1670:
52:
1259:
1408:
1025:
2255:
2010:
2067:
1105:
3452:
Dixon, M. R.; Kirichenko, V. V.; Kurdachenko, L. A.; Otal, J.; Semko, N. N.; Shemetkov, L. A.; Subbotin, I. Ya. (2012). "S. N. Chernikov and the development of infinite group theory".
1563:
2325:
3563:
3529:
2134:
497:
472:
435:
1452:
1163:
1319:
1485:
1629:
799:
2365:
2345:
2295:
2275:
2198:
2178:
2158:
1971:
1951:
1931:
1187:
1336:
953:
1653:
It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class
1898:
3492:
3546:
357:
3509:
3691:
307:
3475:
792:
302:
3794:
3768:
3749:
3714:
3665:
3643:
2203:
718:
3818:
3741:
3683:
1976:
785:
900:
402:
216:
2386:
An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).
3706:
2015:
1032:
3813:
1793:
2532:
2380:
1638:
equals the length of the lower central series or upper central series. If a group has nilpotency class at most
600:
334:
211:
99:
3420:
is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in
1490:
2303:
2886:
2458:
The following statements are equivalent for finite groups, revealing some useful properties of nilpotency:
750:
540:
1811:, an example of a non-abelian infinite nilpotent group. It has nilpotency class 2 with central series 1,
1704:
905:
624:
3361:
is a direct product of its Sylow subgroups, and normality is preserved upon direct product of groups,
3653:
2372:
2072:
1328:
1172:
870:
866:
844:
832:
564:
552:
170:
104:
888:, as well as in the classification of groups. They also appear prominently in the classification of
874:
828:
139:
34:
480:
455:
418:
3786:
3432:
2440:
1764:
1331:
terminating in the whole group after finitely many steps. That is, a series of normal subgroups
124:
96:
3579:
1415:
1110:
3790:
3764:
3745:
3710:
3687:
3661:
3639:
3635:
1778:
1266:
695:
529:
372:
266:
1457:
3425:
2543:
2371:, and need not be nilpotent in general. They are proven to be nilpotent if they have finite
1805:
1756:
1708:
1690:
1678:
878:
863:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
134:
3778:
2555:
2482:
1254:{\displaystyle G=G_{0}\triangleright G_{1}\triangleright \dots \triangleright G_{n}=\{1\}}
946:
764:
757:
743:
700:
588:
511:
341:
255:
75:
1608:
1631:
different subgroups in the series, including the trivial subgroup and the whole group.)
3622:
2518:
2421:
2417:
2350:
2330:
2280:
2260:
2183:
2163:
2143:
1956:
1936:
1916:
1881:
1842:
1827:
1760:
1722:
1403:{\displaystyle \{1\}=Z_{0}\triangleleft Z_{1}\triangleleft \dots \triangleleft Z_{n}=G}
1020:{\displaystyle \{1\}=G_{0}\triangleleft G_{1}\triangleleft \dots \triangleleft G_{n}=G}
942:
926:
855:
840:
771:
707:
397:
377:
314:
279:
200:
190:
175:
160:
114:
91:
1669:
3807:
3675:
1176:
885:
851:
690:
612:
446:
319:
185:
2747:) is a proper subgroup of it. Therefore, pullback this subgroup to the subgroup in
2395:
1674:
859:
817:
545:
244:
233:
180:
155:
150:
109:
80:
43:
17:
3730:
3392:
2368:
896:
813:
2499:
2376:
2137:
1909:
Nilpotent groups are called so because the "adjoint action" of any element is
712:
440:
2300:
This is not a defining characteristic of nilpotent groups: groups for which
1910:
929:
for a group. The following are equivalent definitions for a nilpotent group
889:
533:
51:
1179:
70:
877:. The concept is credited to work in the 1930s by Russian mathematician
3338:
1736:
1729:
412:
326:
3254:
are relatively prime. Lagrange's
Theorem implies the intersection of
3399:
is normal. Thus we can apply (c) (since we already proved (c)→(e)).
1770:
Furthermore, every finite nilpotent group is the direct product of
3494:
Topics in Group Theory (Springer
Undergraduate Mathematics Series)
1668:
1182:
after finitely many steps. That is, a series of normal subgroups
2420:
is abelian, and the series is finite, every nilpotent group is a
854:". This idea is motivated by the fact that nilpotent groups are
3728:. Lecture Notes in Mathematics. Vol. 359. Springer-Verlag.
3634:. De Gruyter Expositions in Mathematics. Vol. 36. Berlin:
3548:
Automatic
Sequences (De Gruyter Expositions in Mathematics, 36)
1755:
is abelian). The 2-groups of maximal class are the generalised
1864:) is abelian has nilpotency class 2, with central series {1},
2250:{\displaystyle \left(\operatorname {ad} _{g}\right)^{n}(x)=e}
1893:
850:
Intuitively, a nilpotent group is a group that is "almost
2711:) is a nilpotent group. Thus, there exists a subgroup of
1743:> 1, the maximal nilpotency class of a group of order
2759:-groups – the only fact we needed was if
3763:. Springer Undergraduate Mathematics Series. Springer.
3403:
Statement (d) can be extended to infinite groups: if
2353:
2333:
2306:
2283:
2263:
2206:
2186:
2166:
2146:
2075:
2018:
1979:
1959:
1939:
1919:
1611:
1493:
1460:
1418:
1339:
1269:
1190:
1113:
1035:
956:
483:
458:
421:
3703:
Banach
Algebras and the General Theory of *-algebras
3565:
Banach algebras and the general theory of *-algebras
2005:{\displaystyle \operatorname {ad} _{g}\colon G\to G}
873:. It is also true that finite nilpotent groups are
3431:Many properties of nilpotent groups are shared by
3006:be the distinct primes dividing its order and let
2804:be the distinct primes dividing its order and let
2359:
2339:
2319:
2289:
2269:
2249:
2192:
2172:
2152:
2128:
2061:
2004:
1965:
1945:
1925:
1623:
1557:
1479:
1446:
1402:
1313:
1253:
1157:
1099:
1019:
491:
466:
429:
1711:2, and its upper central series is {1}, {1, −1},
1686:As noted above, every abelian group is nilpotent.
3407:is a nilpotent group, then every Sylow subgroup
2514:and can be phrased simply as "normalizers grow".
3469:
3467:
2775:) – so the details are omitted.)
1891:is nilpotent have been characterized (sequence
1689:For a small non-abelian example, consider the
2427:Every subgroup of a nilpotent group of class
793:
8:
3523:
3521:
2062:{\displaystyle \operatorname {ad} _{g}(x):=}
1661:are exactly the non-trivial abelian groups.
1346:
1340:
1248:
1242:
1100:{\displaystyle G_{i+1}/G_{i}\leq Z(G/G_{i})}
963:
957:
3590:
3588:
2755:. (This proof is the same argument as for
800:
786:
238:
64:
29:
3759:Tabachnikova, Olga; Smith, Geoff (2000).
3323:. This completes the induction. Now take
2352:
2332:
2311:
2305:
2282:
2262:
2226:
2216:
2205:
2200:th iteration of the function is trivial:
2185:
2165:
2145:
2111:
2098:
2074:
2023:
2017:
1984:
1978:
1958:
1938:
1918:
1610:
1546:
1537:
1519:
1510:
1498:
1492:
1465:
1459:
1423:
1417:
1388:
1369:
1356:
1338:
1296:
1274:
1268:
1233:
1214:
1201:
1189:
1149:
1127:
1112:
1088:
1079:
1061:
1052:
1040:
1034:
1005:
986:
973:
955:
485:
484:
482:
460:
459:
457:
423:
422:
420:
1558:{\displaystyle Z_{i+1}/Z_{i}=Z(G/Z_{i})}
3444:
2320:{\displaystyle \operatorname {ad} _{g}}
945:of finite length. That is, a series of
356:
122:
32:
1657:, and groups of nilpotency class
1634:Equivalently, the nilpotency class of
862:nilpotent groups, two elements having
358:Classification of finite simple groups
27:Concept in group theory of mathematics
3594:Bechtell (1971), p. 51, Theorem 5.1.3
2180:) is nilpotent in the sense that the
1913:, meaning that for a nilpotent group
1725:of two nilpotent groups is nilpotent.
7:
2424:with a relatively simple structure.
2379:to be nilpotent as long as they are
1841:is not in general nilpotent, but is
1570:For a nilpotent group, the smallest
1751:- 1 (for example, a group of order
1777:The multiplicative group of upper
1699:, which is a smallest non-abelian
925:The definition uses the idea of a
25:
3491:Tabachnikova & Smith (2000).
3630:Von Haeseler, Friedrich (2002).
3454:Algebra and Discrete Mathematics
2969:. By (b) we must therefore have
2347:(in the sense above) are called
1718:; so it is nilpotent of class 2.
1642:, then it is sometimes called a
1601:. (By definition, the length is
50:
3365:has a normal subgroup of order
3345:has a normal subgroup of order
2129:{\displaystyle =g^{-1}x^{-1}gx}
1681:, a well-known nilpotent group.
1578:has a central series of length
2877:is a normal Sylow subgroup of
2451:is nilpotent of class at most
2443:of a nilpotent group of class
2431:is nilpotent of class at most
2238:
2232:
2088:
2076:
2056:
2044:
2038:
2032:
1996:
1552:
1531:
1441:
1435:
1308:
1289:
1139:
1114:
1094:
1073:
719:Infinite dimensional Lie group
1:
3742:American Mathematical Society
3684:American Mathematical Society
3262:is equal to 1. By definition,
1887:for which any group of order
895:Analogous terms are used for
3740:. Providence, Rhode Island:
3701:Palmer, Theodore W. (1994).
1826:The multiplicative group of
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
1828:invertible upper triangular
1800:− 1. In particular, taking
217:List of group theory topics
3835:
3736:Suprunenko, D. A. (1976).
3707:Cambridge University Press
1487:is the subgroup such that
1447:{\displaystyle Z_{1}=Z(G)}
1158:{\displaystyle \leq G_{i}}
884:Nilpotent groups arise in
3053:we show inductively that
2582:is abelian, then for any
839:. Equivalently, it has a
3726:Homology in Group Theory
3617:Bechtell, Homer (1971).
3603:Isaacs (2008), Thm. 1.26
2922:is a normal subgroup of
2905:is a normal subgroup of
2763:is nilpotent then so is
2473:is a proper subgroup of
1788:matrices over any field
1314:{\displaystyle G_{i+1}=}
843:of finite length or its
335:Elementary abelian group
212:Glossary of group theory
3724:Stammbach, Urs (1973).
3098:. Note first that each
2535:of its Sylow subgroups.
2510:). This is called the
2327:is nilpotent of degree
1735:are in fact nilpotent (
1480:{\displaystyle Z_{i+1}}
3761:Topics in Group Theory
3578:For the term, compare
2611:) is not contained in
2394:Since each successive
2361:
2341:
2321:
2291:
2271:
2251:
2194:
2174:
2154:
2130:
2063:
2006:
1967:
1947:
1927:
1837:matrices over a field
1682:
1625:
1559:
1481:
1448:
1404:
1315:
1255:
1159:
1101:
1021:
751:Linear algebraic group
493:
468:
431:
3654:Hungerford, Thomas W.
3582:, also on nilpotency.
2466:is a nilpotent group.
2362:
2342:
2322:
2292:
2272:
2252:
2195:
2175:
2155:
2131:
2064:
2007:
1968:
1948:
1933:of nilpotence degree
1928:
1848:Any nonabelian group
1672:
1626:
1560:
1482:
1449:
1405:
1316:
1256:
1160:
1102:
1022:
847:terminates with {1}.
835:that terminates with
494:
469:
432:
3819:Properties of groups
3783:The Theory of Groups
3619:The Theory of Groups
3531:The theory of groups
2447:, then the image of
2418:upper central series
2351:
2331:
2304:
2281:
2261:
2204:
2184:
2164:
2144:
2073:
2016:
1977:
1957:
1937:
1917:
1796:of nilpotency class
1609:
1491:
1458:
1416:
1337:
1329:upper central series
1267:
1188:
1173:lower central series
1111:
1033:
954:
915:upper central series
911:lower central series
845:lower central series
833:upper central series
481:
456:
419:
3680:Finite Group Theory
3660:. Springer-Verlag.
3632:Automatic Sequences
3528:Zassenhaus (1999).
3508:Hungerford (1974).
3474:Suprunenko (1976).
3433:hypercentral groups
3223:|⋅···⋅|
2948:) is a subgroup of
2723:) which normalizes
2512:normalizer property
1905:Explanation of term
1765:semidihedral groups
1624:{\displaystyle n+1}
1596:nilpotent of class
1175:terminating in the
125:Group homomorphisms
35:Algebraic structure
18:Nilpotent Lie group
3787:Dover Publications
3369:for every divisor
3300:which is equal to
3177:, so by induction
2977:, which gives (c).
2751:and it normalizes
2687:) is contained in
2671:) is contained in
2435:; in addition, if
2381:finitely generated
2357:
2337:
2317:
2287:
2267:
2247:
2190:
2170:
2150:
2126:
2059:
2002:
1963:
1943:
1923:
1683:
1621:
1555:
1477:
1444:
1400:
1311:
1251:
1155:
1107:, or equivalently
1097:
1017:
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
3693:978-0-8218-4344-4
3676:Isaacs, I. Martin
3636:Walter de Gruyter
3545:Haeseler (2002).
3292:is isomorphic to
3246:|, the orders of
3205:. In particular,|
3181:is isomorphic to
3133:is a subgroup of
3075:is isomorphic to
2574:By induction on |
2360:{\displaystyle n}
2340:{\displaystyle n}
2290:{\displaystyle G}
2270:{\displaystyle x}
2193:{\displaystyle n}
2173:{\displaystyle x}
2153:{\displaystyle g}
1966:{\displaystyle g}
1946:{\displaystyle n}
1926:{\displaystyle G}
1757:quaternion groups
1673:A portion of the
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
3826:
3814:Nilpotent groups
3800:
3779:Zassenhaus, Hans
3774:
3755:
3729:
3720:
3697:
3671:
3649:
3626:
3604:
3601:
3595:
3592:
3583:
3576:
3570:
3569:
3559:
3553:
3552:
3542:
3536:
3535:
3525:
3516:
3515:
3505:
3499:
3498:
3488:
3482:
3481:
3471:
3462:
3461:
3449:
3426:torsion subgroup
2366:
2364:
2363:
2358:
2346:
2344:
2343:
2338:
2326:
2324:
2323:
2318:
2316:
2315:
2296:
2294:
2293:
2288:
2276:
2274:
2273:
2268:
2256:
2254:
2253:
2248:
2231:
2230:
2225:
2221:
2220:
2199:
2197:
2196:
2191:
2179:
2177:
2176:
2171:
2159:
2157:
2156:
2151:
2135:
2133:
2132:
2127:
2119:
2118:
2106:
2105:
2068:
2066:
2065:
2060:
2028:
2027:
2011:
2009:
2008:
2003:
1989:
1988:
1972:
1970:
1969:
1964:
1952:
1950:
1949:
1944:
1932:
1930:
1929:
1924:
1896:
1806:Heisenberg group
1691:quaternion group
1679:Heisenberg group
1677:of the discrete
1660:
1656:
1647:
1641:
1637:
1630:
1628:
1627:
1622:
1604:
1599:
1593:
1589:
1584:nilpotency class
1581:
1577:
1573:
1564:
1562:
1561:
1556:
1551:
1550:
1541:
1524:
1523:
1514:
1509:
1508:
1486:
1484:
1483:
1478:
1476:
1475:
1453:
1451:
1450:
1445:
1428:
1427:
1409:
1407:
1406:
1401:
1393:
1392:
1374:
1373:
1361:
1360:
1326:
1320:
1318:
1317:
1312:
1301:
1300:
1285:
1284:
1260:
1258:
1257:
1252:
1238:
1237:
1219:
1218:
1206:
1205:
1170:
1164:
1162:
1161:
1156:
1154:
1153:
1138:
1137:
1106:
1104:
1103:
1098:
1093:
1092:
1083:
1066:
1065:
1056:
1051:
1050:
1026:
1024:
1023:
1018:
1010:
1009:
991:
990:
978:
977:
947:normal subgroups
940:
932:
879:Sergei Chernikov
864:relatively prime
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:
65:
54:
41:
30:
21:
3834:
3833:
3829:
3828:
3827:
3825:
3824:
3823:
3804:
3803:
3797:
3777:
3771:
3758:
3752:
3735:
3723:
3717:
3700:
3694:
3674:
3668:
3652:
3646:
3629:
3616:
3613:
3608:
3607:
3602:
3598:
3593:
3586:
3580:Engel's theorem
3577:
3573:
3568:. p. 1283.
3562:Palmer (2001).
3561:
3560:
3556:
3544:
3543:
3539:
3527:
3526:
3519:
3507:
3506:
3502:
3490:
3489:
3485:
3473:
3472:
3465:
3451:
3450:
3446:
3441:
3415:
3322:
3313:
3306:
3283:
3274:
3268:
3245:
3232:
3222:
3215:
3204:
3194:
3187:
3176:
3163:
3153:
3147:
3141:be the product
3132:
3123:
3117:
3106:
3097:
3088:
3081:
3074:
3065:
3059:
3028:
3027:
3014:
3005:
2996:
2989:
2960:
2943:
2930:
2918:), we get that
2913:
2868:
2851:
2826:
2825:
2812:
2803:
2794:
2787:
2631:
2623:
2594:
2565:
2556:normal subgroup
2493:
2483:normal subgroup
2415:
2406:
2392:
2349:
2348:
2329:
2328:
2307:
2302:
2301:
2279:
2278:
2259:
2258:
2212:
2208:
2207:
2202:
2201:
2182:
2181:
2162:
2161:
2142:
2141:
2107:
2094:
2071:
2070:
2019:
2014:
2013:
1980:
1975:
1974:
1973:, the function
1955:
1954:
1953:and an element
1935:
1934:
1915:
1914:
1907:
1892:
1882:natural numbers
1804:= 3 yields the
1794:nilpotent group
1761:dihedral groups
1717:
1703:-group. It has
1698:
1667:
1658:
1654:
1645:
1639:
1635:
1607:
1606:
1602:
1597:
1591:
1587:
1579:
1575:
1571:
1568:
1542:
1515:
1494:
1489:
1488:
1461:
1456:
1455:
1419:
1414:
1413:
1384:
1365:
1352:
1335:
1334:
1324:
1292:
1270:
1265:
1264:
1229:
1210:
1197:
1186:
1185:
1168:
1145:
1123:
1109:
1108:
1084:
1057:
1036:
1031:
1030:
1001:
982:
969:
952:
951:
938:
930:
923:
822:nilpotent group
816:, specifically
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:
479:
478:
454:
453:
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:
3832:
3830:
3822:
3821:
3816:
3806:
3805:
3802:
3801:
3795:
3775:
3769:
3756:
3750:
3733:
3721:
3715:
3698:
3692:
3672:
3666:
3650:
3644:
3627:
3623:Addison-Wesley
3612:
3609:
3606:
3605:
3596:
3584:
3571:
3554:
3537:
3534:. p. 143.
3517:
3514:. p. 100.
3500:
3497:. p. 169.
3483:
3480:. p. 205.
3463:
3443:
3442:
3440:
3437:
3411:
3401:
3400:
3383:For any prime
3381:
3378:
3335:
3332:
3331:to obtain (d).
3318:
3311:
3304:
3279:
3272:
3266:
3241:
3227:
3220:
3213:
3199:
3192:
3185:
3172:
3158:
3151:
3145:
3128:
3121:
3115:
3102:
3093:
3086:
3079:
3070:
3063:
3057:
3023:
3019:
3010:
3001:
2994:
2987:
2981:
2978:
2956:
2939:
2935:). This means
2926:
2909:
2887:characteristic
2864:
2847:
2821:
2817:
2808:
2799:
2792:
2785:
2779:
2776:
2659:) normalizers
2627:
2619:
2590:
2572:
2564:
2563:
2536:
2533:direct product
2526:
2519:Sylow subgroup
2515:
2489:
2467:
2460:
2422:solvable group
2411:
2401:
2391:
2388:
2356:
2336:
2314:
2310:
2286:
2266:
2246:
2243:
2240:
2237:
2234:
2229:
2224:
2219:
2215:
2211:
2189:
2169:
2149:
2125:
2122:
2117:
2114:
2110:
2104:
2101:
2097:
2093:
2090:
2087:
2084:
2081:
2078:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2026:
2022:
2001:
1998:
1995:
1992:
1987:
1983:
1962:
1942:
1922:
1906:
1903:
1878:
1877:
1846:
1824:
1775:
1768:
1726:
1723:direct product
1719:
1715:
1696:
1687:
1666:
1663:
1620:
1617:
1614:
1594:is said to be
1582:is called the
1567:
1566:
1554:
1549:
1545:
1540:
1536:
1533:
1530:
1527:
1522:
1518:
1513:
1507:
1504:
1501:
1497:
1474:
1471:
1468:
1464:
1443:
1440:
1437:
1434:
1431:
1426:
1422:
1411:
1410:
1399:
1396:
1391:
1387:
1383:
1380:
1377:
1372:
1368:
1364:
1359:
1355:
1351:
1348:
1345:
1342:
1322:
1310:
1307:
1304:
1299:
1295:
1291:
1288:
1283:
1280:
1277:
1273:
1262:
1261:
1250:
1247:
1244:
1241:
1236:
1232:
1228:
1225:
1222:
1217:
1213:
1209:
1204:
1200:
1196:
1193:
1166:
1152:
1148:
1144:
1141:
1136:
1133:
1130:
1126:
1122:
1119:
1116:
1096:
1091:
1087:
1082:
1078:
1075:
1072:
1069:
1064:
1060:
1055:
1049:
1046:
1043:
1039:
1028:
1027:
1016:
1013:
1008:
1004:
1000:
997:
994:
989:
985:
981:
976:
972:
968:
965:
962:
959:
943:central series
935:
927:central series
922:
919:
841:central series
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3831:
3820:
3817:
3815:
3812:
3811:
3809:
3798:
3796:0-486-40922-8
3792:
3788:
3784:
3780:
3776:
3772:
3770:1-85233-235-2
3766:
3762:
3757:
3753:
3751:0-8218-1341-2
3747:
3743:
3739:
3738:Matrix Groups
3734:
3732:
3727:
3722:
3718:
3716:0-521-36638-0
3712:
3708:
3704:
3699:
3695:
3689:
3685:
3681:
3677:
3673:
3669:
3667:0-387-90518-9
3663:
3659:
3655:
3651:
3647:
3645:3-11-015629-6
3641:
3637:
3633:
3628:
3624:
3620:
3615:
3614:
3610:
3600:
3597:
3591:
3589:
3585:
3581:
3575:
3572:
3567:
3566:
3558:
3555:
3551:. p. 15.
3550:
3549:
3541:
3538:
3533:
3532:
3524:
3522:
3518:
3513:
3512:
3504:
3501:
3496:
3495:
3487:
3484:
3479:
3478:
3477:Matrix Groups
3470:
3468:
3464:
3460:(2): 169–208.
3459:
3455:
3448:
3445:
3438:
3436:
3434:
3429:
3427:
3423:
3419:
3414:
3410:
3406:
3398:
3396:
3390:
3386:
3382:
3379:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3336:
3333:
3330:
3326:
3321:
3317:
3310:
3303:
3299:
3295:
3291:
3287:
3282:
3278:
3271:
3265:
3261:
3257:
3253:
3249:
3244:
3240:
3236:
3230:
3226:
3219:
3212:
3208:
3202:
3198:
3191:
3184:
3180:
3175:
3171:
3167:
3161:
3157:
3150:
3144:
3140:
3136:
3131:
3127:
3120:
3114:
3110:
3107:is normal in
3105:
3101:
3096:
3092:
3085:
3078:
3073:
3069:
3062:
3056:
3052:
3048:
3044:
3040:
3036:
3032:
3026:
3022:
3018:
3013:
3009:
3004:
3000:
2993:
2986:
2982:
2979:
2976:
2972:
2968:
2964:
2959:
2955:
2951:
2947:
2942:
2938:
2934:
2929:
2925:
2921:
2917:
2912:
2908:
2904:
2900:
2896:
2892:
2888:
2884:
2880:
2876:
2872:
2867:
2863:
2859:
2855:
2850:
2846:
2842:
2838:
2834:
2830:
2824:
2820:
2816:
2811:
2807:
2802:
2798:
2791:
2784:
2780:
2777:
2774:
2770:
2766:
2762:
2758:
2754:
2750:
2746:
2742:
2738:
2734:
2730:
2726:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2662:
2658:
2654:
2650:
2646:
2642:
2640:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2606:
2603:. If not, if
2602:
2598:
2593:
2589:
2585:
2581:
2577:
2573:
2570:
2569:
2568:
2561:
2557:
2553:
2549:
2545:
2541:
2537:
2534:
2530:
2527:
2524:
2520:
2516:
2513:
2509:
2505:
2501:
2497:
2492:
2488:
2484:
2480:
2476:
2472:
2468:
2465:
2462:
2461:
2459:
2456:
2454:
2450:
2446:
2442:
2438:
2434:
2430:
2425:
2423:
2419:
2414:
2410:
2404:
2400:
2397:
2389:
2387:
2384:
2382:
2378:
2374:
2370:
2354:
2334:
2312:
2308:
2298:
2284:
2264:
2244:
2241:
2235:
2227:
2222:
2217:
2213:
2209:
2187:
2167:
2147:
2139:
2123:
2120:
2115:
2112:
2108:
2102:
2099:
2095:
2091:
2085:
2082:
2079:
2053:
2050:
2047:
2041:
2035:
2029:
2024:
2020:
1999:
1993:
1990:
1985:
1981:
1960:
1940:
1920:
1912:
1904:
1902:
1900:
1895:
1890:
1886:
1883:
1875:
1871:
1867:
1863:
1859:
1855:
1851:
1847:
1844:
1840:
1836:
1832:
1829:
1825:
1822:
1818:
1814:
1810:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1780:
1779:unitriangular
1776:
1773:
1769:
1766:
1762:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1732:
1727:
1724:
1720:
1714:
1710:
1706:
1702:
1695:
1692:
1688:
1685:
1684:
1680:
1676:
1671:
1664:
1662:
1651:
1649:
1632:
1618:
1615:
1612:
1605:if there are
1600:
1585:
1547:
1543:
1538:
1534:
1528:
1525:
1520:
1516:
1511:
1505:
1502:
1499:
1495:
1472:
1469:
1466:
1462:
1438:
1432:
1429:
1424:
1420:
1397:
1394:
1389:
1385:
1381:
1378:
1375:
1370:
1366:
1362:
1357:
1353:
1349:
1343:
1333:
1332:
1330:
1323:
1305:
1302:
1297:
1293:
1286:
1281:
1278:
1275:
1271:
1245:
1239:
1234:
1230:
1226:
1223:
1220:
1215:
1211:
1207:
1202:
1198:
1194:
1191:
1184:
1183:
1181:
1178:
1174:
1167:
1150:
1146:
1142:
1134:
1131:
1128:
1124:
1120:
1117:
1089:
1085:
1080:
1076:
1070:
1067:
1062:
1058:
1053:
1047:
1044:
1041:
1037:
1014:
1011:
1006:
1002:
998:
995:
992:
987:
983:
979:
974:
970:
966:
960:
950:
949:
948:
944:
937:
936:
934:
928:
920:
918:
916:
912:
908:
907:
902:
898:
893:
891:
887:
886:Galois theory
882:
880:
876:
875:supersolvable
872:
868:
865:
861:
857:
853:
848:
846:
842:
838:
834:
830:
826:
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:
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:
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:
31:
19:
3785:. New York:
3782:
3760:
3737:
3725:
3702:
3679:
3657:
3631:
3618:
3599:
3574:
3564:
3557:
3547:
3540:
3530:
3510:
3503:
3493:
3486:
3476:
3457:
3453:
3447:
3430:
3421:
3417:
3412:
3408:
3404:
3402:
3394:
3388:
3384:
3374:
3370:
3366:
3362:
3358:
3354:
3350:
3346:
3342:
3337:Note that a
3328:
3324:
3319:
3315:
3308:
3301:
3297:
3293:
3289:
3285:
3280:
3276:
3269:
3263:
3259:
3255:
3251:
3247:
3242:
3238:
3234:
3228:
3224:
3217:
3210:
3206:
3200:
3196:
3189:
3182:
3178:
3173:
3169:
3165:
3159:
3155:
3148:
3142:
3138:
3134:
3129:
3125:
3118:
3112:
3108:
3103:
3099:
3094:
3090:
3083:
3076:
3071:
3067:
3060:
3054:
3050:
3046:
3042:
3038:
3034:
3030:
3024:
3020:
3016:
3011:
3007:
3002:
2998:
2991:
2984:
2974:
2970:
2966:
2962:
2957:
2953:
2949:
2945:
2940:
2936:
2932:
2927:
2923:
2919:
2915:
2910:
2906:
2902:
2898:
2894:
2890:
2882:
2878:
2874:
2870:
2865:
2861:
2857:
2853:
2848:
2844:
2840:
2836:
2832:
2828:
2822:
2818:
2814:
2809:
2805:
2800:
2796:
2789:
2782:
2772:
2768:
2764:
2760:
2756:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2672:
2668:
2664:
2660:
2656:
2652:
2648:
2644:
2638:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2596:
2591:
2587:
2583:
2579:
2575:
2566:
2559:
2551:
2547:
2542:divides the
2539:
2528:
2522:
2511:
2507:
2503:
2495:
2490:
2486:
2481:is a proper
2478:
2474:
2470:
2463:
2457:
2452:
2448:
2444:
2441:homomorphism
2436:
2432:
2428:
2426:
2412:
2408:
2402:
2398:
2396:factor group
2393:
2385:
2369:Engel groups
2299:
1908:
1888:
1884:
1879:
1873:
1869:
1865:
1861:
1857:
1853:
1849:
1838:
1834:
1830:
1820:
1816:
1812:
1808:
1801:
1797:
1789:
1785:
1781:
1771:
1752:
1748:
1744:
1740:
1730:
1712:
1700:
1693:
1675:Cayley graph
1652:
1643:
1633:
1595:
1583:
1569:
924:
914:
910:
904:
903:) including
897:Lie algebras
894:
883:
849:
836:
831:that has an
824:
821:
818:group theory
811:
640:
628:
616:
604:
592:
580:
568:
556:
327:
284:
271:
260:
249:
245:Cyclic group
195:
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
2377:conjectured
2012:defined by
1728:All finite
1707:{1, −1} of
901:Lie bracket
899:(using the
814:mathematics
530:Topological
369:alternating
3808:Categories
3611:References
3387:dividing |
3349:for all 1≤
3233:|. Since |
3041:. For any
2952:and hence
2525:is normal.
2500:normalizer
2390:Properties
2375:, and are
2138:commutator
1852:such that
1763:, and the
1574:such that
921:Definition
890:Lie groups
858:, and for
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
3397:-subgroup
3341:of order
3216:|⋅|
2873:). Since
2852:for some
2699:). Note,
2558:of order
2113:−
2100:−
2030:
1997:→
1991::
1911:nilpotent
1382:◃
1379:⋯
1376:◃
1363:◃
1227:▹
1224:⋯
1221:▹
1208:▹
1143:≤
1068:≤
999:◃
996:⋯
993:◃
980:◃
906:nilpotent
701:Conformal
589:Euclidean
196:nilpotent
3781:(1999).
3678:(2008).
3656:(1974).
3357:. Since
3288:, hence
3164:and let
2893:. Since
2856:and let
2257:for all
1843:solvable
1774:-groups.
1665:Examples
1180:subgroup
856:solvable
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
3658:Algebra
3511:Algebra
3391:|, the
3380:(e)→(a)
3339:p-group
3334:(d)→(e)
3033:), 1 ≤
2980:(c)→(d)
2831:), 1 ≤
2778:(b)→(c)
2735:) and
2675:, then
2615:, then
2571:(a)→(b)
2567:Proof:
2550:, then
2531:is the
2498:) (the
2477:, then
2416:in the
2136:is the
2069:(where
1897:in the
1894:A056867
1739:). For
1733:-groups
1327:has an
1177:trivial
871:commute
852:abelian
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
3793:
3767:
3748:
3731:review
3713:
3690:
3664:
3642:
3393:Sylow
3137:. Let
3045:, 1 ≤
2839:. Let
2578:|. If
2554:has a
2517:Every
1759:, the
1705:center
1590:; and
1412:where
1263:where
1171:has a
1029:where
941:has a
913:, and
867:orders
860:finite
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
3439:Notes
3424:(see
3314:×···×
3237:| = |
3209:| = |
3195:×···×
3089:×···×
2997:,...,
2897:char
2795:,...,
2663:. If
2647:, so
2544:order
2439:is a
2373:order
1792:is a
1737:proof
1709:order
1648:group
869:must
829:group
827:is a
730:Sp(∞)
727:SU(∞)
140:image
3791:ISBN
3765:ISBN
3746:ISBN
3711:ISBN
3688:ISBN
3662:ISBN
3640:ISBN
3373:of |
3258:and
3250:and
2983:Let
2965:) =
2901:and
2781:Let
2599:) =
2160:and
1899:OEIS
1880:The
1721:The
1644:nil-
1454:and
820:, a
724:O(∞)
713:Loop
532:and
3428:).
3416:of
3275:···
3154:···
3124:···
3111:so
3066:···
3017:Syl
3015:in
2889:in
2885:is
2815:Syl
2813:in
2546:of
2538:If
2521:of
2506:in
2502:of
2485:of
2469:If
2277:in
2140:of
1901:).
1872:),
1819:),
1747:is
1586:of
812:In
639:Sp(
627:SU(
603:SO(
567:SL(
555:GL(
3810::
3789:.
3744:.
3709:.
3705:.
3686:.
3682:.
3638:.
3621:.
3587:^
3520:^
3466:^
3458:13
3456:.
3435:.
3377:|.
3327:=
3290:HK
3286:HK
3284:=
3231:−1
3203:−1
3168:=
3162:−1
3049:≤
3037:≤
2973:=
2881:,
2860:=
2843:=
2835:≤
2643:=
2639:H'
2637:h'
2635:=
2586:,
2455:.
2405:+1
2383:.
2309:ad
2297:.
2214:ad
2042::=
2021:ad
1982:ad
1833:×
1784:×
1650:.
917:.
909:,
892:.
881:.
615:U(
591:E(
579:O(
37:→
3799:.
3773:.
3754:.
3719:.
3696:.
3670:.
3648:.
3625:.
3422:G
3418:G
3413:p
3409:G
3405:G
3395:p
3389:G
3385:p
3375:G
3371:d
3367:d
3363:G
3359:G
3355:k
3353:≤
3351:m
3347:p
3343:p
3329:s
3325:t
3320:t
3316:P
3312:2
3309:P
3307:×
3305:1
3302:P
3298:K
3296:×
3294:H
3281:t
3277:P
3273:2
3270:P
3267:1
3264:P
3260:K
3256:H
3252:K
3248:H
3243:t
3239:P
3235:K
3229:t
3225:P
3221:2
3218:P
3214:1
3211:P
3207:H
3201:t
3197:P
3193:2
3190:P
3188:×
3186:1
3183:P
3179:H
3174:t
3170:P
3166:K
3160:t
3156:P
3152:2
3149:P
3146:1
3143:P
3139:H
3135:G
3130:t
3126:P
3122:2
3119:P
3116:1
3113:P
3109:G
3104:i
3100:P
3095:t
3091:P
3087:2
3084:P
3082:×
3080:1
3077:P
3072:t
3068:P
3064:2
3061:P
3058:1
3055:P
3051:s
3047:t
3043:t
3039:s
3035:i
3031:G
3029:(
3025:i
3021:p
3012:i
3008:P
3003:s
2999:p
2995:2
2992:p
2990:,
2988:1
2985:p
2975:G
2971:N
2967:N
2963:N
2961:(
2958:G
2954:N
2950:N
2946:N
2944:(
2941:G
2937:N
2933:N
2931:(
2928:G
2924:N
2920:P
2916:N
2914:(
2911:G
2907:N
2903:N
2899:N
2895:P
2891:N
2883:P
2879:N
2875:P
2871:P
2869:(
2866:G
2862:N
2858:N
2854:i
2849:i
2845:P
2841:P
2837:s
2833:i
2829:G
2827:(
2823:i
2819:p
2810:i
2806:P
2801:s
2797:p
2793:2
2790:p
2788:,
2786:1
2783:p
2773:G
2771:(
2769:Z
2767:/
2765:G
2761:G
2757:p
2753:H
2749:G
2745:G
2743:(
2741:Z
2739:/
2737:H
2733:G
2731:(
2729:Z
2727:/
2725:H
2721:G
2719:(
2717:Z
2715:/
2713:G
2709:G
2707:(
2705:Z
2703:/
2701:G
2697:G
2695:(
2693:Z
2691:/
2689:G
2685:G
2683:(
2681:Z
2679:/
2677:H
2673:H
2669:G
2667:(
2665:Z
2661:H
2657:G
2655:(
2653:Z
2651:·
2649:H
2645:H
2641:h
2633:h
2629:Z
2625:H
2621:Z
2617:h
2613:H
2609:G
2607:(
2605:Z
2601:G
2597:H
2595:(
2592:G
2588:N
2584:H
2580:G
2576:G
2562:.
2560:d
2552:G
2548:G
2540:d
2529:G
2523:G
2508:G
2504:H
2496:H
2494:(
2491:G
2487:N
2479:H
2475:G
2471:H
2464:G
2453:n
2449:f
2445:n
2437:f
2433:n
2429:n
2413:i
2409:Z
2407:/
2403:i
2399:Z
2367:-
2355:n
2335:n
2313:g
2285:G
2265:x
2245:e
2242:=
2239:)
2236:x
2233:(
2228:n
2223:)
2218:g
2210:(
2188:n
2168:x
2148:g
2124:x
2121:g
2116:1
2109:x
2103:1
2096:g
2092:=
2089:]
2086:x
2083:,
2080:g
2077:[
2057:]
2054:x
2051:,
2048:g
2045:[
2039:)
2036:x
2033:(
2025:g
2000:G
1994:G
1986:g
1961:g
1941:n
1921:G
1889:k
1885:k
1876:.
1874:G
1870:G
1868:(
1866:Z
1862:G
1860:(
1858:Z
1856:/
1854:G
1850:G
1845:.
1839:F
1835:n
1831:n
1823:.
1821:H
1817:H
1815:(
1813:Z
1809:H
1802:n
1798:n
1790:F
1786:n
1782:n
1772:p
1767:.
1753:p
1749:n
1745:p
1741:n
1731:p
1716:8
1713:Q
1701:p
1697:8
1694:Q
1659:1
1655:0
1646:n
1640:n
1636:G
1619:1
1616:+
1613:n
1603:n
1598:n
1592:G
1588:G
1580:n
1576:G
1572:n
1565:.
1553:)
1548:i
1544:Z
1539:/
1535:G
1532:(
1529:Z
1526:=
1521:i
1517:Z
1512:/
1506:1
1503:+
1500:i
1496:Z
1473:1
1470:+
1467:i
1463:Z
1442:)
1439:G
1436:(
1433:Z
1430:=
1425:1
1421:Z
1398:G
1395:=
1390:n
1386:Z
1371:1
1367:Z
1358:0
1354:Z
1350:=
1347:}
1344:1
1341:{
1325:G
1321:.
1309:]
1306:G
1303:,
1298:i
1294:G
1290:[
1287:=
1282:1
1279:+
1276:i
1272:G
1249:}
1246:1
1243:{
1240:=
1235:n
1231:G
1216:1
1212:G
1203:0
1199:G
1195:=
1192:G
1169:G
1165:.
1151:i
1147:G
1140:]
1135:1
1132:+
1129:i
1125:G
1121:,
1118:G
1115:[
1095:)
1090:i
1086:G
1081:/
1077:G
1074:(
1071:Z
1063:i
1059:G
1054:/
1048:1
1045:+
1042:i
1038:G
1015:G
1012:=
1007:n
1003:G
988:1
984:G
975:0
971:G
967:=
964:}
961:1
958:{
939:G
933::
931:G
837:G
825: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
631:)
629:n
619:)
617:n
607:)
605:n
595:)
593:n
583:)
581:n
571:)
569:n
559:)
557:n
499:)
486:Z
474:)
461:Z
437:)
424:Z
415:(
328:p
293:Q
285:n
282:D
272:n
269:A
261:n
258:S
250:n
247:Z
20:)
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