946:
41:
821:. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include
1287:), 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
1837:. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the
1132:
An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The
486:
461:
424:
1635:
The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.
2471:
788:
1624:
is that such "building blocks" do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same
2441:
1144:
and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of
1136:
of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of
902:
of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of
1374:
This provides a partial converse to
Lagrange's theorem giving information about how many subgroups of a given order are contained in
1279:. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (
852:
During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the
1291:, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their
2374:
2344:
1838:
1561:
1299:
865:
837:
346:
1620:
is a more precise way of stating this fact about finite groups. However, a significant difference with respect to the case of
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1608:
The finite simple groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the
1679:
1328:
296:
2359:
781:
291:
1247:â 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(
1236:
1220:
2541:
707:
2655:
2379:
1640:
1490:
774:
2354:
1747:
1699:
1565:
1617:
391:
205:
1423:
123:
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implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. If
1811:
1629:
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323:
200:
88:
1686:
is the square of a prime, then there are exactly two possible isomorphism types of group of order
469:
444:
407:
1118:
1086:
1283:). Although it was known since 19th century that other finite simple groups exist (for example,
2650:
1306:
include all the finite simple groups other than the cyclic groups, the alternating groups, the
1179:
739:
529:
954:
2399:
1766:
1702:
give asymptotically correct estimates for the number of isomorphism types of groups of order
1621:
1443:
1357:
1311:
1207:
Finite groups of Lie type were among the first groups to be considered in mathematics, after
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2404:
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1541:
1529:
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1141:
1098:
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998:
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since it arose in the 19th century. One major area of study has been classification: the
8:
2384:
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1525:
1505:
1400:
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1114:
982:
919:
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23:
2433:
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1625:
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113:
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1302:. Inspection of the list of finite simple groups shows that groups of Lie type over a
1228:
945:
2578:
2537:
2455:
1636:
1587:
1580:
1407:
1216:
1157:
1126:
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gives an isomorphism between the two. This can be done with any finite cyclic group.
940:
914:. These are finite groups generated by reflections which act on a finite-dimensional
826:
518:
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in the 1830s. The systematic exploration of finite groups of Lie type started with
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918:. The properties of finite groups can thus play a role in subjects such as
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869:
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233:
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98:
69:
32:
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are gradually publishing a simplified and revised version of the proof.
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During the second half of the twentieth century, mathematicians such as
1601:
1307:
974:
911:
887:, and other related groups. One such family of groups is the family of
818:
701:
429:
1713:, some restrictions may be placed on the structure of groups of order
1267:
realized that after an appropriate reformulation, many theorems about
1069:, the identity. A typical realization of this group is as the complex
1013:
1002:
923:
903:
522:
2623:
sequence A060689 (Number of non-Abelian groups of order n)
1054:
is a group all of whose elements are powers of a particular element
1411:
1342:
1121:
to two group elements does not depend on their order (the axiom of
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59:
2619:
2607:
2599:
2364:
1493:
401:
315:
1814:, which has a long and complicated proof, every group of order
40:
2611:
sequence A000688 (Number of
Abelian groups of order
2434:"The Status of the Classification of the Finite Simple Groups"
1853:
for which there are two non-isomorphic simple groups of order
1271:
admit analogues for algebraic groups over an arbitrary field
1706:, and the number grows very rapidly as the power increases.
2622:
2610:
2602:
1877:
1873:
1869:
1781:
is divisible by fewer than three distinct primes, i.e. if
1659:, it is not at all a routine matter to determine how many
1555:
872:
from which all finite groups can be built are now known.
16:
Mathematical group based upon a finite number of elements
2628:
2603:
sequence A000001 (Number of groups of order n)
1717:, as a consequence, for example, of results such as the
1192:. Finite groups of Lie type give the bulk of nonabelian
832:
The study of finite groups has been an integral part of
1508:
has order divisible by at least three distinct primes.
2462:, December 1, 1985, vol. 253, no. 6, pp. 104â115.
1860:
883:
also increased our understanding of finite analogs of
472:
447:
410:
1650:
1204:, the Steinberg groups, and the SuzukiâRee groups.
1005:from the set of symbols to itself. Since there are
1849:, and there are infinitely many positive integers
480:
455:
418:
1275:, leading to construction of what are now called
1024:(the number of elements) of the symmetric group S
2642:
1746:. For a necessary and sufficient condition, see
1604:(sometimes considered as a 27th sporadic group).
1263:in the beginning of 20th century. In the 1950s
1694:is a higher power of a prime, then results of
1422:. This can be understood as an example of the
2536:. Oxford University Press. pp. 238â242.
2472:Group Theory and its Application to Chemistry
2445:. Vol. 51, no. 7. pp. 736â740.
1845:there are at most two simple groups of order
782:
2442:Notices of the American Mathematical Society
1549:
1545:
910:", is strongly influenced by the associated
2477:
1001:of such permutations, which are treated as
898:Finite groups often occur when considering
2428:
1568:belongs to one of the following families:
1298:The belief has now become a theorem â the
1117:in which the result of applying the group
789:
775:
2531:
1259:. Other classical groups were studied by
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2519:
2507:
2495:
2483:
1709:Depending on the prime factorization of
1511:
1092:
944:
906:, which may be viewed as dealing with "
2643:
2375:Representation theory of finite groups
2345:Classification of finite simple groups
1839:classification of finite simple groups
1562:classification of finite simple groups
1556:Classification of finite simple groups
1300:classification of finite simple groups
866:classification of finite simple groups
838:classification of finite simple groups
347:Classification of finite simple groups
1612:are the basic building blocks of the
1437:
1322:
1151:
934:
868:was achieved, meaning that all those
1721:. For example, every group of order
1178:) of rational points of a reductive
1016:) possible permutations of a set of
1773:, states that every group of order
1381:
856:of finite groups and the theory of
13:
2561:
1861:Table of distinct groups of order
1810:are non-negative integers. By the
864:. As a consequence, the complete
14:
2667:
2593:
1651:Number of groups of a given order
1632:does not have a unique solution.
1363:
1329:Lagrange's theorem (group theory)
1223:over prime finite fields, PSL(2,
2474:The Chemistry LibreTexts library
2458:(1985), "The Enormous Theorem",
1690:, both of which are abelian. If
1564:is a theorem stating that every
1317:
1221:projective special linear groups
1037:
39:
2360:Cauchy's theorem (group theory)
1237:projective special linear group
2525:
2513:
2501:
2489:
2465:
2449:
2422:
1356:. The theorem is named after
1341:(number of elements) of every
1170:closely related to the group
708:Infinite dimensional Lie group
1:
2416:
2380:Modular representation theory
1196:. Special cases include the
1020:symbols, it follows that the
2355:List of finite simple groups
1628:or, put in another way, the
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
7:
2532:Humphreys, John F. (1996).
2337:
1841:. For any positive integer
1825:For every positive integer
1524:, states that every finite
985:whose elements are all the
929:
206:List of group theory topics
10:
2672:
1867:
1761:, then any group of order
1670:there are. Every group of
1441:
1385:
1367:
1326:
1155:
1096:
1041:
938:
847:
840:(those with no nontrivial
2637:for groups of small order
2569:Jacobson, Nathan (2009).
1655:Given a positive integer
1504:. Hence each non-Abelian
844:) was completed in 2004.
2534:A Course in Group Theory
1588:simple group of Lie type
1125:). They are named after
324:Elementary abelian group
201:Glossary of group theory
1829:, most groups of order
1802:are prime numbers, and
1281:Tits simplicity theorem
1227:) being constructed by
1087:primitive root of unity
953:of the symmetric group
1595:sporadic simple groups
1312:sporadic simple groups
1295:in the sense of Tits.
1180:linear algebraic group
960:
740:Linear algebraic group
482:
457:
420:
2400:Commuting probability
1812:FeitâThompson theorem
1622:integer factorization
1618:JordanâHölder theorem
1583:of degree at least 5;
1518:FeitâThompson theorem
1512:FeitâThompson theorem
1459:is a finite group of
1395:, named in honour of
1358:Joseph-Louis Lagrange
1352:divides the order of
1333:For any finite group
1269:semisimple Lie groups
1093:Finite abelian groups
948:
889:general linear groups
483:
458:
421:
2656:Properties of groups
2405:Finite State Machine
2370:List of small groups
1542:John Griggs Thompson
1399:, states that every
1257:finite simple groups
1235:'s theorem that the
1194:finite simple groups
1142:Ludwig Stickelberger
1099:Finite abelian group
470:
445:
408:
2522:, p. 72, ex. 1
2460:Scientific American
2430:Aschbacher, Michael
2385:Monstrous moonshine
1663:types of groups of
1566:finite simple group
1536:. It was proved by
1506:finite simple group
1430:on the elements of
1185:with values in the
1003:bijective functions
993:symbols, and whose
920:theoretical physics
908:continuous symmetry
114:Group homomorphisms
24:Algebraic structure
2575:Dover Publications
2350:Association scheme
1767:Burnside's theorem
1680:Lagrange's theorem
1626:composition series
1449:Burnside's theorem
1444:Burnside's theorem
1438:Burnside's theorem
1323:Lagrange's theorem
1152:Groups of Lie type
1134:automorphism group
961:
935:Permutation groups
827:permutation groups
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
2584:978-0-486-47189-1
2456:Daniel Gorenstein
2335:
2334:
1818:is solvable when
1777:is solvable when
1742:not divisible by
1630:extension problem
1581:alternating group
1576:with prime order;
1522:odd order theorem
1219:groups, with the
1164:group of Lie type
1158:Group of Lie type
1127:Niels Henrik Abel
1111:commutative group
941:Permutation group
799:
798:
374:
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256:Alternating group
213:
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2627:Small groups on
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2573:(2nd ed.).
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1793:
1771:group characters
1741:
1735:are primes with
1734:
1393:Cayley's theorem
1388:Cayley's theorem
1382:Cayley's theorem
1277:Chevalley groups
1265:Claude Chevalley
1243:) is simple for
1202:Chevalley groups
1198:classical groups
1109:, also called a
1078:
1077:
1068:
1048:A cyclic group Z
885:classical groups
862:nilpotent groups
803:abstract algebra
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
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280:Quaternion group
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2571:Basic Algebra I
2564:
2562:Further reading
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2502:
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2390:Profinite group
2340:
1880:
1868:Main articles:
1866:
1782:
1769:, proved using
1736:
1726:
1725:is cyclic when
1653:
1614:natural numbers
1558:
1538:Walter Feit
1514:
1455:states that if
1446:
1440:
1416:symmetric group
1390:
1384:
1372:
1366:
1331:
1325:
1320:
1289:sporadic groups
1261:Leonard Dickson
1229:Ăvariste Galois
1160:
1154:
1138:Georg Frobenius
1101:
1095:
1075:
1071:
1059:
1053:
1046:
1040:
1029:
995:group operation
981:symbols is the
972:
965:symmetric group
958:
943:
937:
932:
916:Euclidean space
850:
842:normal subgroup
795:
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765:
754:Abelian variety
747:Reductive group
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614:Special unitary
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331:Frobenius group
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65:Normal subgroup
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1719:Sylow theorems
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1598:
1593:One of the 26
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1557:
1554:
1513:
1510:
1442:Main article:
1439:
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1386:Main article:
1383:
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1370:Sylow theorems
1368:Main article:
1365:
1364:Sylow theorems
1362:
1327:Main article:
1324:
1321:
1319:
1316:
1285:Mathieu groups
1233:Camille Jordan
1156:Main article:
1153:
1150:
1146:linear algebra
1097:Main article:
1094:
1091:
1079:roots of unity
1049:
1042:Main article:
1039:
1036:
1025:
968:
956:
939:Main article:
936:
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931:
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849:
846:
815:underlying set
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436:Modular groups
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309:Hall's theorem
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304:Sylow theorems
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160:multiplicative
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89:direct product
83:
81:Quotient group
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2651:Finite groups
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2604:
2598:
2597:
2586:
2580:
2576:
2572:
2567:
2566:
2553:
2549:
2545:
2539:
2535:
2528:
2521:
2520:Jacobson 2009
2516:
2509:
2508:Jacobson 2009
2504:
2497:
2496:Jacobson 2009
2492:
2485:
2484:Jacobson 2009
2480:
2473:
2468:
2461:
2457:
2452:
2444:
2443:
2435:
2431:
2425:
2421:
2411:
2408:
2406:
2403:
2401:
2398:
2396:
2393:
2391:
2388:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2342:
2330:
2327:
2324:
2321:
2320:
2316:
2313:
2310:
2307:
2306:
2302:
2299:
2296:
2293:
2292:
2288:
2285:
2282:
2279:
2278:
2274:
2271:
2268:
2265:
2264:
2260:
2257:
2254:
2251:
2250:
2246:
2243:
2240:
2237:
2236:
2232:
2229:
2226:
2223:
2222:
2218:
2215:
2212:
2209:
2208:
2204:
2201:
2198:
2195:
2194:
2190:
2187:
2184:
2181:
2180:
2176:
2173:
2170:
2167:
2166:
2162:
2159:
2156:
2153:
2152:
2148:
2145:
2142:
2139:
2138:
2134:
2131:
2128:
2125:
2124:
2120:
2117:
2114:
2111:
2110:
2106:
2103:
2100:
2097:
2096:
2092:
2089:
2086:
2083:
2082:
2078:
2075:
2072:
2069:
2068:
2064:
2061:
2058:
2055:
2054:
2050:
2047:
2044:
2041:
2040:
2036:
2033:
2030:
2027:
2026:
2022:
2019:
2016:
2013:
2012:
2008:
2005:
2002:
1999:
1998:
1994:
1991:
1988:
1985:
1984:
1980:
1977:
1974:
1971:
1970:
1966:
1963:
1960:
1957:
1956:
1952:
1949:
1946:
1943:
1942:
1938:
1935:
1932:
1929:
1928:
1924:
1921:
1918:
1915:
1914:
1910:
1907:
1904:
1901:
1900:
1896:
1893:
1890:
1888:
1884:
1883:
1879:
1875:
1871:
1864:
1858:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1828:
1823:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1792:
1789:
1785:
1780:
1776:
1772:
1768:
1765:is solvable.
1764:
1760:
1756:
1751:
1749:
1748:cyclic number
1745:
1739:
1733:
1729:
1724:
1720:
1716:
1712:
1707:
1705:
1701:
1697:
1696:Graham Higman
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1666:
1662:
1658:
1648:
1646:
1642:
1638:
1633:
1631:
1627:
1623:
1619:
1615:
1611:
1610:prime numbers
1603:
1599:
1596:
1592:
1589:
1585:
1582:
1578:
1575:
1571:
1570:
1569:
1567:
1563:
1553:
1551:
1547:
1543:
1540: and
1539:
1535:
1531:
1527:
1523:
1519:
1509:
1507:
1503:
1499:
1495:
1492:
1488:
1484:
1480:
1479:prime numbers
1476:
1472:
1468:
1465:
1462:
1458:
1454:
1450:
1445:
1435:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1402:
1398:
1397:Arthur Cayley
1394:
1389:
1379:
1377:
1371:
1361:
1359:
1355:
1351:
1347:
1344:
1340:
1336:
1330:
1318:Main theorems
1315:
1313:
1310:, and the 26
1309:
1305:
1301:
1296:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1205:
1203:
1199:
1195:
1191:
1188:
1184:
1181:
1177:
1173:
1169:
1165:
1159:
1149:
1147:
1143:
1139:
1135:
1130:
1128:
1124:
1123:commutativity
1120:
1116:
1112:
1108:
1107:
1106:abelian group
1100:
1090:
1088:
1084:
1080:
1074:
1066:
1062:
1057:
1052:
1045:
1038:Cyclic groups
1035:
1033:
1028:
1023:
1019:
1015:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
980:
976:
971:
966:
959:
952:
947:
942:
927:
925:
921:
917:
913:
909:
905:
901:
896:
894:
893:finite fields
890:
886:
882:
878:
873:
871:
870:simple groups
867:
863:
859:
855:
845:
843:
839:
835:
830:
828:
824:
823:cyclic groups
820:
816:
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
738:
737:
734:
729:
728:
718:
715:
712:
711:
709:
703:
700:
698:
695:
694:
691:
688:
686:
683:
681:
678:
677:
674:
668:
666:
660:
658:
652:
650:
644:
642:
636:
635:
631:
627:
624:
623:
619:
615:
612:
611:
607:
603:
600:
599:
595:
591:
588:
587:
583:
579:
576:
575:
571:
567:
564:
563:
559:
555:
552:
551:
547:
543:
540:
539:
536:
533:
531:
528:
527:
524:
520:
515:
514:
507:
504:
502:
499:
497:
494:
493:
465:
440:
439:
437:
431:
428:
403:
400:
399:
393:
390:
388:
385:
384:
380:
379:
368:
365:
363:
360:
357:
354:
353:
352:
351:
348:
345:
344:
339:
336:
335:
332:
329:
328:
325:
322:
320:
318:
314:
313:
310:
307:
305:
302:
301:
298:
295:
293:
290:
289:
288:
287:
281:
278:
275:
270:
267:
266:
262:
257:
254:
251:
246:
243:
240:
235:
232:
231:
230:
229:
224:
223:Finite groups
219:
218:
207:
204:
202:
199:
198:
197:
196:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
137:
136:
135:
130:
127:
125:
122:
121:
120:
119:
116:
115:
111:
110:
105:
102:
100:
97:
95:
92:
90:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
57:
56:
55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
21:
20:
2612:
2570:
2533:
2527:
2515:
2510:, p. 38
2503:
2498:, p. 41
2491:
2486:, p. 31
2479:
2467:
2459:
2451:
2440:
2424:
1897:Non-Abelian
1886:
1878:oeis:A060689
1874:oeis:A000688
1870:oeis:A000001
1862:
1854:
1850:
1846:
1842:
1830:
1826:
1824:
1819:
1815:
1807:
1803:
1799:
1795:
1790:
1787:
1783:
1778:
1774:
1762:
1754:
1752:
1743:
1737:
1731:
1727:
1722:
1714:
1710:
1708:
1703:
1700:Charles Sims
1691:
1687:
1683:
1667:
1656:
1654:
1634:
1607:
1574:cyclic group
1559:
1521:
1517:
1515:
1497:
1491:non-negative
1486:
1482:
1474:
1470:
1466:
1463:
1456:
1453:group theory
1448:
1447:
1431:
1427:
1424:group action
1419:
1403:
1392:
1391:
1375:
1373:
1353:
1349:
1345:
1334:
1332:
1304:finite field
1297:
1292:
1280:
1276:
1272:
1252:
1248:
1244:
1240:
1224:
1206:
1189:
1182:
1175:
1171:
1163:
1161:
1131:
1110:
1104:
1102:
1082:
1072:
1064:
1060:
1055:
1050:
1047:
1044:Cyclic group
1031:
1026:
1017:
1010:
1006:
990:
987:permutations
978:
969:
964:
962:
951:Cayley graph
897:
874:
854:local theory
851:
834:group theory
831:
807:finite group
806:
800:
629:
617:
605:
593:
581:
569:
557:
545:
316:
273:
260:
249:
238:
234:Cyclic group
222:
144:
112:
99:Free product
70:Group action
33:Group theory
28:Group theory
27:
2395:Finite ring
1661:isomorphism
1217:alternating
999:composition
912:Weyl groups
519:Topological
358:alternating
2645:Categories
2635:classifier
2629:GroupNames
2552:0843.20001
2543:0198534590
2417:References
1759:squarefree
1678:, because
1639:(d.1992),
1637:Gorenstein
1602:Tits group
1418:acting on
1408:isomorphic
1308:Tits group
1081:. Sending
975:finite set
904:Lie groups
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
1891:# Groups
1674:order is
1213:symmetric
1119:operation
1014:factorial
924:chemistry
881:Steinberg
877:Chevalley
690:Conformal
578:Euclidean
185:nilpotent
2432:(2004).
2338:See also
1894:Abelian
1835:solvable
1822:is odd.
1794:, where
1534:solvable
1502:solvable
1494:integers
1469:, where
1412:subgroup
1343:subgroup
1293:geometry
930:Examples
900:symmetry
858:solvable
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
2365:P-group
1645:Solomon
1616:. The
1544: (
1528:of odd
1496:, then
1414:of the
1239:PSL(2,
1113:, is a
997:is the
989:of the
848:History
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
2581:
2550:
2540:
1885:Order
1876:, and
1676:cyclic
1643:, and
1481:, and
1337:, the
1209:cyclic
1200:, the
1058:where
819:finite
813:whose
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
2437:(PDF)
1730:<
1672:prime
1665:order
1641:Lyons
1530:order
1526:group
1520:, or
1461:order
1410:to a
1401:group
1339:order
1255:) of
1187:field
1168:group
1166:is a
1115:group
1085:to a
1022:order
983:group
973:on a
891:over
811:group
809:is a
719:Sp(â)
716:SU(â)
129:image
2620:OEIS
2608:OEIS
2600:OEIS
2579:ISBN
2538:ISBN
1833:are
1806:and
1798:and
1698:and
1600:The
1560:The
1550:1963
1546:1962
1516:The
1489:are
1485:and
1477:are
1473:and
1215:and
1140:and
963:The
922:and
879:and
860:and
825:and
805:, a
713:O(â)
702:Loop
521:and
2548:Zbl
2322:30
2308:29
2294:28
2280:27
2266:26
2252:25
2247:12
2241:15
2238:24
2224:23
2210:22
2196:21
2182:20
2168:19
2154:18
2140:17
2129:14
2126:16
2112:15
2098:14
2084:13
2070:12
2056:11
2042:10
1757:is
1753:If
1740:â 1
1579:An
1532:is
1500:is
1451:in
1426:of
1406:is
1348:of
1103:An
1067:= e
1034:!.
1030:is
1009:! (
977:of
817:is
801:In
628:Sp(
616:SU(
592:SO(
556:SL(
544:GL(
2647::
2633:A
2577:.
2546:.
2439:.
2331:3
2328:1
2325:4
2317:0
2314:1
2311:1
2303:2
2300:2
2297:4
2289:2
2286:3
2283:5
2275:1
2272:1
2269:2
2261:0
2258:2
2255:2
2244:3
2233:0
2230:1
2227:1
2219:1
2216:1
2213:2
2205:1
2202:1
2199:2
2191:3
2188:2
2185:5
2177:0
2174:1
2171:1
2163:3
2160:2
2157:5
2149:0
2146:1
2143:1
2135:9
2132:5
2121:0
2118:1
2115:1
2107:1
2104:1
2101:2
2093:0
2090:1
2087:1
2079:3
2076:2
2073:5
2065:0
2062:1
2059:1
2051:1
2048:1
2045:2
2037:0
2034:2
2031:2
2028:9
2023:2
2020:3
2017:5
2014:8
2009:0
2006:1
2003:1
2000:7
1995:1
1992:1
1989:2
1986:6
1981:0
1978:1
1975:1
1972:5
1967:0
1964:2
1961:2
1958:4
1953:0
1950:1
1947:1
1944:3
1939:0
1936:1
1933:1
1930:2
1925:0
1922:1
1919:1
1916:1
1911:0
1908:0
1905:0
1902:0
1872:,
1857:.
1786:=
1750:.
1723:pq
1586:A
1572:A
1552:)
1548:,
1434:.
1378:.
1360:.
1314:.
1251:,
1211:,
1162:A
1148:.
1129:.
1076:th
1063:=
949:A
926:.
895:.
829:.
604:U(
580:E(
568:O(
26:â
2615:)
2613:n
2587:.
2554:.
1887:n
1863:n
1855:n
1851:n
1847:n
1843:n
1831:n
1827:n
1820:n
1816:n
1808:b
1804:a
1800:q
1796:p
1791:q
1788:p
1784:n
1779:n
1775:n
1763:n
1755:n
1744:q
1738:p
1732:p
1728:q
1715:n
1711:n
1704:n
1692:n
1688:n
1684:n
1668:n
1657:n
1597:;
1590:;
1498:G
1487:b
1483:a
1475:q
1471:p
1467:q
1464:p
1457:G
1432:G
1428:G
1420:G
1404:G
1376:G
1354:G
1350:G
1346:H
1335:G
1273:k
1253:q
1249:n
1245:q
1241:q
1225:p
1190:k
1183:G
1176:k
1174:(
1172:G
1083:a
1073:n
1065:a
1061:a
1056:a
1051:n
1032:n
1027:n
1018:n
1011:n
1007:n
991:n
979:n
970:n
967:S
957:4
955:S
790:e
783:t
776:v
672:8
670:E
664:7
662:E
656:6
654:E
648:4
646:F
640:2
638:G
632:)
630:n
620:)
618:n
608:)
606:n
596:)
594:n
584:)
582:n
572:)
570:n
560:)
558:n
548:)
546:n
488:)
475:Z
463:)
450:Z
426:)
413:Z
404:(
317:p
282:Q
274:n
271:D
261:n
258:A
250:n
247:S
239:n
236:Z
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