937:
58:
176:
1983:
1907:
2272:
2506:, φ(3) = 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2) = 1, and is only of limited utility for composite
1143:
2581:). This can often be used to prove that there are no homomorphisms or no injective homomorphisms, between two explicitly given groups. (For example, there can be no nontrivial homomorphism
1764:
2719:
1841:
2078:
As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord(S
2384:
621:
596:
559:
1929:
2813:
2780:
923:
1853:
2221:
2905:
1623:
and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the
122:
481:
94:
1107:
75:
2059:
1148:
431:
101:
2873:
2853:
2114:
916:
426:
2884:
141:
108:
17:
1683:
2645:
1786:
842:
90:
79:
909:
2491:
526:
340:
2352:
258:
2820:
2787:
1916:
936:
724:
458:
335:
223:
2450:
denotes the maximum of all the orders of the group's elements, then every element's order divides
604:
579:
542:
2185:
68:
42:
988:) is the order of the subgroup generated by the element. If the group operation is denoted as a
115:
874:
664:
35:
2900:
2435:
748:
46:
989:
949:
688:
676:
294:
228:
8:
2739:
of the representatives of the non-trivial conjugacy classes. For example, the center of S
2047:
1678:
969:
263:
158:
31:
2526:
1777:
248:
220:
2880:
2869:
2849:
2628:
653:
496:
390:
819:
2760:
2122:
2118:
1022:
997:
804:
796:
788:
780:
772:
760:
700:
640:
630:
472:
414:
289:
2735:| bigger than one, and they are also equal to the indices of the centralizers in
2731:
are the sizes of the non-trivial conjugacy classes; these are proper divisors of |
2636:
2608:
2126:
1847:
The relationship between the two concepts of order is the following: if we write
1216:
888:
881:
867:
824:
712:
635:
465:
379:
319:
199:
1978:{\displaystyle \operatorname {ord} (a)=\operatorname {ord} (\langle a\rangle ).}
30:
This article is about order in group theory. For other uses in mathematics, see
2620:
2168:
has finite order, we have the following formula for the order of the powers of
895:
831:
521:
501:
438:
403:
324:
314:
299:
284:
238:
215:
27:
Cardinality of a mathematical group, or of the subgroup generated by an element
2894:
1675:
1643:
1624:
945:
814:
736:
570:
443:
309:
2102:
2086:
1767:
1224:
965:
669:
368:
357:
304:
279:
274:
233:
204:
167:
2603:
has order 5, which does not divide the orders 1, 2, and 3 of elements in S
957:
2446:)). As a consequence, one can prove that in a finite abelian group, if
941:
836:
564:
2566:
1163:, the order of the subgroup divides the order of the group; that is,
657:
57:
1913:
1902:{\displaystyle \langle a\rangle =\{a^{k}\colon k\in \mathbb {Z} \}}
194:
2082:) = 6, the possible orders of the elements are 1, 2, 3 or 6.
2499:
2267:{\displaystyle \operatorname {ord} (ab)=\operatorname {ord} (ba)}
2008:
1174:
536:
450:
2514:= 6, since φ(6) = 2, and there are zero elements of order 6 in S
2129:. The consequences of the theorem include: the order of a group
2062:. (This is, however, only true when G has finite order. If ord(
175:
2125:
does not have an element of order four. This can be shown by
2277:
There is no general formula relating the order of a product
2494:, giving the number of positive integers no larger than
1766:. The converse is not true; for example, the (additive)
1138:{\displaystyle \operatorname {ord} (\langle a\rangle ),}
2648:
2355:
2224:
1932:
1856:
1789:
1686:
1110:
607:
582:
545:
82:. Unsourced material may be challenged and removed.
2743:is just the trivial group with the single element
2713:
2378:
2266:
1977:
1901:
1835:
1758:
1137:
944:with different orders: 90° rotation with order 4,
615:
590:
553:
2868:Dummit, David; Foote, Richard. Abstract Algebra,
2892:
2457:
2160:has infinite order, then all non-zero powers of
2521:
1145:where the brackets denote the generated group.
2313:has finite order. An example of the former is
917:
1966:
1960:
1896:
1869:
1863:
1857:
1780:6 is abelian, but the number 2 has order 3:
1759:{\displaystyle ab=(ab)^{-1}=b^{-1}a^{-1}=ba}
1515:. By definition, the order of the identity,
1126:
1120:
2714:{\displaystyle |G|=|Z(G)|+\sum _{i}d_{i}\;}
2085:The following partial converse is true for
2710:
2529:tend to reduce the orders of elements: if
1836:{\displaystyle 2+2+2=6\equiv 0{\pmod {6}}}
1646:. In any group, only the identity element
1555:, so these group elements have order two:
972:is not finite, one says that its order is
924:
910:
41:For groups with an ordering relation, see
2623:; it relates the order of a finite group
2369:
2018:In general, the order of any subgroup of
1892:
1631:|, the more complicated the structure of
609:
584:
547:
142:Learn how and when to remove this message
2619:An important result about orders is the
2105:, then there exists an element of order
935:
14:
2893:
2596:, because every number except zero in
1654:= 1. If every non-identity element in
1614:
980:of an element of a group (also called
482:Classification of finite simple groups
2502:to it. For example, in the case of S
2289:. In fact, it is possible that both
968:is the number of its elements. If a
80:adding citations to reliable sources
51:
2635:) and the sizes of its non-trivial
2117:). The statement does not hold for
1825:
24:
2814:"Consequences of Cauchy's Theorem"
2811:
2778:
25:
2917:
2614:
2607:.) A further consequence is that
2379:{\displaystyle Sym(\mathbb {Z} )}
2301:has infinite order, or that both
2003: if and only if ord(
1658:is equal to its inverse (so that
996:of a group, is thus the smallest
2906:Algebraic properties of elements
2164:have infinite order as well. If
1507:This group has six elements, so
174:
56:
2844:Dummit, David; Foote, Richard.
2430:, we can at least say that ord(
1818:
1197:of any element is a divisor of
948:with infinite order, and their
67:needs additional citations for
2838:
2805:
2772:
2683:
2679:
2673:
2666:
2658:
2650:
2490:) (possibly zero), where φ is
2386:. An example of the latter is
2373:
2365:
2261:
2252:
2240:
2231:
1969:
1957:
1945:
1939:
1829:
1819:
1706:
1696:
1129:
1117:
1080:, and the order of an element
843:Infinite dimensional Lie group
13:
1:
2862:
2458:Counting by order of elements
2093:divides the order of a group
1151:states that for any subgroup
2522:In relation to homomorphisms
616:{\displaystyle \mathbb {Z} }
591:{\displaystyle \mathbb {Z} }
554:{\displaystyle \mathbb {Z} }
7:
2781:"Proof of Cauchy's Theorem"
2754:
2747:, and the equation reads |S
2466:is a finite group of order
2046:) = , where is called the
1187:. In particular, the order
341:List of group theory topics
10:
2922:
2549:of finite order, then ord(
2309:have infinite order while
2113:(this is sometimes called
1210:
992:, the order of an element
40:
29:
2879:Artin, Michael. Algebra,
91:"Order" group theory
2766:
2492:Euler's totient function
2297:have finite order while
2066:) = ∞, the quotient ord(
459:Elementary abelian group
336:Glossary of group theory
2541:is a homomorphism, and
2478:. The number of order
2074:) does not make sense.)
1031:denotes the product of
43:partially ordered group
2715:
2380:
2268:
2133:is a power of a prime
2058:, an integer. This is
1979:
1903:
1837:
1760:
1139:
953:
875:Linear algebraic group
617:
592:
555:
36:Order (disambiguation)
34:. For other uses, see
2716:
2611:have the same order.
2381:
2269:
2212:have the same order.
2026:. More precisely: if
2022:divides the order of
1980:
1904:
1838:
1761:
1619:The order of a group
1140:
1058:The order of a group
1049:exists, the order of
939:
618:
593:
556:
47:totally ordered group
2646:
2627:to the order of its
2353:
2222:
1930:
1854:
1787:
1684:
1670:) = 2; this implies
1642:| = 1, the group is
1583:have order 3, since
1225:multiplication table
1108:
605:
580:
543:
76:improve this article
2876:, pp. 20, 54–59, 90
2527:Group homomorphisms
2486:is a multiple of φ(
2141:) is some power of
2137:if and only if ord(
1615:Order and structure
249:Group homomorphisms
159:Algebraic structure
32:Order (mathematics)
2711:
2699:
2609:conjugate elements
2376:
2264:
2200:for every integer
2060:Lagrange's theorem
1975:
1899:
1833:
1756:
1223:has the following
1157:of a finite group
1149:Lagrange's theorem
1135:
1025:of the group, and
954:
725:Special orthogonal
613:
588:
551:
432:Lagrange's theorem
2690:
2637:conjugacy classes
2545:is an element of
2281:to the orders of
2204:. In particular,
2121:orders, e.g. the
2030:is a subgroup of
1503:
1502:
934:
933:
509:
508:
391:Alternating group
348:
347:
152:
151:
144:
126:
16:(Redirected from
2913:
2857:
2846:Abstract Algebra
2842:
2836:
2835:
2833:
2831:
2825:
2819:. Archived from
2818:
2809:
2803:
2802:
2800:
2798:
2792:
2786:. Archived from
2785:
2776:
2761:Torsion subgroup
2751:| = 1+2+3.
2720:
2718:
2717:
2712:
2709:
2708:
2698:
2686:
2669:
2661:
2653:
2474:is a divisor of
2385:
2383:
2382:
2377:
2372:
2349:−1 in the group
2273:
2271:
2270:
2265:
2208:and its inverse
2123:Klein four-group
2115:Cauchy's theorem
1988:For any integer
1984:
1982:
1981:
1976:
1908:
1906:
1905:
1900:
1895:
1881:
1880:
1842:
1840:
1839:
1834:
1832:
1765:
1763:
1762:
1757:
1746:
1745:
1733:
1732:
1717:
1716:
1610:
1596:
1582:
1576:
1570:
1554:
1548:
1542:
1536:
1530:
1521:, is one, since
1520:
1514:
1499:
1455:
1411:
1367:
1323:
1279:
1232:
1231:
1206:
1204:
1196:
1194:
1186:
1184:
1172:
1170:
1162:
1156:
1144:
1142:
1141:
1136:
1103:
1101:
1093:
1085:
1079:
1077:
1069:
1061:
1054:
1048:
1042:
1036:
1030:
1023:identity element
1020:
1014:
1004:
998:positive integer
995:
926:
919:
912:
868:Algebraic groups
641:Hyperbolic group
631:Arithmetic group
622:
620:
619:
614:
612:
597:
595:
594:
589:
587:
560:
558:
557:
552:
550:
473:Schur multiplier
427:Cauchy's theorem
415:Quaternion group
363:
362:
189:
188:
178:
165:
154:
153:
147:
140:
136:
133:
127:
125:
84:
60:
52:
21:
18:Order of a group
2921:
2920:
2916:
2915:
2914:
2912:
2911:
2910:
2891:
2890:
2865:
2860:
2843:
2839:
2829:
2827:
2823:
2816:
2812:Conrad, Keith.
2810:
2806:
2796:
2794:
2790:
2783:
2779:Conrad, Keith.
2777:
2773:
2769:
2757:
2750:
2742:
2729:
2704:
2700:
2694:
2682:
2665:
2657:
2649:
2647:
2644:
2643:
2617:
2606:
2602:
2595:
2588:
2557:)) divides ord(
2524:
2517:
2505:
2460:
2368:
2354:
2351:
2350:
2223:
2220:
2219:
2215:In any group,
2127:inductive proof
2081:
1931:
1928:
1927:
1891:
1876:
1872:
1855:
1852:
1851:
1817:
1788:
1785:
1784:
1775:
1738:
1734:
1725:
1721:
1709:
1705:
1685:
1682:
1681:
1617:
1598:
1584:
1578:
1572:
1556:
1550:
1544:
1538:
1532:
1522:
1516:
1512:
1508:
1495:
1451:
1407:
1363:
1319:
1275:
1222:
1217:symmetric group
1213:
1200:
1198:
1190:
1188:
1180:
1178:
1166:
1164:
1158:
1152:
1109:
1106:
1105:
1097:
1095:
1087:
1081:
1073:
1071:
1063:
1059:
1050:
1044:
1038:
1032:
1026:
1016:
1006:
1000:
993:
942:transformations
930:
901:
900:
889:Abelian variety
882:Reductive group
870:
860:
859:
858:
857:
808:
800:
792:
784:
776:
749:Special unitary
660:
646:
645:
627:
626:
608:
606:
603:
602:
583:
581:
578:
577:
546:
544:
541:
540:
532:
531:
522:Discrete groups
511:
510:
466:Frobenius group
411:
398:
387:
380:Symmetric group
376:
360:
350:
349:
200:Normal subgroup
186:
166:
157:
148:
137:
131:
128:
85:
83:
73:
61:
50:
39:
28:
23:
22:
15:
12:
11:
5:
2919:
2909:
2908:
2903:
2889:
2888:
2877:
2874:978-0471433347
2864:
2861:
2859:
2858:
2854:978-0471433347
2837:
2804:
2770:
2768:
2765:
2764:
2763:
2756:
2753:
2748:
2740:
2727:
2722:
2721:
2707:
2703:
2697:
2693:
2689:
2685:
2681:
2678:
2675:
2672:
2668:
2664:
2660:
2656:
2652:
2621:class equation
2616:
2615:Class equation
2613:
2604:
2600:
2593:
2586:
2577:)) = ord(
2523:
2520:
2515:
2503:
2459:
2456:
2375:
2371:
2367:
2364:
2361:
2358:
2275:
2274:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2198:
2197:
2079:
2076:
2075:
2016:
2015:
1986:
1985:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1910:
1909:
1898:
1894:
1890:
1887:
1884:
1879:
1875:
1871:
1868:
1865:
1862:
1859:
1845:
1844:
1831:
1828:
1824:
1821:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1773:
1755:
1752:
1749:
1744:
1741:
1737:
1731:
1728:
1724:
1720:
1715:
1712:
1708:
1704:
1701:
1698:
1695:
1692:
1689:
1616:
1613:
1510:
1505:
1504:
1501:
1500:
1493:
1488:
1483:
1478:
1473:
1468:
1462:
1461:
1456:
1449:
1444:
1439:
1434:
1429:
1423:
1422:
1417:
1412:
1405:
1400:
1395:
1390:
1384:
1383:
1378:
1373:
1368:
1361:
1356:
1351:
1345:
1344:
1339:
1334:
1329:
1324:
1317:
1312:
1306:
1305:
1300:
1295:
1290:
1285:
1280:
1273:
1267:
1266:
1261:
1256:
1251:
1246:
1241:
1236:
1220:
1212:
1209:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1086:is denoted by
1062:is denoted by
1055:is infinite.
990:multiplication
932:
931:
929:
928:
921:
914:
906:
903:
902:
899:
898:
896:Elliptic curve
892:
891:
885:
884:
878:
877:
871:
866:
865:
862:
861:
856:
855:
852:
849:
845:
841:
840:
839:
834:
832:Diffeomorphism
828:
827:
822:
817:
811:
810:
806:
802:
798:
794:
790:
786:
782:
778:
774:
769:
768:
757:
756:
745:
744:
733:
732:
721:
720:
709:
708:
697:
696:
689:Special linear
685:
684:
677:General linear
673:
672:
667:
661:
652:
651:
648:
647:
644:
643:
638:
633:
625:
624:
611:
599:
586:
573:
571:Modular groups
569:
568:
567:
562:
549:
533:
530:
529:
524:
518:
517:
516:
513:
512:
507:
506:
505:
504:
499:
494:
491:
485:
484:
478:
477:
476:
475:
469:
468:
462:
461:
456:
447:
446:
444:Hall's theorem
441:
439:Sylow theorems
435:
434:
429:
421:
420:
419:
418:
412:
407:
404:Dihedral group
400:
399:
394:
388:
383:
377:
372:
361:
356:
355:
352:
351:
346:
345:
344:
343:
338:
330:
329:
328:
327:
322:
317:
312:
307:
302:
297:
295:multiplicative
292:
287:
282:
277:
269:
268:
267:
266:
261:
253:
252:
244:
243:
242:
241:
239:Wreath product
236:
231:
226:
224:direct product
218:
216:Quotient group
210:
209:
208:
207:
202:
197:
187:
184:
183:
180:
179:
171:
170:
150:
149:
64:
62:
55:
26:
9:
6:
4:
3:
2:
2918:
2907:
2904:
2902:
2899:
2898:
2896:
2886:
2885:0-13-004763-5
2882:
2878:
2875:
2871:
2867:
2866:
2855:
2851:
2847:
2841:
2826:on 2018-07-12
2822:
2815:
2808:
2793:on 2018-11-23
2789:
2782:
2775:
2771:
2762:
2759:
2758:
2752:
2746:
2738:
2734:
2730:
2705:
2701:
2695:
2691:
2687:
2676:
2670:
2662:
2654:
2642:
2641:
2640:
2638:
2634:
2630:
2626:
2622:
2612:
2610:
2599:
2592:
2589: →
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2537: →
2536:
2532:
2528:
2519:
2513:
2509:
2501:
2497:
2493:
2489:
2485:
2481:
2477:
2473:
2469:
2465:
2455:
2453:
2449:
2445:
2441:
2437:
2433:
2429:
2425:
2421:
2417:
2413:
2409:
2405:
2401:
2397:
2393:
2389:
2362:
2359:
2356:
2348:
2344:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2308:
2304:
2300:
2296:
2292:
2288:
2284:
2280:
2258:
2255:
2249:
2246:
2243:
2237:
2234:
2228:
2225:
2218:
2217:
2216:
2213:
2211:
2207:
2203:
2195:
2191:
2187:
2183:
2179:
2175:
2174:
2173:
2171:
2167:
2163:
2159:
2154:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2087:finite groups
2083:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2036:
2035:
2033:
2029:
2025:
2021:
2013:
2010:
2006:
2002:
1998:
1995:
1994:
1993:
1991:
1972:
1963:
1954:
1951:
1948:
1942:
1936:
1933:
1926:
1925:
1924:
1922:
1918:
1915:
1888:
1885:
1882:
1877:
1873:
1866:
1860:
1850:
1849:
1848:
1826:
1822:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1783:
1782:
1781:
1779:
1772:
1769:
1753:
1750:
1747:
1742:
1739:
1735:
1729:
1726:
1722:
1718:
1713:
1710:
1702:
1699:
1693:
1690:
1687:
1680:
1677:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1645:
1641:
1636:
1634:
1630:
1626:
1625:factorization
1622:
1612:
1609:
1605:
1601:
1595:
1591:
1587:
1581:
1575:
1568:
1564:
1560:
1553:
1547:
1541:
1535:
1529:
1525:
1519:
1498:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1464:
1463:
1460:
1457:
1454:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1424:
1421:
1418:
1416:
1413:
1410:
1406:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1385:
1382:
1379:
1377:
1374:
1372:
1369:
1366:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1346:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1322:
1318:
1316:
1313:
1311:
1308:
1307:
1304:
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1278:
1274:
1272:
1269:
1268:
1265:
1262:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1234:
1233:
1230:
1229:
1228:
1226:
1218:
1208:
1203:
1193:
1183:
1176:
1169:
1161:
1155:
1150:
1146:
1132:
1123:
1114:
1111:
1104:, instead of
1100:
1091:
1084:
1076:
1067:
1056:
1053:
1047:
1043:. If no such
1041:
1035:
1029:
1024:
1019:
1013:
1009:
1003:
999:
991:
987:
983:
982:period length
979:
975:
971:
967:
963:
959:
952:with order 3.
951:
947:
943:
938:
927:
922:
920:
915:
913:
908:
907:
905:
904:
897:
894:
893:
890:
887:
886:
883:
880:
879:
876:
873:
872:
869:
864:
863:
853:
850:
847:
846:
844:
838:
835:
833:
830:
829:
826:
823:
821:
818:
816:
813:
812:
809:
803:
801:
795:
793:
787:
785:
779:
777:
771:
770:
766:
762:
759:
758:
754:
750:
747:
746:
742:
738:
735:
734:
730:
726:
723:
722:
718:
714:
711:
710:
706:
702:
699:
698:
694:
690:
687:
686:
682:
678:
675:
674:
671:
668:
666:
663:
662:
659:
655:
650:
649:
642:
639:
637:
634:
632:
629:
628:
600:
575:
574:
572:
566:
563:
538:
535:
534:
528:
525:
523:
520:
519:
515:
514:
503:
500:
498:
495:
492:
489:
488:
487:
486:
483:
480:
479:
474:
471:
470:
467:
464:
463:
460:
457:
455:
453:
449:
448:
445:
442:
440:
437:
436:
433:
430:
428:
425:
424:
423:
422:
416:
413:
410:
405:
402:
401:
397:
392:
389:
386:
381:
378:
375:
370:
367:
366:
365:
364:
359:
358:Finite groups
354:
353:
342:
339:
337:
334:
333:
332:
331:
326:
323:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
272:
271:
270:
265:
262:
260:
257:
256:
255:
254:
251:
250:
246:
245:
240:
237:
235:
232:
230:
227:
225:
222:
219:
217:
214:
213:
212:
211:
206:
203:
201:
198:
196:
193:
192:
191:
190:
185:Basic notions
182:
181:
177:
173:
172:
169:
164:
160:
156:
155:
146:
143:
135:
124:
121:
117:
114:
110:
107:
103:
100:
96:
93: –
92:
88:
87:Find sources:
81:
77:
71:
70:
65:This article
63:
59:
54:
53:
48:
44:
37:
33:
19:
2901:Group theory
2845:
2840:
2828:. Retrieved
2821:the original
2807:
2795:. Retrieved
2788:the original
2774:
2744:
2736:
2732:
2725:
2723:
2632:
2624:
2618:
2597:
2590:
2582:
2578:
2574:
2570:
2562:
2558:
2554:
2550:
2546:
2542:
2538:
2534:
2530:
2525:
2511:
2507:
2495:
2487:
2483:
2482:elements in
2479:
2475:
2471:
2467:
2463:
2461:
2451:
2447:
2443:
2439:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2391:
2387:
2346:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2286:
2282:
2278:
2276:
2214:
2209:
2205:
2201:
2199:
2193:
2189:
2181:
2177:
2169:
2165:
2161:
2157:
2155:
2150:
2146:
2142:
2138:
2134:
2130:
2110:
2106:
2103:prime number
2098:
2094:
2090:
2084:
2077:
2071:
2067:
2063:
2055:
2051:
2043:
2039:
2031:
2027:
2023:
2019:
2017:
2011:
2004:
2000:
1996:
1989:
1987:
1920:
1911:
1846:
1776:of integers
1770:
1768:cyclic group
1671:
1667:
1666:), then ord(
1663:
1659:
1655:
1651:
1647:
1639:
1637:
1632:
1628:
1620:
1618:
1607:
1603:
1599:
1593:
1589:
1585:
1579:
1573:
1566:
1562:
1558:
1551:
1545:
1539:
1533:
1527:
1523:
1517:
1506:
1496:
1490:
1485:
1480:
1475:
1470:
1465:
1458:
1452:
1446:
1441:
1436:
1431:
1426:
1419:
1414:
1408:
1402:
1397:
1392:
1387:
1380:
1375:
1370:
1364:
1358:
1353:
1348:
1341:
1336:
1331:
1326:
1320:
1314:
1309:
1302:
1297:
1292:
1287:
1282:
1276:
1270:
1263:
1258:
1253:
1248:
1243:
1238:
1214:
1201:
1191:
1181:
1167:
1159:
1153:
1147:
1098:
1089:
1082:
1074:
1065:
1057:
1051:
1045:
1039:
1033:
1027:
1021:denotes the
1017:
1011:
1007:
1001:
985:
981:
977:
973:
966:finite group
961:
955:
950:compositions
940:Examples of
764:
752:
740:
728:
716:
704:
692:
680:
451:
408:
395:
384:
373:
369:Cyclic group
247:
234:Free product
205:Group action
168:Group theory
163:Group theory
162:
138:
129:
119:
112:
105:
98:
86:
74:Please help
69:verification
66:
2887:, pp. 46–47
2569:, then ord(
1571:. Finally,
1549:squares to
958:mathematics
654:Topological
493:alternating
2895:Categories
2863:References
2724:where the
2434:) divides
2145:for every
1992:, we have
1531:. Each of
1513:) = 6
1037:copies of
1005:such that
761:Symplectic
701:Orthogonal
658:Lie groups
565:Free group
290:continuous
229:Direct sum
102:newspapers
2692:∑
2567:injective
2250:
2229:
2119:composite
1967:⟩
1961:⟨
1955:
1937:
1917:generated
1889:∈
1883::
1864:⟩
1858:⟨
1812:≡
1740:−
1727:−
1711:−
1127:⟩
1121:⟨
1115:
825:Conformal
713:Euclidean
320:nilpotent
2856:, pp. 57
2755:See also
2585:: S
2510:such as
2462:Suppose
2410:−1 with
2180:) = ord(
2070:) / ord(
2042:) / ord(
1914:subgroup
1912:for the
1650:has ord(
1606: =
1602: =
1592: =
1588: =
1015:, where
974:infinite
946:shearing
820:Poincaré
665:Solenoid
537:Integers
527:Lattices
502:sporadic
497:Lie type
325:solvable
315:dihedral
300:additive
285:infinite
195:Subgroup
132:May 2011
2830:May 14,
2797:May 14,
2533::
2500:coprime
2442:), ord(
2034:, then
2009:divides
1923:, then
1676:abelian
1644:trivial
1211:Example
1175:divisor
815:Lorentz
737:Unitary
636:Lattice
576:PSL(2,
310:abelian
221:(Semi-)
116:scholar
2883:
2872:
2852:
2629:center
2561:). If
2470:, and
2333:) = 1−
2321:) = 2−
1778:modulo
1638:For |
1597:, and
1543:, and
1205:|
1199:|
1195:|
1189:|
1185:|
1179:|
1171:|
1165:|
1102:|
1096:|
1078:|
1072:|
986:period
976:. The
960:, the
670:Circle
601:SL(2,
490:cyclic
454:-group
305:cyclic
280:finite
275:simple
259:kernel
118:
111:
104:
97:
89:
2824:(PDF)
2817:(PDF)
2791:(PDF)
2784:(PDF)
2767:Notes
2438:(ord(
2422:. If
2337:with
2188:(ord(
2101:is a
2089:: if
2048:index
1679:since
1648:a = e
1569:| = 2
1565:| = |
1561:| = |
1509:ord(S
1173:is a
978:order
970:group
964:of a
962:order
854:Sp(∞)
851:SU(∞)
264:image
123:JSTOR
109:books
2881:ISBN
2870:ISBN
2850:ISBN
2832:2011
2799:2011
2498:and
2418:) =
2406:) =
2398:+1,
2394:) =
2345:) =
2305:and
2293:and
2285:and
2184:) /
2176:ord(
2097:and
2038:ord(
1627:of |
1577:and
1215:The
1088:ord(
1064:ord(
848:O(∞)
837:Loop
656:and
95:news
45:and
2565:is
2436:lcm
2247:ord
2226:ord
2192:),
2186:gcd
2156:If
2149:in
2109:in
2054:in
2050:of
1952:ord
1934:ord
1919:by
1823:mod
1674:is
1177:of
1112:ord
1094:or
1070:or
984:or
956:In
763:Sp(
751:SU(
727:SO(
691:SL(
679:GL(
78:by
2897::
2848:,
2639::
2631:Z(
2518:.
2454:.
2432:ab
2428:ba
2426:=
2424:ab
2412:ab
2339:ab
2325:,
2311:ab
2299:ab
2279:ab
2172::
2153:.
2007:)
1999:=
1662:=
1652:a)
1635:.
1611:.
1604:uv
1590:vu
1537:,
1526:=
1235:•
1227:.
1207:.
1010:=
739:U(
715:E(
703:O(
161:→
2834:.
2801:.
2749:3
2745:e
2741:3
2737:G
2733:G
2728:i
2726:d
2706:i
2702:d
2696:i
2688:+
2684:|
2680:)
2677:G
2674:(
2671:Z
2667:|
2663:=
2659:|
2655:G
2651:|
2633:G
2625:G
2605:3
2601:5
2598:Z
2594:5
2591:Z
2587:3
2583:h
2579:a
2575:a
2573:(
2571:f
2563:f
2559:a
2555:a
2553:(
2551:f
2547:G
2543:a
2539:H
2535:G
2531:f
2516:3
2512:d
2508:d
2504:3
2496:d
2488:d
2484:G
2480:d
2476:n
2472:d
2468:n
2464:G
2452:m
2448:m
2444:b
2440:a
2420:x
2416:x
2414:(
2408:x
2404:x
2402:(
2400:b
2396:x
2392:x
2390:(
2388:a
2374:)
2370:Z
2366:(
2363:m
2360:y
2357:S
2347:x
2343:x
2341:(
2335:x
2331:x
2329:(
2327:b
2323:x
2319:x
2317:(
2315:a
2307:b
2303:a
2295:b
2291:a
2287:b
2283:a
2262:)
2259:a
2256:b
2253:(
2244:=
2241:)
2238:b
2235:a
2232:(
2210:a
2206:a
2202:k
2196:)
2194:k
2190:a
2182:a
2178:a
2170:a
2166:a
2162:a
2158:a
2151:G
2147:a
2143:p
2139:a
2135:p
2131:G
2111:G
2107:d
2099:d
2095:G
2091:d
2080:3
2072:H
2068:G
2064:G
2056:G
2052:H
2044:H
2040:G
2032:G
2028:H
2024:G
2020:G
2014:.
2012:k
2005:a
2001:e
1997:a
1990:k
1973:.
1970:)
1964:a
1958:(
1949:=
1946:)
1943:a
1940:(
1921:a
1897:}
1893:Z
1886:k
1878:k
1874:a
1870:{
1867:=
1861:a
1843:.
1830:)
1827:6
1820:(
1815:0
1809:6
1806:=
1803:2
1800:+
1797:2
1794:+
1791:2
1774:6
1771:Z
1754:a
1751:b
1748:=
1743:1
1736:a
1730:1
1723:b
1719:=
1714:1
1707:)
1703:b
1700:a
1697:(
1694:=
1691:b
1688:a
1672:G
1668:a
1664:e
1660:a
1656:G
1640:G
1633:G
1629:G
1621:G
1608:e
1600:v
1594:e
1586:u
1580:v
1574:u
1567:w
1563:t
1559:s
1557:|
1552:e
1546:w
1540:t
1534:s
1528:e
1524:e
1518:e
1511:3
1497:e
1491:s
1486:t
1481:u
1476:v
1471:w
1466:w
1459:t
1453:u
1447:e
1442:s
1437:w
1432:v
1427:v
1420:s
1415:e
1409:v
1403:w
1398:t
1393:u
1388:u
1381:v
1376:w
1371:s
1365:e
1359:u
1354:t
1349:t
1342:u
1337:t
1332:w
1327:v
1321:e
1315:s
1310:s
1303:w
1298:v
1293:u
1288:t
1283:s
1277:e
1271:e
1264:w
1259:v
1254:u
1249:t
1244:s
1239:e
1221:3
1219:S
1202:G
1192:a
1182:G
1168:H
1160:G
1154:H
1133:,
1130:)
1124:a
1118:(
1099:a
1092:)
1090:a
1083:a
1075:G
1068:)
1066:G
1060:G
1052:a
1046:m
1040:a
1034:m
1028:a
1018:e
1012:e
1008:a
1002:m
994:a
925:e
918:t
911:v
807:8
805:E
799:7
797:E
791:6
789:E
783:4
781:F
775:2
773:G
767:)
765:n
755:)
753:n
743:)
741:n
731:)
729:n
719:)
717:n
707:)
705:n
695:)
693:n
683:)
681:n
623:)
610:Z
598:)
585:Z
561:)
548:Z
539:(
452:p
417:Q
409:n
406:D
396:n
393:A
385:n
382:S
374:n
371:Z
145:)
139:(
134:)
130:(
120:·
113:·
106:·
99:·
72:.
49:.
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
