48:
2773:
An algebraic group is said to be affine if its underlying algebraic variety is an affine variety. Among the examples above the additive, multiplicative groups and the general and special linear groups are affine. Using the action of an affine algebraic group on its
3517:
by an infinite normal discrete subgroup. An algebraic group over the real or complex numbers may have closed subgroups (in the analytic topology) that do not have the same connected component of the identity as any algebraic subgroup.
2872:
2962:
Abelian varieties are connected projective algebraic groups, for instance elliptic curves. They are always commutative. They arise naturally in various situations in algebraic geometry and number theory, for example as the
1386:
2590:
2102:
1048:
2461:
2283:
2248:
2162:
2132:
1456:
1078:
2415:
2652:. As a vector space the Lie algebra is isomorphic to the tangent space at the identity element. The Lie bracket can be constructed from its interpretation as a space of derivations.
1920:
1772:
1003:
3380:
1634:
2332:
3233:
2811:
1854:
1701:
3118:
1956:
1663:
1546:
1485:
1286:
1228:
1192:
1123:
3495:
3411:
3342:
3442:
3464:
3291:
3051:
2483:
2383:
2361:
2305:
2206:
2184:
2064:
2042:
2020:
1995:
946:
493:
468:
431:
1736:
3198:
3171:
3068:
if the underlying algebraic variety is connected for the
Zariski topology. For an algebraic group this means that it is not the union of two proper algebraic subsets.
3061:, i.e. the group operations may not be continuous for this topology (because Zariski topology on the product is not the product of Zariski topologies on the factors).
3144:
3239:). In addition it is both affine and projective. Thus, in particular for classification purposes, it is natural to restrict statements to connected algebraic group.
2778:
it can be shown that every affine algebraic group is a linear (or matrix group), meaning that it is isomorphic to an algebraic subgroup of the general linear group.
3599:
1517:
2503:
1257:
2816:
3557:
3311:
3265:
3089:
2936:. The classification over arbitrary fields is more involved but still well-understood. If can be made very explicit in some cases, for example over the real or
2757:
2729:
2702:
2678:
2650:
2626:
2523:
1878:
1812:
1792:
1586:
1566:
1163:
1143:
966:
924:
2975:
Not all algebraic groups are linear groups or abelian varieties, for instance some group schemes occurring naturally in arithmetic geometry are neither.
2337:
Quotients in the category of algebraic groups are more delicate to deal with. An algebraic subgroup is said to be normal if it is stable under every
795:
4115:
3848:
2528:
2069:
1015:
2420:
2605:
3601:; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the
3203:
More generally every finite group is an algebraic group (it can be realised as a finite, hence
Zariski-closed, subgroup of some
353:
4081:
4046:
4020:
3946:
3913:
824:
structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to
1125:
endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its
303:
3979:
2928:. In turn reductive groups are decomposed as (again essentially) a product of their center (an algebraic torus) with a
788:
298:
1291:
3173:. Another non-connected group are orthogonal group in even dimension (the determinant gives a surjective morphism to
2137:
2976:
2253:
1010:
895:
2107:
1391:
1053:
2218:
2388:
714:
2933:
4136:
4067:
3901:
3634:
781:
1883:
3413:
into a topological group. Such groups are important examples in the general theory of topological groups.
1741:
971:
3955:
3347:
1591:
398:
212:
20:
3559:, and the number of elements of the general linear group over a finite field is (up to some factor) the
3963:
3206:
2784:
1827:
1674:
130:
3094:
1932:
1639:
1522:
1461:
1262:
1204:
1168:
1099:
4131:
3469:
3385:
3316:
3419:
2310:
836:
596:
330:
207:
95:
3447:
3274:
3034:
2466:
2366:
2344:
2288:
2189:
2167:
2047:
2025:
2003:
1978:
929:
476:
451:
414:
4012:
1958:
can be endowed with a geometrically defined group law that makes it into an algebraic group (see
1814:
matrices), multiplication of matrices is regular and the formula for the inverse in terms of the
848:
3823:
3605:, which considers Coxeter groups to be simple algebraic groups over the field with one element.
1998:
3862:
3644:
3639:
3602:
3527:
2984:
2945:
2768:
2736:
1706:
1636: ; note that the subset of invertible elements does not define an algebraic subvariety in
871:
746:
536:
3176:
3149:
4063:
3123:
620:
3501:. Not all Lie groups can be obtained via this procedure, for example the universal cover of
2867:{\displaystyle x\mapsto \left({\begin{smallmatrix}1&x\\0&1\end{smallmatrix}}\right)}
4056:
3989:
3923:
3858:
3568:
1822:
1669:
1490:
879:
844:
560:
548:
166:
100:
3888:
3785:
2488:
1233:
859:
are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as
8:
3236:
1857:
821:
135:
30:
3539:
3344:
is endowed with the analytic topology coming from any embedding into a projective space
898:
states that every algebraic group can be constructed from groups in those two families.
3296:
3250:
3074:
2920:
states that every such is (essentially) a semidirect product of a unipotent group (its
2897:
2742:
2714:
2687:
2663:
2635:
2611:
2508:
2338:
1863:
1797:
1777:
1571:
1551:
1148:
1128:
951:
909:
891:
864:
825:
120:
92:
4041:, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: BirkhÀuser Boston,
3998:
Affine Group
Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups
4098:
4077:
4042:
4026:
4016:
3975:
3942:
3909:
3614:
3058:
2921:
2917:
2909:
2593:
887:
817:
525:
368:
262:
691:
3967:
3884:
3781:
3772:
Conrad, Brian (2002). "A modern proof of
Chevalley's theorem on algebraic groups".
3514:
3054:
2964:
2929:
2705:
840:
676:
668:
660:
652:
644:
632:
572:
512:
502:
344:
286:
161:
3997:
4073:
4052:
3985:
3938:
3919:
3905:
3854:
3850:
Séminaire C. Chevalley, 1956--1958. Classification des groupes de Lie algébriques
3071:
Examples of groups that are not connected are given by the algebraic subgroup of
2980:
2957:
2925:
2890:
2884:
2775:
1815:
1006:
852:
760:
753:
696:
584:
507:
337:
251:
191:
71:
2832:
3619:
2905:
1959:
875:
860:
767:
703:
393:
373:
310:
275:
196:
186:
171:
156:
110:
87:
4125:
4004:
3533:
2992:
2711:
Yet another definition of the concept is to say that an algebraic group over
686:
608:
442:
315:
181:
4090:
3876:
2941:
2732:
2681:
856:
829:
541:
240:
229:
176:
151:
146:
105:
76:
39:
4102:
4030:
3971:
2285:
that is also a group homomorphism. Its kernel is an algebraic subgroup of
3960:
Algebraic Groups: The Theory of Group
Schemes of Finite Type over a Field
3629:
3560:
3268:
2629:
1926:
1922:
in much the same way as the multiplicative group in the previous example.
1094:
809:
3930:
3861:, Reprinted as volume 3 of Chevalley's collected works., archived from
3624:
2932:. The latter are classified over algebraically closed fields via their
2877:
There are many examples of such groups beyond those given previously:
1818:
shows that inversion is regular as well on matrices with determinant 1.
708:
436:
3382:
as a quasi-projective variety. This is a group topology, and it makes
3532:
There are a number of analogous results between algebraic groups and
3498:
2901:
529:
3693:
3242:
2979:
asserts that every connected algebraic group is an extension of an
2660:
A more sophisticated definition of an algebraic group over a field
66:
3883:. Graduate Texts in Mathematics. Springer-Verlag. pp. x+288.
3705:
3271:(for instance the real or complex numbers, or a p-adic field) and
3120:(each point is a Zariski-closed subset so it is not connected for
2592:
may not be surjective (the default of surjectivity is measured by
2585:{\displaystyle \mathrm {G} (k)\to \mathrm {G} (k)/\mathrm {H} (k)}
4072:, Graduate Texts in Mathematics, vol. 66, Berlin, New York:
3536:â for instance, the number of elements of the symmetric group is
2970:
408:
322:
1568:-points are isomorphic to the multiplicative group of the field
3502:
2937:
2916:
Linear algebraic groups can be classified to a certain extent.
2097:{\displaystyle \mathrm {G} \times \mathrm {G} \to \mathrm {G} }
1043:{\displaystyle \mathrm {G} \times \mathrm {G} \to \mathrm {G} }
890:, which are the algebraic groups whose underlying variety is a
47:
2456:{\displaystyle \pi :\mathrm {G} \to \mathrm {G} /\mathrm {H} }
3681:
2881:
orthogonal and symplectic groups are affine algebraic groups.
1880:
is an algebraic group. It can be realised as a subvariety in
1703:
is an algebraic group: it is given by the algebraic equation
1145:-points are isomorphic as a group to the additive group of
3717:
1487:, and they satisfy the group axioms (with neutral element
3729:
1080:(the inversion operation) that satisfy the group axioms.
870:
An important class of algebraic groups is given by the
3804:
3792:
3521:
3669:
3571:
3542:
3472:
3450:
3422:
3388:
3350:
3319:
3299:
3277:
3253:
3209:
3179:
3152:
3126:
3097:
3077:
3037:
2819:
2787:
2745:
2717:
2690:
2666:
2638:
2614:
2531:
2511:
2491:
2469:
2423:
2391:
2369:
2347:
2313:
2291:
2256:
2221:
2192:
2170:
2140:
2110:
2072:
2050:
2028:
2006:
1981:
1935:
1886:
1866:
1830:
1800:
1780:
1744:
1709:
1677:
1642:
1594:
1574:
1554:
1525:
1493:
1464:
1394:
1294:
1265:
1236:
1207:
1171:
1151:
1131:
1102:
1056:
1018:
974:
954:
932:
912:
479:
454:
417:
3753:
3741:
3657:
2599:
2525:
is not algebraically closed, the morphism of groups
3593:
3551:
3489:
3458:
3436:
3405:
3374:
3336:
3305:
3285:
3259:
3227:
3192:
3165:
3138:
3112:
3083:
3045:
2866:
2805:
2781:For example the additive group can be embedded in
2751:
2723:
2704:(group schemes can more generally be defined over
2696:
2672:
2644:
2620:
2584:
2517:
2497:
2477:
2455:
2409:
2377:
2355:
2326:
2299:
2277:
2242:
2200:
2178:
2156:
2126:
2096:
2058:
2036:
2014:
1989:
1950:
1914:
1872:
1848:
1806:
1786:
1766:
1730:
1695:
1657:
1628:
1580:
1560:
1540:
1511:
1479:
1450:
1380:
1280:
1251:
1222:
1186:
1157:
1137:
1117:
1072:
1042:
997:
960:
940:
918:
878:; they are exactly the algebraic subgroups of the
487:
462:
425:
3243:Algebraic groups over local fields and Lie groups
874:, those whose underlying algebraic variety is an
4123:
1710:
1381:{\displaystyle ((x,y),(x',y'))\mapsto (xx',yy')}
3003:, there exists a unique normal closed subgroup
2157:{\displaystyle \mathrm {H} \times \mathrm {H} }
3091:th roots of unity in the multiplicative group
2971:Structure theorem for general algebraic groups
1230:be the affine variety defined by the equation
2278:{\displaystyle \mathrm {G} \to \mathrm {G} '}
789:
2127:{\displaystyle \mathrm {G} \to \mathrm {G} }
1548:is called multiplicative group, because its
1451:{\displaystyle (x,y)\mapsto (x^{-1},y^{-1})}
1073:{\displaystyle \mathrm {G} \to \mathrm {G} }
3853:, 2 vols, Paris: Secrétariat Mathématique,
4095:Courbes algébriques et variétés abéliennes
4062:
3015:is a connected linear algebraic group and
2762:
2655:
2243:{\displaystyle \mathrm {G} ,\mathrm {G} '}
906:Formally, an algebraic group over a field
796:
782:
3895:
3846:
3452:
3430:
3353:
2410:{\displaystyle \mathrm {G} /\mathrm {H} }
1938:
1889:
1747:
1645:
1268:
1105:
481:
456:
419:
4036:
3881:Linear algebraic groups. 2nd enlarged ed
2307:, its image is an algebraic subgroup of
968:, together with a distinguished element
16:Algebraic variety with a group structure
4116:Algebraic groups and their Lie algebras
4003:
4124:
3771:
3146:). This group is generally denoted by
1966:
1915:{\displaystyle \mathbb {A} ^{n^{2}+1}}
354:Classification of finite simple groups
3954:
3875:
3810:
3798:
3759:
3747:
3735:
3723:
3711:
3699:
3687:
3675:
3663:
2608:, to an algebraic group over a field
2385:then there exists an algebraic group
4089:
4069:Introduction to affine group schemes
3929:
2951:
2606:Lie groupâLie algebra correspondence
1767:{\displaystyle \mathbb {A} ^{n^{2}}}
998:{\displaystyle e\in \mathrm {G} (k)}
3522:Coxeter groups and algebraic groups
3375:{\displaystyle \mathbb {P} ^{n}(k)}
1629:{\displaystyle x\mapsto (x,x^{-1})}
1050:(the multiplication operation) and
839:are algebraic groups; for example,
13:
4109:
3904:, vol. 21, Berlin, New York:
3474:
3390:
3321:
3279:
3215:
3212:
3100:
3039:
2793:
2790:
2569:
2550:
2533:
2471:
2449:
2439:
2431:
2403:
2393:
2371:
2363:is a normal algebraic subgroup of
2349:
2316:
2293:
2267:
2258:
2232:
2223:
2194:
2172:
2150:
2142:
2120:
2112:
2090:
2082:
2074:
2052:
2030:
2008:
1983:
1836:
1833:
1683:
1680:
1528:
1467:
1210:
1174:
1066:
1058:
1036:
1028:
1020:
982:
934:
14:
4148:
3228:{\displaystyle \mathrm {GL} _{n}}
3064:An algebraic group is said to be
2999:a connected algebraic group over
2831:
2806:{\displaystyle \mathrm {GL} _{2}}
2600:Lie algebra of an algebraic group
2134:defining the group structure map
1849:{\displaystyle \mathrm {GL} _{n}}
1696:{\displaystyle \mathrm {SL} _{n}}
886:. Another class is formed by the
3113:{\displaystyle \mathrm {G} _{m}}
3026:
1951:{\displaystyle \mathbb {P} ^{2}}
1658:{\displaystyle \mathbb {A} ^{1}}
1541:{\displaystyle \mathrm {G} _{m}}
1480:{\displaystyle \mathrm {G} _{m}}
1281:{\displaystyle \mathbb {A} ^{2}}
1223:{\displaystyle \mathrm {G} _{m}}
1187:{\displaystyle \mathrm {G} _{a}}
1118:{\displaystyle \mathbb {A} ^{1}}
882:, and are therefore also called
46:
3847:Chevalley, Claude, ed. (1958),
3816:
3765:
3490:{\displaystyle \mathrm {G} (k)}
3406:{\displaystyle \mathrm {G} (k)}
3337:{\displaystyle \mathrm {G} (k)}
3579:
3572:
3484:
3478:
3437:{\displaystyle k=\mathbb {R} }
3400:
3394:
3369:
3363:
3331:
3325:
2823:
2579:
2573:
2560:
2554:
2546:
2543:
2537:
2435:
2262:
2116:
2086:
1774:(identified with the space of
1719:
1713:
1623:
1601:
1598:
1506:
1494:
1445:
1413:
1410:
1407:
1395:
1375:
1347:
1344:
1341:
1338:
1316:
1310:
1298:
1295:
1062:
1032:
992:
986:
901:
715:Infinite dimensional Lie group
1:
3902:Graduate Texts in Mathematics
3650:
2977:Chevalley's structure theorem
2341:(which are regular maps). If
2327:{\displaystyle \mathrm {G} '}
2215:between two algebraic groups
896:Chevalley's structure theorem
3896:Humphreys, James E. (1972),
3459:{\displaystyle \mathbb {C} }
3286:{\displaystyle \mathrm {G} }
3046:{\displaystyle \mathrm {G} }
2739:of algebraic varieties over
2478:{\displaystyle \mathrm {H} }
2378:{\displaystyle \mathrm {G} }
2356:{\displaystyle \mathrm {H} }
2300:{\displaystyle \mathrm {G} }
2201:{\displaystyle \mathrm {H} }
2179:{\displaystyle \mathrm {H} }
2059:{\displaystyle \mathrm {G} }
2037:{\displaystyle \mathrm {G} }
2015:{\displaystyle \mathrm {H} }
1990:{\displaystyle \mathrm {G} }
1588:(an isomorphism is given by
941:{\displaystyle \mathrm {G} }
488:{\displaystyle \mathbb {Z} }
463:{\displaystyle \mathbb {Z} }
426:{\displaystyle \mathbb {Z} }
19:Not to be confused with the
7:
4037:Springer, Tonny A. (1998),
3608:
2908:such as that of invertible
2044:that is also a subgroup of
1083:
213:List of group theory topics
10:
4153:
3964:Cambridge University Press
3525:
2955:
2766:
2417:and a surjective morphism
1165:), and usually denoted by
18:
3635:CherlinâZilber conjecture
3513:, or the quotient of the
3057:. It is not in general a
2940:fields, and thereby over
2505:. Note that if the field
1731:{\displaystyle \det(g)=1}
837:geometric transformations
3193:{\displaystyle \mu _{2}}
3166:{\displaystyle \mu _{n}}
3031:As an algebraic variety
2680:is that it is that of a
1929:in the projective plane
926:is an algebraic variety
331:Elementary abelian group
208:Glossary of group theory
4039:Linear algebraic groups
4013:Oxford University Press
3898:Linear Algebraic Groups
3702:, Corollary 1.4, p. 47.
3139:{\displaystyle n\geq 1}
2946:local-global principles
2763:Affine algebraic groups
2656:Alternative definitions
1519:). The algebraic group
884:linear algebraic groups
872:affine algebraic groups
4064:Waterhouse, William C.
3824:"Non-linear Lie group"
3774:J. Ramanujan Math. Soc
3645:Pseudo-reductive group
3640:Adelic algebraic group
3603:field with one element
3595:
3553:
3528:Field with one element
3491:
3460:
3438:
3407:
3376:
3338:
3313:-group then the group
3307:
3287:
3261:
3229:
3194:
3167:
3140:
3114:
3085:
3047:
2985:linear algebraic group
2868:
2807:
2769:Linear algebraic group
2753:
2725:
2698:
2674:
2646:
2622:
2586:
2519:
2499:
2479:
2457:
2411:
2379:
2357:
2328:
2301:
2279:
2244:
2202:
2180:
2158:
2128:
2098:
2060:
2038:
2016:
1991:
1975:of an algebraic group
1952:
1916:
1874:
1850:
1808:
1788:
1768:
1732:
1697:
1659:
1630:
1582:
1562:
1542:
1513:
1481:
1452:
1382:
1282:
1253:
1224:
1188:
1159:
1139:
1119:
1074:
1044:
999:
962:
942:
920:
747:Linear algebraic group
489:
464:
427:
3972:10.1017/9781316711736
3714:, Theorem 6.8, p. 98.
3596:
3594:{\displaystyle _{q}!}
3554:
3526:Further information:
3492:
3461:
3439:
3408:
3377:
3339:
3308:
3288:
3262:
3230:
3195:
3168:
3141:
3115:
3086:
3048:
2987:. More precisely, if
2869:
2808:
2754:
2726:
2699:
2675:
2647:
2623:
2587:
2520:
2500:
2480:
2458:
2412:
2380:
2358:
2329:
2302:
2280:
2245:
2203:
2186:, respectively, into
2181:
2159:
2129:
2099:
2061:
2039:
2017:
1992:
1953:
1917:
1875:
1851:
1809:
1789:
1769:
1733:
1698:
1660:
1631:
1583:
1563:
1543:
1514:
1512:{\displaystyle (1,1)}
1482:
1453:
1383:
1283:
1254:
1225:
1189:
1160:
1140:
1120:
1075:
1045:
1000:
963:
943:
921:
845:general linear groups
490:
465:
428:
4137:Properties of groups
3937:, Berlin, New York:
3569:
3540:
3470:
3448:
3420:
3386:
3348:
3317:
3297:
3275:
3251:
3207:
3177:
3150:
3124:
3095:
3075:
3035:
3023:an abelian variety.
2817:
2785:
2743:
2715:
2688:
2664:
2636:
2612:
2529:
2509:
2498:{\displaystyle \pi }
2489:
2467:
2421:
2389:
2367:
2345:
2311:
2289:
2254:
2219:
2190:
2168:
2138:
2108:
2070:
2048:
2026:
2004:
1979:
1933:
1884:
1864:
1828:
1823:general linear group
1798:
1778:
1742:
1738:in the affine space
1707:
1675:
1670:special linear group
1640:
1592:
1572:
1552:
1523:
1491:
1462:
1392:
1292:
1263:
1259:in the affine plane
1252:{\displaystyle xy=1}
1234:
1205:
1199:multiplicative group
1169:
1149:
1129:
1100:
1054:
1016:
972:
952:
930:
910:
880:general linear group
477:
452:
415:
2910:triangular matrices
2898:semidirect products
2066:(that is, the maps
1967:Related definitions
1858:invertible matrices
121:Group homomorphisms
31:Algebraic structure
4097:, Paris: Hermann,
3591:
3552:{\displaystyle n!}
3549:
3487:
3456:
3434:
3403:
3372:
3334:
3303:
3283:
3257:
3225:
3190:
3163:
3136:
3110:
3081:
3043:
2864:
2858:
2857:
2803:
2749:
2721:
2694:
2670:
2642:
2618:
2582:
2515:
2495:
2475:
2453:
2407:
2375:
2353:
2339:inner automorphism
2324:
2297:
2275:
2240:
2198:
2176:
2154:
2124:
2094:
2056:
2034:
2012:
1987:
1973:algebraic subgroup
1948:
1912:
1870:
1846:
1804:
1784:
1764:
1728:
1693:
1655:
1626:
1578:
1558:
1538:
1509:
1477:
1448:
1378:
1278:
1249:
1220:
1184:
1155:
1135:
1115:
1070:
1040:
995:
958:
938:
916:
892:projective variety
865:Jacobian varieties
826:algebraic geometry
597:Special orthogonal
485:
460:
423:
304:Lagrange's theorem
4083:978-0-387-90421-4
4048:978-0-8176-4021-7
4022:978-0-19-560528-0
4009:Abelian varieties
3948:978-0-387-90875-5
3935:Abelian varieties
3915:978-0-387-90108-4
3615:Character variety
3306:{\displaystyle k}
3260:{\displaystyle k}
3084:{\displaystyle n}
2952:Abelian varieties
2922:unipotent radical
2752:{\displaystyle k}
2724:{\displaystyle k}
2706:commutative rings
2697:{\displaystyle k}
2673:{\displaystyle k}
2645:{\displaystyle k}
2621:{\displaystyle k}
2604:Similarly to the
2594:Galois cohomology
2518:{\displaystyle k}
2485:is the kernel of
2250:is a regular map
1873:{\displaystyle k}
1807:{\displaystyle n}
1787:{\displaystyle n}
1581:{\displaystyle k}
1561:{\displaystyle k}
1158:{\displaystyle k}
1138:{\displaystyle k}
961:{\displaystyle k}
919:{\displaystyle k}
888:abelian varieties
849:projective groups
841:orthogonal groups
818:algebraic variety
806:
805:
381:
380:
263:Alternating group
220:
219:
21:variety of groups
4144:
4132:Algebraic groups
4118:by Daniel Miller
4105:
4086:
4059:
4033:
3992:
3951:
3926:
3892:
3872:
3871:
3870:
3839:
3838:
3836:
3834:
3820:
3814:
3808:
3802:
3796:
3790:
3789:
3769:
3763:
3757:
3751:
3745:
3739:
3733:
3727:
3721:
3715:
3709:
3703:
3697:
3691:
3690:, 1.6(2), p. 49.
3685:
3679:
3673:
3667:
3661:
3600:
3598:
3597:
3592:
3587:
3586:
3558:
3556:
3555:
3550:
3515:Heisenberg group
3496:
3494:
3493:
3488:
3477:
3466:then this makes
3465:
3463:
3462:
3457:
3455:
3443:
3441:
3440:
3435:
3433:
3412:
3410:
3409:
3404:
3393:
3381:
3379:
3378:
3373:
3362:
3361:
3356:
3343:
3341:
3340:
3335:
3324:
3312:
3310:
3309:
3304:
3292:
3290:
3289:
3284:
3282:
3266:
3264:
3263:
3258:
3237:Cayley's theorem
3234:
3232:
3231:
3226:
3224:
3223:
3218:
3199:
3197:
3196:
3191:
3189:
3188:
3172:
3170:
3169:
3164:
3162:
3161:
3145:
3143:
3142:
3137:
3119:
3117:
3116:
3111:
3109:
3108:
3103:
3090:
3088:
3087:
3082:
3055:Zariski topology
3052:
3050:
3049:
3044:
3042:
2965:Jacobian variety
2930:semisimple group
2885:unipotent groups
2873:
2871:
2870:
2865:
2863:
2859:
2813:by the morphism
2812:
2810:
2809:
2804:
2802:
2801:
2796:
2758:
2756:
2755:
2750:
2730:
2728:
2727:
2722:
2703:
2701:
2700:
2695:
2679:
2677:
2676:
2671:
2651:
2649:
2648:
2643:
2628:is associated a
2627:
2625:
2624:
2619:
2591:
2589:
2588:
2583:
2572:
2567:
2553:
2536:
2524:
2522:
2521:
2516:
2504:
2502:
2501:
2496:
2484:
2482:
2481:
2476:
2474:
2462:
2460:
2459:
2454:
2452:
2447:
2442:
2434:
2416:
2414:
2413:
2408:
2406:
2401:
2396:
2384:
2382:
2381:
2376:
2374:
2362:
2360:
2359:
2354:
2352:
2333:
2331:
2330:
2325:
2323:
2319:
2306:
2304:
2303:
2298:
2296:
2284:
2282:
2281:
2276:
2274:
2270:
2261:
2249:
2247:
2246:
2241:
2239:
2235:
2226:
2207:
2205:
2204:
2199:
2197:
2185:
2183:
2182:
2177:
2175:
2163:
2161:
2160:
2155:
2153:
2145:
2133:
2131:
2130:
2125:
2123:
2115:
2103:
2101:
2100:
2095:
2093:
2085:
2077:
2065:
2063:
2062:
2057:
2055:
2043:
2041:
2040:
2035:
2033:
2021:
2019:
2018:
2013:
2011:
1996:
1994:
1993:
1988:
1986:
1957:
1955:
1954:
1949:
1947:
1946:
1941:
1921:
1919:
1918:
1913:
1911:
1910:
1903:
1902:
1892:
1879:
1877:
1876:
1871:
1855:
1853:
1852:
1847:
1845:
1844:
1839:
1813:
1811:
1810:
1805:
1793:
1791:
1790:
1785:
1773:
1771:
1770:
1765:
1763:
1762:
1761:
1760:
1750:
1737:
1735:
1734:
1729:
1702:
1700:
1699:
1694:
1692:
1691:
1686:
1664:
1662:
1661:
1656:
1654:
1653:
1648:
1635:
1633:
1632:
1627:
1622:
1621:
1587:
1585:
1584:
1579:
1567:
1565:
1564:
1559:
1547:
1545:
1544:
1539:
1537:
1536:
1531:
1518:
1516:
1515:
1510:
1486:
1484:
1483:
1478:
1476:
1475:
1470:
1457:
1455:
1454:
1449:
1444:
1443:
1428:
1427:
1387:
1385:
1384:
1379:
1374:
1360:
1337:
1326:
1288:. The functions
1287:
1285:
1284:
1279:
1277:
1276:
1271:
1258:
1256:
1255:
1250:
1229:
1227:
1226:
1221:
1219:
1218:
1213:
1193:
1191:
1190:
1185:
1183:
1182:
1177:
1164:
1162:
1161:
1156:
1144:
1142:
1141:
1136:
1124:
1122:
1121:
1116:
1114:
1113:
1108:
1079:
1077:
1076:
1071:
1069:
1061:
1049:
1047:
1046:
1041:
1039:
1031:
1023:
1004:
1002:
1001:
996:
985:
967:
965:
964:
959:
947:
945:
944:
939:
937:
925:
923:
922:
917:
853:Euclidean groups
798:
791:
784:
740:Algebraic groups
513:Hyperbolic group
503:Arithmetic group
494:
492:
491:
486:
484:
469:
467:
466:
461:
459:
432:
430:
429:
424:
422:
345:Schur multiplier
299:Cauchy's theorem
287:Quaternion group
235:
234:
61:
60:
50:
37:
26:
25:
4152:
4151:
4147:
4146:
4145:
4143:
4142:
4141:
4122:
4121:
4112:
4110:Further reading
4084:
4074:Springer-Verlag
4049:
4023:
3982:
3949:
3939:Springer-Verlag
3916:
3906:Springer-Verlag
3868:
3866:
3843:
3842:
3832:
3830:
3822:
3821:
3817:
3809:
3805:
3797:
3793:
3770:
3766:
3758:
3754:
3746:
3742:
3734:
3730:
3722:
3718:
3710:
3706:
3698:
3694:
3686:
3682:
3674:
3670:
3662:
3658:
3653:
3611:
3582:
3578:
3570:
3567:
3566:
3541:
3538:
3537:
3530:
3524:
3506:
3473:
3471:
3468:
3467:
3451:
3449:
3446:
3445:
3429:
3421:
3418:
3417:
3389:
3387:
3384:
3383:
3357:
3352:
3351:
3349:
3346:
3345:
3320:
3318:
3315:
3314:
3298:
3295:
3294:
3278:
3276:
3273:
3272:
3252:
3249:
3248:
3245:
3219:
3211:
3210:
3208:
3205:
3204:
3184:
3180:
3178:
3175:
3174:
3157:
3153:
3151:
3148:
3147:
3125:
3122:
3121:
3104:
3099:
3098:
3096:
3093:
3092:
3076:
3073:
3072:
3038:
3036:
3033:
3032:
3029:
2981:abelian variety
2973:
2960:
2958:Abelian variety
2954:
2926:reductive group
2906:solvable groups
2900:, for instance
2856:
2855:
2850:
2844:
2843:
2838:
2830:
2826:
2818:
2815:
2814:
2797:
2789:
2788:
2786:
2783:
2782:
2776:coordinate ring
2771:
2765:
2744:
2741:
2740:
2716:
2713:
2712:
2689:
2686:
2685:
2665:
2662:
2661:
2658:
2637:
2634:
2633:
2613:
2610:
2609:
2602:
2568:
2563:
2549:
2532:
2530:
2527:
2526:
2510:
2507:
2506:
2490:
2487:
2486:
2470:
2468:
2465:
2464:
2448:
2443:
2438:
2430:
2422:
2419:
2418:
2402:
2397:
2392:
2390:
2387:
2386:
2370:
2368:
2365:
2364:
2348:
2346:
2343:
2342:
2315:
2314:
2312:
2309:
2308:
2292:
2290:
2287:
2286:
2266:
2265:
2257:
2255:
2252:
2251:
2231:
2230:
2222:
2220:
2217:
2216:
2193:
2191:
2188:
2187:
2171:
2169:
2166:
2165:
2149:
2141:
2139:
2136:
2135:
2119:
2111:
2109:
2106:
2105:
2089:
2081:
2073:
2071:
2068:
2067:
2051:
2049:
2046:
2045:
2029:
2027:
2024:
2023:
2007:
2005:
2002:
2001:
1982:
1980:
1977:
1976:
1969:
1942:
1937:
1936:
1934:
1931:
1930:
1925:A non-singular
1898:
1894:
1893:
1888:
1887:
1885:
1882:
1881:
1865:
1862:
1861:
1840:
1832:
1831:
1829:
1826:
1825:
1816:adjugate matrix
1799:
1796:
1795:
1779:
1776:
1775:
1756:
1752:
1751:
1746:
1745:
1743:
1740:
1739:
1708:
1705:
1704:
1687:
1679:
1678:
1676:
1673:
1672:
1649:
1644:
1643:
1641:
1638:
1637:
1614:
1610:
1593:
1590:
1589:
1573:
1570:
1569:
1553:
1550:
1549:
1532:
1527:
1526:
1524:
1521:
1520:
1492:
1489:
1488:
1471:
1466:
1465:
1463:
1460:
1459:
1458:are regular on
1436:
1432:
1420:
1416:
1393:
1390:
1389:
1367:
1353:
1330:
1319:
1293:
1290:
1289:
1272:
1267:
1266:
1264:
1261:
1260:
1235:
1232:
1231:
1214:
1209:
1208:
1206:
1203:
1202:
1178:
1173:
1172:
1170:
1167:
1166:
1150:
1147:
1146:
1130:
1127:
1126:
1109:
1104:
1103:
1101:
1098:
1097:
1086:
1065:
1057:
1055:
1052:
1051:
1035:
1027:
1019:
1017:
1014:
1013:
1007:neutral element
981:
973:
970:
969:
953:
950:
949:
933:
931:
928:
927:
911:
908:
907:
904:
861:elliptic curves
835:Many groups of
820:endowed with a
814:algebraic group
802:
773:
772:
761:Abelian variety
754:Reductive group
742:
732:
731:
730:
729:
680:
672:
664:
656:
648:
621:Special unitary
532:
518:
517:
499:
498:
480:
478:
475:
474:
455:
453:
450:
449:
418:
416:
413:
412:
404:
403:
394:Discrete groups
383:
382:
338:Frobenius group
283:
270:
259:
252:Symmetric group
248:
232:
222:
221:
72:Normal subgroup
58:
38:
29:
24:
17:
12:
11:
5:
4150:
4140:
4139:
4134:
4120:
4119:
4111:
4108:
4107:
4106:
4087:
4082:
4060:
4047:
4034:
4021:
4005:Mumford, David
4001:
3995:Milne, J. S.,
3993:
3981:978-1107167483
3980:
3952:
3947:
3927:
3914:
3893:
3873:
3841:
3840:
3815:
3803:
3791:
3764:
3752:
3740:
3728:
3716:
3704:
3692:
3680:
3668:
3655:
3654:
3652:
3649:
3648:
3647:
3642:
3637:
3632:
3627:
3622:
3620:Borel subgroup
3617:
3610:
3607:
3590:
3585:
3581:
3577:
3574:
3548:
3545:
3534:Coxeter groups
3523:
3520:
3504:
3486:
3483:
3480:
3476:
3454:
3432:
3428:
3425:
3402:
3399:
3396:
3392:
3371:
3368:
3365:
3360:
3355:
3333:
3330:
3327:
3323:
3302:
3281:
3256:
3244:
3241:
3222:
3217:
3214:
3187:
3183:
3160:
3156:
3135:
3132:
3129:
3107:
3102:
3080:
3059:group topology
3041:
3028:
3025:
2972:
2969:
2956:Main article:
2953:
2950:
2918:Levi's theorem
2914:
2913:
2894:
2891:algebraic tori
2888:
2882:
2862:
2854:
2851:
2849:
2846:
2845:
2842:
2839:
2837:
2834:
2833:
2829:
2825:
2822:
2800:
2795:
2792:
2767:Main article:
2764:
2761:
2748:
2720:
2693:
2669:
2657:
2654:
2641:
2617:
2601:
2598:
2581:
2578:
2575:
2571:
2566:
2562:
2559:
2556:
2552:
2548:
2545:
2542:
2539:
2535:
2514:
2494:
2473:
2451:
2446:
2441:
2437:
2433:
2429:
2426:
2405:
2400:
2395:
2373:
2351:
2322:
2318:
2295:
2273:
2269:
2264:
2260:
2238:
2234:
2229:
2225:
2196:
2174:
2152:
2148:
2144:
2122:
2118:
2114:
2092:
2088:
2084:
2080:
2076:
2054:
2032:
2010:
1985:
1968:
1965:
1964:
1963:
1960:elliptic curve
1945:
1940:
1923:
1909:
1906:
1901:
1897:
1891:
1869:
1843:
1838:
1835:
1819:
1803:
1783:
1759:
1755:
1749:
1727:
1724:
1721:
1718:
1715:
1712:
1690:
1685:
1682:
1666:
1652:
1647:
1625:
1620:
1617:
1613:
1609:
1606:
1603:
1600:
1597:
1577:
1557:
1535:
1530:
1508:
1505:
1502:
1499:
1496:
1474:
1469:
1447:
1442:
1439:
1435:
1431:
1426:
1423:
1419:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1377:
1373:
1370:
1366:
1363:
1359:
1356:
1352:
1349:
1346:
1343:
1340:
1336:
1333:
1329:
1325:
1322:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1275:
1270:
1248:
1245:
1242:
1239:
1217:
1212:
1195:
1181:
1176:
1154:
1134:
1112:
1107:
1091:additive group
1085:
1082:
1068:
1064:
1060:
1038:
1034:
1030:
1026:
1022:
994:
991:
988:
984:
980:
977:
957:
936:
915:
903:
900:
876:affine variety
804:
803:
801:
800:
793:
786:
778:
775:
774:
771:
770:
768:Elliptic curve
764:
763:
757:
756:
750:
749:
743:
738:
737:
734:
733:
728:
727:
724:
721:
717:
713:
712:
711:
706:
704:Diffeomorphism
700:
699:
694:
689:
683:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
641:
640:
629:
628:
617:
616:
605:
604:
593:
592:
581:
580:
569:
568:
561:Special linear
557:
556:
549:General linear
545:
544:
539:
533:
524:
523:
520:
519:
516:
515:
510:
505:
497:
496:
483:
471:
458:
445:
443:Modular groups
441:
440:
439:
434:
421:
405:
402:
401:
396:
390:
389:
388:
385:
384:
379:
378:
377:
376:
371:
366:
363:
357:
356:
350:
349:
348:
347:
341:
340:
334:
333:
328:
319:
318:
316:Hall's theorem
313:
311:Sylow theorems
307:
306:
301:
293:
292:
291:
290:
284:
279:
276:Dihedral group
272:
271:
266:
260:
255:
249:
244:
233:
228:
227:
224:
223:
218:
217:
216:
215:
210:
202:
201:
200:
199:
194:
189:
184:
179:
174:
169:
167:multiplicative
164:
159:
154:
149:
141:
140:
139:
138:
133:
125:
124:
116:
115:
114:
113:
111:Wreath product
108:
103:
98:
96:direct product
90:
88:Quotient group
82:
81:
80:
79:
74:
69:
59:
56:
55:
52:
51:
43:
42:
15:
9:
6:
4:
3:
2:
4149:
4138:
4135:
4133:
4130:
4129:
4127:
4117:
4114:
4113:
4104:
4100:
4096:
4092:
4088:
4085:
4079:
4075:
4071:
4070:
4065:
4061:
4058:
4054:
4050:
4044:
4040:
4035:
4032:
4028:
4024:
4018:
4014:
4010:
4006:
4002:
4000:
3999:
3994:
3991:
3987:
3983:
3977:
3973:
3969:
3965:
3961:
3957:
3953:
3950:
3944:
3940:
3936:
3932:
3928:
3925:
3921:
3917:
3911:
3907:
3903:
3899:
3894:
3890:
3886:
3882:
3878:
3877:Borel, Armand
3874:
3865:on 2014-11-04
3864:
3860:
3856:
3852:
3851:
3845:
3844:
3829:
3825:
3819:
3812:
3807:
3800:
3795:
3787:
3783:
3779:
3775:
3768:
3761:
3756:
3749:
3744:
3737:
3732:
3726:, 3.5, p. 65.
3725:
3720:
3713:
3708:
3701:
3696:
3689:
3684:
3677:
3672:
3665:
3660:
3656:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3612:
3606:
3604:
3588:
3583:
3575:
3565:
3563:
3546:
3543:
3535:
3529:
3519:
3516:
3512:
3510:
3500:
3481:
3426:
3423:
3414:
3397:
3366:
3358:
3328:
3300:
3270:
3254:
3247:If the field
3240:
3238:
3220:
3201:
3185:
3181:
3158:
3154:
3133:
3130:
3127:
3105:
3078:
3069:
3067:
3062:
3060:
3056:
3027:Connectedness
3024:
3022:
3018:
3014:
3010:
3006:
3002:
2998:
2994:
2993:perfect field
2990:
2986:
2982:
2978:
2968:
2966:
2959:
2949:
2947:
2943:
2942:number fields
2939:
2935:
2931:
2927:
2923:
2919:
2911:
2907:
2903:
2899:
2895:
2892:
2889:
2886:
2883:
2880:
2879:
2878:
2875:
2860:
2852:
2847:
2840:
2835:
2827:
2820:
2798:
2779:
2777:
2770:
2760:
2746:
2738:
2734:
2718:
2709:
2707:
2691:
2683:
2667:
2653:
2639:
2631:
2615:
2607:
2597:
2595:
2576:
2564:
2557:
2540:
2512:
2492:
2444:
2427:
2424:
2398:
2340:
2335:
2320:
2271:
2236:
2227:
2214:
2209:
2146:
2078:
2000:
1974:
1961:
1943:
1928:
1924:
1907:
1904:
1899:
1895:
1867:
1860:over a field
1859:
1841:
1824:
1820:
1817:
1801:
1781:
1757:
1753:
1725:
1722:
1716:
1688:
1671:
1667:
1650:
1618:
1615:
1611:
1607:
1604:
1595:
1575:
1555:
1533:
1503:
1500:
1497:
1472:
1440:
1437:
1433:
1429:
1424:
1421:
1417:
1404:
1401:
1398:
1371:
1368:
1364:
1361:
1357:
1354:
1350:
1334:
1331:
1327:
1323:
1320:
1313:
1307:
1304:
1301:
1273:
1246:
1243:
1240:
1237:
1215:
1200:
1196:
1179:
1152:
1132:
1110:
1096:
1092:
1088:
1087:
1081:
1024:
1012:
1008:
989:
978:
975:
955:
913:
899:
897:
893:
889:
885:
881:
877:
873:
868:
866:
862:
858:
857:matrix groups
854:
850:
846:
842:
838:
833:
831:
827:
823:
819:
815:
811:
799:
794:
792:
787:
785:
780:
779:
777:
776:
769:
766:
765:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
736:
735:
725:
722:
719:
718:
716:
710:
707:
705:
702:
701:
698:
695:
693:
690:
688:
685:
684:
681:
675:
673:
667:
665:
659:
657:
651:
649:
643:
642:
638:
634:
631:
630:
626:
622:
619:
618:
614:
610:
607:
606:
602:
598:
595:
594:
590:
586:
583:
582:
578:
574:
571:
570:
566:
562:
559:
558:
554:
550:
547:
546:
543:
540:
538:
535:
534:
531:
527:
522:
521:
514:
511:
509:
506:
504:
501:
500:
472:
447:
446:
444:
438:
435:
410:
407:
406:
400:
397:
395:
392:
391:
387:
386:
375:
372:
370:
367:
364:
361:
360:
359:
358:
355:
352:
351:
346:
343:
342:
339:
336:
335:
332:
329:
327:
325:
321:
320:
317:
314:
312:
309:
308:
305:
302:
300:
297:
296:
295:
294:
288:
285:
282:
277:
274:
273:
269:
264:
261:
258:
253:
250:
247:
242:
239:
238:
237:
236:
231:
230:Finite groups
226:
225:
214:
211:
209:
206:
205:
204:
203:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
144:
143:
142:
137:
134:
132:
129:
128:
127:
126:
123:
122:
118:
117:
112:
109:
107:
104:
102:
99:
97:
94:
91:
89:
86:
85:
84:
83:
78:
75:
73:
70:
68:
65:
64:
63:
62:
57:Basic notions
54:
53:
49:
45:
44:
41:
36:
32:
28:
27:
22:
4094:
4068:
4038:
4008:
3996:
3959:
3956:Milne, J. S.
3934:
3897:
3880:
3867:, retrieved
3863:the original
3849:
3831:. Retrieved
3828:MathOverflow
3827:
3818:
3806:
3794:
3777:
3773:
3767:
3755:
3743:
3738:, pp. 55-56.
3731:
3719:
3707:
3695:
3683:
3671:
3659:
3561:
3531:
3508:
3415:
3246:
3202:
3070:
3065:
3063:
3030:
3020:
3016:
3012:
3011:, such that
3008:
3004:
3000:
2996:
2988:
2974:
2967:of a curve.
2961:
2915:
2876:
2780:
2772:
2733:group object
2710:
2682:group scheme
2659:
2603:
2336:
2212:
2210:
1972:
1970:
1198:
1090:
1011:regular maps
905:
883:
869:
855:, etc. Many
834:
830:group theory
813:
807:
739:
636:
624:
612:
600:
588:
576:
564:
552:
323:
280:
267:
256:
245:
241:Cyclic group
119:
106:Free product
77:Group action
40:Group theory
35:Group theory
34:
4091:Weil, André
3931:Lang, Serge
3780:(1): 1â18.
3630:Morley rank
3269:local field
2934:Lie algebra
2630:Lie algebra
1927:cubic curve
1095:affine line
902:Definitions
810:mathematics
526:Topological
365:alternating
4126:Categories
3889:0726.20030
3869:2012-06-25
3811:Borel 1991
3799:Borel 1991
3786:1007.14005
3760:Borel 1991
3748:Borel 1991
3736:Borel 1991
3724:Borel 1991
3712:Borel 1991
3700:Borel 1991
3688:Borel 1991
3676:Borel 1991
3664:Borel 1991
3651:References
3625:Tame group
3564:-factorial
3053:carries a
2904:, or some
2902:Jet groups
2463:such that
1999:subvariety
633:Symplectic
573:Orthogonal
530:Lie groups
437:Free group
162:continuous
101:Direct sum
3499:Lie group
3182:μ
3155:μ
3131:≥
3066:connected
2924:) with a
2824:↦
2547:→
2493:π
2436:→
2425:π
2263:→
2147:×
2117:→
2087:→
2079:×
1616:−
1599:↦
1438:−
1422:−
1411:↦
1345:↦
1063:→
1033:→
1025:×
979:∈
697:Conformal
585:Euclidean
192:nilpotent
4093:(1971),
4066:(1979),
4007:(1970),
3958:(2017),
3933:(1983),
3879:(1991).
3813:, p. 47.
3801:, p. 16.
3678:, p. 46.
3609:See also
2896:certain
2737:category
2321:′
2272:′
2237:′
2213:morphism
1372:′
1358:′
1335:′
1324:′
1084:Examples
692:Poincaré
537:Solenoid
409:Integers
399:Lattices
374:sporadic
369:Lie type
197:solvable
187:dihedral
172:additive
157:infinite
67:Subgroup
4057:1642713
3990:3729270
3924:0396773
3859:0106966
3833:May 13,
3762:, 24.2.
3750:, 24.1.
3666:, p.54.
3497:into a
2735:in the
1009:), and
687:Lorentz
609:Unitary
508:Lattice
448:PSL(2,
182:abelian
93:(Semi-)
4103:322901
4101:
4080:
4055:
4045:
4031:138290
4029:
4019:
3988:
3978:
3945:
3922:
3912:
3887:
3857:
3784:
2995:, and
2938:p-adic
1201:: Let
1093:: the
816:is an
542:Circle
473:SL(2,
362:cyclic
326:-group
177:cyclic
152:finite
147:simple
131:kernel
3293:is a
3267:is a
2991:is a
2983:by a
2731:is a
2684:over
2632:over
1997:is a
1005:(the
948:over
822:group
812:, an
726:Sp(â)
723:SU(â)
136:image
4099:OCLC
4078:ISBN
4043:ISBN
4027:OCLC
4017:ISBN
3976:ISBN
3943:ISBN
3910:ISBN
3835:2022
2944:via
2164:and
2104:and
1821:The
1794:-by-
1668:The
1388:and
1197:The
1089:The
863:and
828:and
720:O(â)
709:Loop
528:and
3968:doi
3885:Zbl
3782:Zbl
3444:or
3416:If
3235:by
3200:).
3007:in
2708:).
2596:).
2208:).
2022:of
1971:An
1856:of
1711:det
808:In
635:Sp(
623:SU(
599:SO(
563:SL(
551:GL(
4128::
4076:,
4053:MR
4051:,
4025:,
4015:,
4011:,
3986:MR
3984:,
3974:,
3966:,
3962:,
3941:,
3920:MR
3918:,
3908:,
3900:,
3855:MR
3826:.
3778:17
3776:.
3503:SL
2948:.
2874:.
2759:.
2334:.
2211:A
1962:).
1665:).
894:.
867:.
851:,
847:,
843:,
832:.
611:U(
587:E(
575:O(
33:â
3970::
3891:.
3837:.
3788:.
3589:!
3584:q
3580:]
3576:n
3573:[
3562:q
3547:!
3544:n
3511:)
3509:R
3507:(
3505:2
3485:)
3482:k
3479:(
3475:G
3453:C
3431:R
3427:=
3424:k
3401:)
3398:k
3395:(
3391:G
3370:)
3367:k
3364:(
3359:n
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3329:k
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3322:G
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3128:n
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3040:G
3021:H
3019:/
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3013:H
3009:G
3005:H
3001:K
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2989:K
2912:.
2893:.
2887:.
2861:)
2853:1
2848:0
2841:x
2836:1
2828:(
2821:x
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2794:L
2791:G
2747:k
2719:k
2692:k
2668:k
2640:k
2616:k
2580:)
2577:k
2574:(
2570:H
2565:/
2561:)
2558:k
2555:(
2551:G
2544:)
2541:k
2538:(
2534:G
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2450:H
2445:/
2440:G
2432:G
2428::
2404:H
2399:/
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2372:G
2350:H
2317:G
2294:G
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2259:G
2233:G
2228:,
2224:G
2195:H
2173:H
2151:H
2143:H
2121:G
2113:G
2091:G
2083:G
2075:G
2053:G
2031:G
2009:H
1984:G
1944:2
1939:P
1908:1
1905:+
1900:2
1896:n
1890:A
1868:k
1842:n
1837:L
1834:G
1802:n
1782:n
1758:2
1754:n
1748:A
1726:1
1723:=
1720:)
1717:g
1714:(
1689:n
1684:L
1681:S
1651:1
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1605:x
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1194:.
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990:k
987:(
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23:.
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