were later discovered, notably in the classification of
1976:, which were eventually replaced in 2004 by a 1,300 page classification of quasithin groups, which is now generally accepted as complete.
1525:
for every sufficiently-large prime characteristic, each simple and having only the cyclic group of that characteristic as its subgroups.
1057:
1692:, which unites this family with the next, and thus all families of non-abelian finite simple groups may be considered to be of Lie type.
870:
1186:
One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of
1708:
is generally considered of this form, though strictly speaking it is not of Lie type, but rather index 2 in a group of Lie type.
1549:
are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way
1577:
1540:
848:
357:
1142:
of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an
2625:
2607:
2589:
2540:
2368:
1937:
was discovered, and the remaining 20 sporadic groups were discovered or conjectured in 1965â1975, culminating in 1981, when
307:
2056:-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.
1190:
order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the
792:
302:
2080:
1822:
not 2 or 3. This is contained in his last letter to
Chevalier, and are the next example of finite simple groups.
1807:
2435:
718:
1320:
2680:
2520:
2062:: A non-Abelian finite simple group has order divisible by at least three distinct primes. This follows from
1942:
785:
17:
2416:) and simplicity discussed on p. 411; exceptional action on 5, 7, or 11 points discussed on pp. 411â412; GL(
1591:
Briefly, finite simple groups are classified as lying in one of 18 families, or being one of 26 exceptions:
2392:
2095:
1969:
1546:
1534:
1558:
841:
402:
216:
2570:
1864:
1845:
134:
1597:
1152:
1962:
1739:
2363:, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI,
1985:
2332:
600:
334:
211:
99:
480:
455:
418:
1266:
1366:
1766:
1689:
750:
540:
2268:
2063:
2025:
1663:
1428:
624:
1953:, meaning that each element of the Monster can be expressed as a 196,883 by 196,883 matrix.
2642:
Silvestri, R. (September 1979), "Simple groups of finite order in the nineteenth century",
2512:
2449:
2378:
2305:
2287:
2197:
1926:
1632:
1522:
1293:
1227:
1196:
967:
564:
552:
170:
104:
2550:
2302:
The classification of the finite simple groups, Number 10. Part V. Chapters 9â17. Theorem
1859:
Later Jordan's results on classical groups were generalized to arbitrary finite fields by
1754:. Therefore, every finite simple group has even order unless it is cyclic of prime order.
8:
2075:
1758:
821:
139:
34:
2659:
2453:
2227:
1762:
1698:
1562:
1492:
1472:
1408:
1125:
1037:
1017:
997:
977:
949:
943:
124:
96:
1799:
2663:
2621:
2603:
2585:
2536:
2429:
2364:
2185:
2085:
1803:
1657:
1581:
1510:
1251:. The second smallest nonabelian simple group is the projective special linear group
1221:
1191:
529:
372:
266:
2231:
1725:
and are referred to as the "Happy Family", while the remaining 6 are referred to as
695:
2651:
2562:
2546:
2528:
2219:
2177:
2090:
1899:
1876:
1853:
1747:
1466:. Explicit examples, which turn out to be finitely presented, include the infinite
939:
936:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
2524:
2374:
2283:
2193:
1973:
1868:
1860:
1834:
1585:
1467:
1458:
infinite simple groups. The first existence result is non-explicit; it is due to
1183:
is not simple; the set of even integers is a non-trivial proper normal subgroup.
825:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
2472:
2033:
1950:
1849:
1826:
1751:
1714:
833:
771:
707:
397:
377:
314:
279:
200:
190:
175:
160:
114:
91:
2532:
2300:
Capdeboscq, Inna; Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald (2023),
2181:
1898:, p. 2). In the 1950s the work on groups of Lie type was continued, with
1840:
At about the same time, it was shown that a family of five groups, called the
2674:
2555:
2457:
2189:
2165:
1946:
1938:
1841:
1795:
1722:
1459:
1255:
of order 168, and every simple group of order 168 is isomorphic to PSL(2,7).
1143:
829:
690:
612:
446:
319:
185:
1830:
1726:
1624:
1550:
1463:
1187:
865:
837:
545:
244:
233:
180:
155:
109:
80:
43:
1965:
of 1962â63, largely lasting until 1983, but only being finished in 2004.
1934:
1743:
1718:
1573:
1569:
851:, completed in 2004, is a major milestone in the history of mathematics.
813:
1765:
of every finite simple group is solvable. This can be proved using the
1688:
The alternating groups may be considered as groups of Lie type over the
2655:
2223:
1705:
1584:, though some problems surfaced (specifically in the classification of
994:
which is 3. Since 3 is prime, its only divisors are 1 and 3, so either
712:
440:
27:
Group without normal subgroups other than the trivial group and itself
1779:
533:
2620:, Springer undergraduate mathematics series (2 ed.), Springer,
840:
one eventually arrives at uniquely determined simple groups, by the
2032:
is not a prime power, then every Sylow subgroup is proper, and, by
1961:
The full classification is generally accepted as starting with the
1363:. Another family of examples of infinite simple groups is given by
1252:
1115:{\displaystyle G=(\mathbb {Z} /12\mathbb {Z} ,+)=\mathbb {Z} _{12}}
70:
1829:
in 1870. Jordan had found 4 families of simple matrix groups over
2491:
1554:
1147:
971:
928:{\displaystyle G=(\mathbb {Z} /3\mathbb {Z} ,+)=\mathbb {Z} _{3}}
412:
326:
2299:
2269:"The classification of finite simple groups: a progress report"
51:
2210:
Burger, M.; Mozes, S. (2000). "Lattices in product of trees".
2247:"The Classification of the Finite Simple Groups: An Overview"
1566:
1513:
infinite simple groups were constructed by Burger and Mozes.
1565:
of a given group have the same length and the same factors,
2602:, Graduate texts in mathematics, vol. 148, Springer,
2168:(1951), "A finitely generated infinite simple group",
2478:
Traité des substitutions et des équations algébriques
2335:
2308:
1871:. Dickson also constructed exception groups of type G
1794:
Simple groups have been studied at least since early
1666:
1635:
1600:
1495:
1475:
1431:
1411:
1369:
1323:
1296:
1269:
1230:
1199:
1155:
1128:
1060:
1040:
1020:
1000:
980:
952:
873:
483:
458:
421:
2009:, then there does not exist a simple group of order
1733:
1772:
1220:of order 60, and every simple group of order 60 is
1054:is the trivial group. On the other hand, the group
2353:
2321:
1678:
1648:
1615:
1501:
1481:
1443:
1417:
1397:
1355:
1309:
1282:
1243:
1212:
1175:
1134:
1114:
1046:
1026:
1006:
986:
958:
927:
491:
466:
429:
2615:
1993:be a positive integer that is not prime, and let
1782:, the tenth of which was published in 2023. See (
2672:
2384:
1146:is normal. Similarly, the additive group of the
2052:. Since 1 is the only such number, the Sylow
1786:) for 19th century history of simple groups.
793:
2398:Journal de Mathématiques Pures et Appliquées
2276:Notices of the American Mathematical Society
2209:
1833:of prime order, which are now known as the
2170:Journal of the London Mathematical Society
1979:
800:
786:
2641:
2616:Smith, Geoff; Tabachnikova, Olga (2000),
2393:"Lettre de Galois Ă M. Auguste Chevalier"
1818:, and remarked that they were simple for
1783:
1603:
1528:
1258:
1160:
1102:
1084:
1071:
915:
897:
884:
485:
460:
423:
2561:
2020:is a prime-power, then a group of order
1750:states that every group of odd order is
1462:and consists of simple quotients of the
1356:{\displaystyle A_{n}\rightarrow A_{n+1}}
836:. This process can be repeated, and for
2600:An introduction to the theory of groups
2266:
1454:It is much more difficult to construct
859:
14:
2673:
2597:
2523:251, vol. 251, Berlin, New York:
2511:
2471:
2448:
2390:
2164:
1895:
1810:of a plane over a prime finite field,
1578:classification of finite simple groups
1576:. In a huge collaborative effort, the
1541:Classification of finite simple groups
849:classification of finite simple groups
358:Classification of finite simple groups
2644:Archive for History of Exact Sciences
2579:
1580:was declared accomplished in 1983 by
1553:are the basic building blocks of the
2244:
1317:with respect to standard embeddings
2442:
2036:, we know that the number of Sylow
1956:
970:(the number of elements) must be a
24:
2028:and, therefore, is not simple. If
1941:announced that he had constructed
1863:, following the classification of
1275:
25:
2692:
2152:Smith & Tabachnikova (2000),
1734:Structure of finite simple groups
1516:
966:is a subgroup of this group, its
2567:Theory of groups of finite order
1802:realized that the fact that the
1773:History for finite simple groups
1616:{\displaystyle \mathbb {Z} _{p}}
1176:{\displaystyle (\mathbb {Z} ,+)}
50:
2484:
2465:
2081:Characteristically simple group
2040:-subgroups of a group of order
1970:listed all finite simple groups
1808:projective special linear group
1789:
1717:, of which 20 are subgroups or
1588:, which were plugged in 2004).
1263:The infinite alternating group
2293:
2260:
2238:
2203:
2158:
2146:
2137:
2125:
2113:
2005:that is congruent to 1 modulo
2001:. If 1 is the only divisor of
1925:)) and by Suzuki and Ree (the
1392:
1386:
1334:
1170:
1156:
1094:
1067:
907:
880:
719:Infinite dimensional Lie group
13:
1:
2521:Graduate Texts in Mathematics
2354:{\displaystyle C_{4}^{\ast }}
2101:
1825:The next discoveries were by
2504:
2096:List of finite simple groups
1535:list of finite simple groups
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
7:
2069:
1882:as well, but not of types F
1865:complex simple Lie algebras
1557:. This is expressed by the
1283:{\displaystyle A_{\infty }}
854:
217:List of group theory topics
10:
2697:
2598:Rotman, Joseph J. (1995),
2580:Knapp, Anthony W. (2006),
2571:Cambridge University Press
2434:: CS1 maint: postscript (
1761:asserts that the group of
1713:One of 26 exceptions, the
1561:which states that any two
1538:
1532:
1398:{\displaystyle PSL_{n}(F)}
2634:
2533:10.1007/978-1-84800-988-2
2454:"Chapter 1: Introduction"
2391:Galois, Ăvariste (1846),
1903:and Herzig (who produced
1425:is an infinite field and
2517:The finite simple groups
2459:The finite simple groups
2267:Solomon, Ronald (2018),
2252:, in Boya, L. J. (ed.),
2106:
335:Elementary abelian group
212:Glossary of group theory
2182:10.1112/jlms/s1-26.1.59
1980:Tests for nonsimplicity
1844:and first described by
1679:{\displaystyle n\geq 5}
1444:{\displaystyle n\geq 2}
1122:is not simple. The set
2618:Topics in group theory
2355:
2323:
1997:be a prime divisor of
1856:in his 1897 textbook.
1767:classification theorem
1697:One of 16 families of
1690:field with one element
1680:
1650:
1617:
1503:
1483:
1445:
1419:
1399:
1357:
1311:
1284:
1259:Infinite simple groups
1245:
1214:
1177:
1136:
1116:
1048:
1028:
1008:
988:
960:
929:
751:Linear algebraic group
493:
468:
431:
2424:) discussed on p. 410
2356:
2324:
2322:{\displaystyle C_{6}}
2254:Problemas del Milenio
2245:Otal, Javier (2004),
2143:Rotman (1995), p. 281
2044:is equal to 1 modulo
2034:Sylow's Third Theorem
1963:FeitâThompson theorem
1846:Ămile LĂ©onard Mathieu
1701:or their derivatives
1681:
1651:
1649:{\displaystyle A_{n}}
1618:
1559:JordanâHölder theorem
1539:Further information:
1523:Tarski monster groups
1509:. Finitely presented
1504:
1484:
1446:
1420:
1400:
1358:
1312:
1310:{\displaystyle A_{n}}
1285:
1246:
1244:{\displaystyle A_{5}}
1215:
1213:{\displaystyle A_{5}}
1178:
1137:
1117:
1049:
1029:
1009:
989:
961:
930:
842:JordanâHölder theorem
494:
469:
432:
2681:Properties of groups
2452:(October 31, 2006),
2333:
2306:
1664:
1633:
1598:
1547:finite simple groups
1529:Finite simple groups
1493:
1473:
1429:
1409:
1367:
1321:
1294:
1267:
1228:
1197:
1153:
1126:
1058:
1038:
1018:
998:
978:
950:
871:
860:Finite simple groups
481:
456:
419:
2350:
2076:Almost simple group
1763:outer automorphisms
1759:Schreier conjecture
125:Group homomorphisms
35:Algebraic structure
2656:10.1007/BF00327738
2351:
2336:
2319:
2224:10.1007/bf02698916
2064:Burnside's theorem
1804:alternating groups
1699:groups of Lie type
1676:
1646:
1613:
1563:composition series
1499:
1479:
1456:finitely generated
1441:
1415:
1395:
1353:
1307:
1280:
1241:
1210:
1173:
1132:
1112:
1044:
1024:
1004:
984:
956:
944:modular arithmetic
937:congruence classes
925:
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
2627:978-1-85233-235-8
2609:978-0-387-94285-8
2591:978-0-8176-3248-9
2563:Burnside, William
2542:978-1-84800-987-5
2513:Wilson, Robert A.
2490:See the proof in
2370:978-1-4704-7553-6
2172:, Second Series,
2086:Quasisimple group
2024:has a nontrivial
1927:SuzukiâRee groups
1658:alternating group
1582:Daniel Gorenstein
1502:{\displaystyle V}
1482:{\displaystyle T}
1418:{\displaystyle F}
1192:alternating group
1135:{\displaystyle H}
1047:{\displaystyle H}
1027:{\displaystyle G}
1007:{\displaystyle H}
987:{\displaystyle G}
959:{\displaystyle H}
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
2688:
2666:
2650:(3â4): 313â356,
2630:
2612:
2594:
2573:
2553:
2498:
2488:
2482:
2481:
2469:
2463:
2462:
2446:
2440:
2439:
2433:
2425:
2411:
2410:
2388:
2382:
2381:
2360:
2358:
2357:
2352:
2349:
2344:
2328:
2326:
2325:
2320:
2318:
2317:
2297:
2291:
2290:
2273:
2264:
2258:
2257:
2251:
2242:
2236:
2235:
2212:Publ. Math. IHĂS
2207:
2201:
2200:
2162:
2156:
2150:
2144:
2141:
2135:
2129:
2123:
2117:
2091:Semisimple group
1974:quasithin groups
1900:Claude Chevalley
1854:William Burnside
1835:classical groups
1817:
1685:
1683:
1682:
1677:
1655:
1653:
1652:
1647:
1645:
1644:
1622:
1620:
1619:
1614:
1612:
1611:
1606:
1586:quasithin groups
1508:
1506:
1505:
1500:
1488:
1486:
1485:
1480:
1450:
1448:
1447:
1442:
1424:
1422:
1421:
1416:
1404:
1402:
1401:
1396:
1385:
1384:
1362:
1360:
1359:
1354:
1352:
1351:
1333:
1332:
1316:
1314:
1313:
1308:
1306:
1305:
1289:
1287:
1286:
1281:
1279:
1278:
1250:
1248:
1247:
1242:
1240:
1239:
1219:
1217:
1216:
1211:
1209:
1208:
1182:
1180:
1179:
1174:
1163:
1141:
1139:
1138:
1133:
1121:
1119:
1118:
1113:
1111:
1110:
1105:
1087:
1079:
1074:
1053:
1051:
1050:
1045:
1033:
1031:
1030:
1025:
1013:
1011:
1010:
1005:
993:
991:
990:
985:
974:of the order of
965:
963:
962:
957:
946:) is simple. If
934:
932:
931:
926:
924:
923:
918:
900:
892:
887:
826:normal subgroups
820:is a nontrivial
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
238:
65:
64:
54:
41:
30:
29:
21:
2696:
2695:
2691:
2690:
2689:
2687:
2686:
2685:
2671:
2670:
2669:
2637:
2628:
2610:
2592:
2576:
2543:
2525:Springer-Verlag
2507:
2502:
2501:
2497:, for instance.
2489:
2485:
2473:Jordan, Camille
2470:
2466:
2447:
2443:
2427:
2426:
2408:
2406:
2389:
2385:
2371:
2345:
2340:
2334:
2331:
2330:
2313:
2309:
2307:
2304:
2303:
2298:
2294:
2271:
2265:
2261:
2249:
2243:
2239:
2208:
2204:
2163:
2159:
2151:
2147:
2142:
2138:
2131:Rotman (1995),
2130:
2126:
2118:
2114:
2109:
2104:
2072:
1982:
1959:
1920:
1909:
1893:
1889:
1885:
1880:
1874:
1869:Wilhelm Killing
1861:Leonard Dickson
1811:
1800:Ăvariste Galois
1792:
1775:
1736:
1715:sporadic groups
1665:
1662:
1661:
1640:
1636:
1634:
1631:
1630:
1607:
1602:
1601:
1599:
1596:
1595:
1543:
1537:
1531:
1519:
1494:
1491:
1490:
1474:
1471:
1470:
1468:Thompson groups
1430:
1427:
1426:
1410:
1407:
1406:
1380:
1376:
1368:
1365:
1364:
1341:
1337:
1328:
1324:
1322:
1319:
1318:
1301:
1297:
1295:
1292:
1291:
1274:
1270:
1268:
1265:
1264:
1261:
1235:
1231:
1229:
1226:
1225:
1204:
1200:
1198:
1195:
1194:
1159:
1154:
1151:
1150:
1127:
1124:
1123:
1106:
1101:
1100:
1083:
1075:
1070:
1059:
1056:
1055:
1039:
1036:
1035:
1019:
1016:
1015:
999:
996:
995:
979:
976:
975:
951:
948:
947:
919:
914:
913:
896:
888:
883:
872:
869:
868:
862:
857:
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
484:
482:
479:
478:
459:
457:
454:
453:
422:
420:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
15:
12:
11:
5:
2694:
2684:
2683:
2668:
2667:
2638:
2636:
2633:
2632:
2631:
2626:
2613:
2608:
2595:
2590:
2575:
2574:
2559:
2541:
2508:
2506:
2503:
2500:
2499:
2483:
2464:
2450:Wilson, Robert
2441:
2383:
2369:
2348:
2343:
2339:
2316:
2312:
2292:
2282:(6): 646â651,
2259:
2237:
2202:
2166:Higman, Graham
2157:
2145:
2136:
2124:
2119:Knapp (2006),
2111:
2110:
2108:
2105:
2103:
2100:
2099:
2098:
2093:
2088:
2083:
2078:
2071:
2068:
1981:
1978:
1958:
1957:Classification
1955:
1951:Griess algebra
1918:
1907:
1891:
1887:
1883:
1878:
1872:
1842:Mathieu groups
1827:Camille Jordan
1791:
1788:
1784:Silvestri 1979
1774:
1771:
1735:
1732:
1731:
1730:
1711:
1710:
1709:
1695:
1694:
1693:
1675:
1672:
1669:
1643:
1639:
1628:
1627:of prime order
1610:
1605:
1533:Main article:
1530:
1527:
1518:
1517:Classification
1515:
1498:
1478:
1440:
1437:
1434:
1414:
1394:
1391:
1388:
1383:
1379:
1375:
1372:
1350:
1347:
1344:
1340:
1336:
1331:
1327:
1304:
1300:
1277:
1273:
1260:
1257:
1238:
1234:
1207:
1203:
1172:
1169:
1166:
1162:
1158:
1131:
1109:
1104:
1099:
1096:
1093:
1090:
1086:
1082:
1078:
1073:
1069:
1066:
1063:
1043:
1023:
1003:
983:
955:
922:
917:
912:
909:
906:
903:
899:
895:
891:
886:
882:
879:
876:
861:
858:
856:
853:
834:quotient group
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
9:
6:
4:
3:
2:
2693:
2682:
2679:
2678:
2676:
2665:
2661:
2657:
2653:
2649:
2645:
2640:
2639:
2629:
2623:
2619:
2614:
2611:
2605:
2601:
2596:
2593:
2587:
2583:
2582:Basic algebra
2578:
2577:
2572:
2568:
2564:
2560:
2557:
2556:2007 preprint
2552:
2548:
2544:
2538:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2509:
2496:
2494:
2487:
2480:
2479:
2474:
2468:
2461:
2460:
2455:
2451:
2445:
2437:
2431:
2423:
2419:
2415:
2404:
2400:
2399:
2394:
2387:
2380:
2376:
2372:
2366:
2362:
2346:
2341:
2337:
2314:
2310:
2296:
2289:
2285:
2281:
2277:
2270:
2263:
2255:
2248:
2241:
2233:
2229:
2225:
2221:
2217:
2213:
2206:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2167:
2161:
2155:
2149:
2140:
2134:
2128:
2122:
2116:
2112:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2073:
2067:
2065:
2061:
2057:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2019:
2014:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1987:
1977:
1975:
1971:
1966:
1964:
1954:
1952:
1948:
1947:Monster group
1944:
1943:Bernd Fischer
1940:
1939:Robert Griess
1936:
1930:
1928:
1924:
1917:
1913:
1906:
1901:
1897:
1881:
1870:
1866:
1862:
1857:
1855:
1851:
1847:
1843:
1838:
1836:
1832:
1831:finite fields
1828:
1823:
1821:
1815:
1809:
1805:
1801:
1797:
1796:Galois theory
1787:
1785:
1781:
1770:
1768:
1764:
1760:
1755:
1753:
1749:
1745:
1741:
1728:
1724:
1723:monster group
1720:
1716:
1712:
1707:
1703:
1702:
1700:
1696:
1691:
1687:
1686:
1673:
1670:
1667:
1659:
1641:
1637:
1629:
1626:
1608:
1594:
1593:
1592:
1589:
1587:
1583:
1579:
1575:
1571:
1568:
1564:
1560:
1556:
1552:
1551:prime numbers
1548:
1542:
1536:
1526:
1524:
1514:
1512:
1496:
1476:
1469:
1465:
1461:
1460:Graham Higman
1457:
1452:
1438:
1435:
1432:
1412:
1389:
1381:
1377:
1373:
1370:
1348:
1345:
1342:
1338:
1329:
1325:
1302:
1298:
1271:
1256:
1254:
1236:
1232:
1223:
1205:
1201:
1193:
1189:
1184:
1167:
1164:
1149:
1145:
1144:abelian group
1129:
1107:
1097:
1091:
1088:
1080:
1076:
1064:
1061:
1041:
1021:
1001:
981:
973:
969:
953:
945:
941:
938:
920:
910:
904:
901:
893:
889:
877:
874:
867:
852:
850:
847:The complete
845:
843:
839:
838:finite groups
835:
831:
830:trivial group
827:
823:
819:
815:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
356:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
122:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
32:
31:
19:
18:Simple groups
2647:
2643:
2617:
2599:
2584:, Springer,
2581:
2566:
2516:
2492:
2486:
2476:
2467:
2458:
2444:
2421:
2417:
2413:
2407:, retrieved
2402:
2396:
2386:
2329:and Theorem
2301:
2295:
2279:
2275:
2262:
2253:
2240:
2215:
2211:
2205:
2176:(1): 61â64,
2173:
2169:
2160:
2148:
2139:
2127:
2115:
2059:
2058:
2053:
2049:
2048:and divides
2045:
2041:
2037:
2029:
2021:
2017:
2015:
2010:
2006:
2002:
1998:
1994:
1990:
1986:Sylow's test
1984:
1983:
1967:
1960:
1931:
1922:
1915:
1911:
1904:
1858:
1839:
1824:
1819:
1813:
1793:
1790:Construction
1776:
1756:
1737:
1719:subquotients
1625:cyclic group
1590:
1544:
1520:
1511:torsion-free
1464:Higman group
1455:
1453:
1262:
1185:
866:cyclic group
863:
846:
818:simple group
817:
811:
640:
628:
616:
604:
592:
580:
568:
556:
327:
284:
271:
260:
249:
245:Cyclic group
150:
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
2218:: 151â194.
1935:Janko group
1896:Wilson 2009
1738:The famous
1574:isomorphism
1570:permutation
824:whose only
814:mathematics
530:Topological
369:alternating
2551:1203.20012
2409:2009-02-04
2102:References
2016:Proof: If
1780:monographs
1706:Tits group
1222:isomorphic
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
2664:120444304
2505:Textbooks
2405:: 408â415
2347:∗
2190:0024-6107
1671:≥
1436:≥
1335:→
1276:∞
701:Conformal
589:Euclidean
196:nilpotent
2675:Category
2565:(1897),
2515:(2009),
2475:(1870),
2430:citation
2412:, PSL(2,
2361:, Case A
2232:55003601
2070:See also
2060:Burnside
1850:sporadic
1798:, where
1752:solvable
1748:Thompson
1555:integers
1405:, where
1253:PSL(2,7)
1148:integers
942:3 (see
855:Examples
828:are the
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
2379:4656413
2288:3792856
2198:0038348
1740:theorem
1727:pariahs
1721:of the
972:divisor
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
2662:
2635:Papers
2624:
2606:
2588:
2549:
2539:
2495:-group
2377:
2367:
2286:
2230:
2196:
2188:
2154:p. 144
2133:p. 226
2121:p. 170
2026:center
1989:: Let
1914:) and
1890:, or E
1812:PSL(2,
940:modulo
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
2660:S2CID
2272:(PDF)
2250:(PDF)
2228:S2CID
2107:Notes
1852:" by
1567:up to
1188:prime
1034:, or
968:order
822:group
730:Sp(â)
727:SU(â)
140:image
2622:ISBN
2604:ISBN
2586:ISBN
2537:ISBN
2436:link
2365:ISBN
2186:ISSN
1945:'s "
1875:and
1757:The
1746:and
1744:Feit
1704:The
1660:for
1572:and
1545:The
1489:and
864:The
816:, a
724:O(â)
713:Loop
532:and
2652:doi
2547:Zbl
2529:doi
2220:doi
2178:doi
1929:).
1886:, E
1867:by
1742:of
1224:to
1014:is
935:of
812:In
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
2677::
2658:,
2648:20
2646:,
2569:,
2554:;
2545:,
2535:,
2527:,
2519:,
2456:,
2432:}}
2428:{{
2403:XI
2401:,
2395:,
2375:MR
2373:,
2284:MR
2280:65
2278:,
2274:,
2226:.
2216:92
2214:.
2194:MR
2192:,
2184:,
2174:26
2066:.
2013:.
1837:.
1769:.
1656:â
1623:â
1451:.
1108:12
1081:12
844:.
615:U(
591:E(
579:O(
37:â
2654::
2558:.
2531::
2493:p
2438:)
2422:p
2420:,
2418:Μ
2414:p
2342:4
2338:C
2315:6
2311:C
2234:.
2222::
2180::
2054:p
2050:n
2046:p
2042:n
2038:p
2030:n
2022:n
2018:n
2011:n
2007:p
2003:n
1999:n
1995:p
1991:n
1923:q
1921:(
1919:6
1916:E
1912:q
1910:(
1908:4
1905:D
1894:(
1892:8
1888:7
1884:4
1879:6
1877:E
1873:2
1820:p
1816:)
1814:p
1729:.
1674:5
1668:n
1642:n
1638:A
1609:p
1604:Z
1497:V
1477:T
1439:2
1433:n
1413:F
1393:)
1390:F
1387:(
1382:n
1378:L
1374:S
1371:P
1349:1
1346:+
1343:n
1339:A
1330:n
1326:A
1303:n
1299:A
1272:A
1237:5
1233:A
1206:5
1202:A
1171:)
1168:+
1165:,
1161:Z
1157:(
1130:H
1103:Z
1098:=
1095:)
1092:+
1089:,
1085:Z
1077:/
1072:Z
1068:(
1065:=
1062:G
1042:H
1022:G
1002:H
982:G
954:H
921:3
916:Z
911:=
908:)
905:+
902:,
898:Z
894:3
890:/
885:Z
881:(
878:=
875:G
801:e
794:t
787:v
683:8
681:E
675:7
673:E
667:6
665:E
659:4
657:F
651:2
649:G
643:)
641:n
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629:n
619:)
617:n
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605:n
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569:n
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486:Z
474:)
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437:)
424:Z
415:(
328:p
293:Q
285:n
282:D
272:n
269:A
261:n
258:S
250:n
247:Z
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