724:
1198:
473:
1759:
2511:
2466:
2597:
384:
768:
2321:
2210:
2139:
1354:
961:
180:
has additional structure, then it may be the case that not all bijections on the set preserve this structure, in which case the automorphism group will be a subgroup of the symmetric group on
2553:
299:
1884:
1428:
1321:
1077:
1031:
925:
2421:
2649:
2357:
2278:
2242:
2171:
1460:
1386:
1113:
993:
116:
812:
2760:
2789:
1617:
2723:
595:
567:
527:
1947:
1806:
2097:
1675:
1648:
1571:
1544:
1517:
1490:
1843:
636:
1283:
883:
217:
2823:. Third, by convention, a Lie group is second countable and has at most coutably many connected components; thus, the general case reduces to the connected case.
2817:
2037:
2009:
1989:
1969:
1924:
1904:
1695:
493:
321:
1146:
396:
1700:
1038:
1034:
387:
2615:. Even better, it is represented by a scheme (since the automorphism groups are defined by polynomials): this scheme is called the
2471:
2426:
2558:
3031:
2987:
2949:
341:
2668:
729:
2287:
2176:
2105:
1326:
933:
2520:
3106:
3066:
2604:
262:
1848:
2959:
1395:
1288:
1044:
998:
892:
2388:
3023:
2971:
2622:
2330:
2251:
2215:
2144:
1433:
1359:
1086:
966:
89:
3138:
773:
3123:
1389:
2728:
3058:
2963:
2765:
1576:
3018:
2704:
1127:
576:
548:
508:
2663:
1225:
56:
1930:
257:
2360:
1769:
1462:, or that each choice of a base point is precisely a choice of a trivialization of the torsor.
1253:
534:
67:
1779:
3092:
849:
719:{\displaystyle G\to \operatorname {Aut} (X),\,g\mapsto \sigma _{g},\,\sigma _{g}(x)=g\cdot x}
2076:
3076:
3041:
2997:
2941:
2370:
Now base extensions applied to the above discussion determines a functor: namely, for each
2048:
1653:
1626:
1549:
1522:
1495:
1468:
832:
328:
75:
44:
3084:
1819:
8:
2056:
1262:
1245:
862:
250:
224:
194:
32:
2877:
2802:
2654:
In general, however, an automorphism group functor may not be represented by a scheme.
2612:
2022:
1994:
1974:
1954:
1909:
1889:
1680:
1201:
1134:
614:
478:
306:
119:
3102:
3062:
3027:
3001:
2983:
2945:
1130:
335:
157:
3080:
3013:
2975:
2371:
242:
231:
3096:
3072:
3052:
3037:
2993:
2100:
1971:
is a module category like the category of finite-dimensional vector spaces, then
1214:
1193:{\displaystyle \operatorname {Aut} (P)\hookrightarrow \operatorname {GL} _{n}(R)}
189:
169:
141:
2674:
130:
2979:
573:
has a structure of a Lie group induced from that on the automorphism group of
468:{\displaystyle {\overline {a}}\mapsto \sigma _{a}\in G,\,\sigma _{a}(x)=x^{a}}
3132:
3005:
2600:
2281:
496:
1392:. In practical terms, this says that a different choice of a base point of
606:
324:
235:
59:
36:
1754:{\displaystyle \operatorname {Aut} (X_{1})\to \operatorname {Aut} (X_{2})}
3048:
2060:
886:
503:
129:
Especially in geometric contexts, an automorphism group is also called a
20:
3124:
https://mathoverflow.net/questions/55042/automorphism-group-of-a-scheme
2881:
2324:
2071:
1241:
2974:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
1141:
928:
848:; these representations are the main object of study in the field of
530:
2888:
1927:
1233:
48:
2327:, and the invertibility is again described by polynomials. Hence,
164:
to itself is an automorphism, and hence the automorphism group of
1620:
140:
Automorphism groups are studied in a general way in the field of
3101:. Graduate Texts in Mathematics. Vol. 66. Springer Verlag.
2868:
Hochschild, G. (1952). "The
Automorphism Group of a Lie Group".
1080:
3057:. Annals of Mathematics Studies. Vol. 72. Princeton, NJ:
2506:{\displaystyle \operatorname {End} _{\text{alg}}(M\otimes R)}
2461:{\displaystyle \operatorname {End} _{\text{alg}}(M\otimes R)}
16:
Mathematical group formed from the automorphisms of an object
133:. A subgroup of an automorphism group is sometimes called a
1761:, as it maps invertible morphisms to invertible morphisms.
2592:{\displaystyle R\mapsto \operatorname {Aut} (M\otimes R)}
2043:
that is equipped with some algebraic structure (that is,
844:
as a group of linear transformations (automorphisms) of
1812:
a category, is called an action or a representation of
379:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}
160:
with no additional structure, then any bijection from
2805:
2768:
2731:
2707:
2625:
2561:
2523:
2474:
2429:
2391:
2333:
2290:
2254:
2218:
2179:
2147:
2108:
2079:
2025:
1997:
1977:
1957:
1933:
1912:
1892:
1851:
1822:
1782:
1703:
1683:
1656:
1629:
1579:
1552:
1525:
1498:
1471:
1436:
1398:
1362:
1329:
1291:
1265:
1149:
1089:
1047:
1001:
969:
936:
895:
865:
855:
Here are some other facts about automorphism groups:
818:
has more structure than just a set. For example, if
776:
763:{\displaystyle \varphi :G\to \operatorname {Aut} (X)}
732:
639:
579:
551:
511:
481:
399:
344:
309:
265:
197:
92:
2795:
is a central subgroup and the automorphism group of
2316:{\displaystyle \operatorname {End} _{\text{alg}}(M)}
2205:{\displaystyle \operatorname {End} _{\text{alg}}(M)}
2134:{\displaystyle \operatorname {End} _{\text{alg}}(M)}
1349:{\displaystyle A\mathrel {\overset {\sim }{\to }} B}
956:{\displaystyle A\mathrel {\overset {\sim }{\to }} B}
502:
The automorphism group of a finite-dimensional real
2725:. Second, every connected Lie group is of the form
2099:that preserve the algebraic structure: they form a
2811:
2783:
2754:
2717:
2643:
2591:
2547:
2505:
2460:
2415:
2351:
2315:
2272:
2236:
2204:
2165:
2133:
2091:
2039:be a finite-dimensional vector space over a field
2031:
2003:
1983:
1963:
1941:
1918:
1898:
1878:
1837:
1800:
1753:
1689:
1669:
1642:
1611:
1565:
1538:
1511:
1484:
1454:
1422:
1380:
1348:
1315:
1277:
1192:
1107:
1071:
1025:
995:, which is a symmetric group (see above), acts on
987:
955:
919:
877:
806:
762:
718:
589:
561:
521:
487:
467:
378:
315:
293:
219:is the group consisting of field automorphisms of
211:
110:
2870:Transactions of the American Mathematical Society
2423:preserving the algebraic structure: denote it by
3130:
2548:{\displaystyle \operatorname {Aut} (M\otimes R)}
184:. Some examples of this include the following:
2697:is simply connected, the automorphism group of
538:
1772:with a single object * or, more generally, if
1232:is the group consisting of all the invertible
1228:in a category, then the automorphism group of
2671:, a technique to remove an automorphism group
2014:
1213:Automorphism groups appear very naturally in
2958:
2936:Dummit, David S.; Foote, Richard M. (2004).
2894:
294:{\displaystyle \operatorname {PGL} _{n}(k).}
3026:, vol. 52, New York: Springer-Verlag,
2935:
2855:
1285:are objects in some category, then the set
3091:
3012:
2918:
2867:
2843:
1879:{\displaystyle F(\operatorname {Obj} (G))}
822:is a vector space, then a group action of
2468:. Then the unit group of the matrix ring
1935:
1423:{\displaystyle \operatorname {Iso} (A,B)}
1316:{\displaystyle \operatorname {Iso} (A,B)}
1072:{\displaystyle \operatorname {Iso} (A,B)}
1026:{\displaystyle \operatorname {Iso} (A,B)}
920:{\displaystyle \operatorname {Iso} (A,B)}
814:. This extends to the case when the set
684:
664:
432:
362:
349:
2416:{\displaystyle M\otimes R\to M\otimes R}
388:multiplicative group of integers modulo
86:is a group, then its automorphism group
2644:{\displaystyle \operatorname {Aut} (M)}
2352:{\displaystyle \operatorname {Aut} (M)}
2273:{\displaystyle \operatorname {End} (M)}
2237:{\displaystyle \operatorname {Aut} (M)}
2166:{\displaystyle \operatorname {End} (M)}
1455:{\displaystyle \operatorname {Aut} (B)}
1430:differs unambiguously by an element of
1381:{\displaystyle \operatorname {Aut} (B)}
1116:
1108:{\displaystyle \operatorname {Aut} (B)}
988:{\displaystyle \operatorname {Aut} (B)}
234:, the automorphism group is called the
111:{\displaystyle \operatorname {Aut} (X)}
3131:
3047:
2906:
2968:Representation theory. A first course
1208:
807:{\displaystyle g\cdot x=\varphi (g)x}
726:, and, conversely, each homomorphism
3098:Introduction to Affine Group Schemes
2791:is a simply connected Lie group and
1886:. Those objects are then said to be
2710:
625:and conversely. Indeed, each left
582:
554:
514:
13:
3054:Introduction to algebraic K-theory
2755:{\displaystyle {\widetilde {G}}/C}
14:
3150:
3117:
1776:is a groupoid, then each functor
569:, then the automorphism group of
62:, then the automorphism group of
2784:{\displaystyle {\widetilde {G}}}
1612:{\displaystyle F:C_{1}\to C_{2}}
545:is a Lie group with Lie algebra
393:, with the isomorphism given by
2718:{\displaystyle {\mathfrak {g}}}
1906:-objects (as they are acted by
885:be two finite sets of the same
590:{\displaystyle {\mathfrak {g}}}
562:{\displaystyle {\mathfrak {g}}}
522:{\displaystyle {\mathfrak {g}}}
118:is the group consisting of all
2912:
2900:
2861:
2849:
2837:
2687:
2638:
2632:
2586:
2574:
2565:
2542:
2530:
2500:
2488:
2455:
2443:
2401:
2346:
2340:
2310:
2304:
2267:
2261:
2231:
2225:
2199:
2193:
2160:
2154:
2128:
2122:
2083:
2055:). It can be, for example, an
1873:
1870:
1864:
1855:
1832:
1826:
1792:
1748:
1735:
1726:
1723:
1710:
1596:
1449:
1443:
1417:
1405:
1375:
1369:
1335:
1310:
1298:
1187:
1181:
1165:
1162:
1156:
1102:
1096:
1066:
1054:
1020:
1008:
982:
976:
942:
914:
902:
798:
792:
757:
751:
742:
701:
695:
668:
658:
652:
643:
529:has the structure of a (real)
449:
443:
410:
367:
345:
285:
279:
241:The automorphism group of the
168:in this case is precisely the
105:
99:
1:
3024:Graduate Texts in Mathematics
2972:Graduate Texts in Mathematics
2928:
2799:is the automorphism group of
2605:category of commutative rings
1697:induces a group homomorphism
621:to the automorphism group of
2830:
1942:{\displaystyle \mathbb {S} }
405:
230:. If the field extension is
188:The automorphism group of a
7:
2657:
147:
66:is the group of invertible
10:
3155:
3059:Princeton University Press
2517:is the automorphism group
2212:is the automorphism group
2015:Automorphism group functor
1519:are objects in categories
1252:. (For some examples, see
613:, the action amounts to a
2980:10.1007/978-1-4612-0979-9
2617:automorphism group scheme
1991:-objects are also called
2895:Fulton & Harris 1991
2846:, Ch. II, Example 7.1.1.
2680:
2664:Outer automorphism group
2323:is the zero set of some
2047:is a finite-dimensional
1801:{\displaystyle F:G\to C}
2856:Dummit & Foote 2004
1768:is a group viewed as a
533:(in fact, it is even a
303:The automorphism group
258:projective linear group
238:of the field extension.
3093:Waterhouse, William C.
2813:
2785:
2756:
2719:
2645:
2593:
2549:
2507:
2462:
2417:
2361:linear algebraic group
2353:
2317:
2274:
2238:
2206:
2167:
2135:
2093:
2092:{\displaystyle M\to M}
2033:
2005:
1985:
1965:
1943:
1920:
1900:
1880:
1839:
1802:
1755:
1691:
1671:
1644:
1613:
1567:
1540:
1513:
1486:
1456:
1424:
1382:
1350:
1317:
1279:
1194:
1109:
1073:
1027:
989:
957:
921:
879:
808:
764:
720:
591:
563:
535:linear algebraic group
523:
489:
469:
380:
317:
295:
213:
112:
68:linear transformations
2858:, § 2.3. Exercise 26.
2814:
2786:
2757:
2720:
2646:
2603:: a functor from the
2594:
2550:
2508:
2463:
2418:
2354:
2318:
2275:
2239:
2207:
2168:
2136:
2094:
2034:
2006:
1986:
1966:
1944:
1921:
1901:
1881:
1840:
1803:
1756:
1692:
1672:
1670:{\displaystyle X_{2}}
1645:
1643:{\displaystyle X_{1}}
1614:
1568:
1566:{\displaystyle C_{2}}
1541:
1539:{\displaystyle C_{1}}
1514:
1512:{\displaystyle X_{2}}
1487:
1485:{\displaystyle X_{1}}
1457:
1425:
1383:
1351:
1318:
1280:
1240:to itself. It is the
1195:
1110:
1074:
1028:
990:
958:
922:
880:
850:representation theory
809:
770:defines an action by
765:
721:
592:
564:
524:
490:
470:
381:
318:
296:
214:
113:
3049:Milnor, John Willard
2803:
2766:
2729:
2705:
2623:
2559:
2521:
2472:
2427:
2389:
2331:
2325:polynomial equations
2288:
2252:
2216:
2177:
2173:. The unit group of
2145:
2106:
2077:
2023:
1995:
1975:
1955:
1931:
1910:
1890:
1849:
1838:{\displaystyle F(*)}
1820:
1780:
1701:
1681:
1654:
1627:
1577:
1550:
1523:
1496:
1469:
1434:
1396:
1360:
1327:
1289:
1263:
1147:
1087:
1045:
999:
967:
934:
893:
863:
833:group representation
774:
730:
637:
577:
549:
509:
479:
397:
342:
307:
263:
195:
135:transformation group
90:
76:general linear group
3139:Group automorphisms
2057:associative algebra
1278:{\displaystyle A,B}
1246:endomorphism monoid
1202:inner automorphisms
1140:. Then there is an
1117:#In category theory
878:{\displaystyle A,B}
212:{\displaystyle L/K}
120:group automorphisms
51:. For example, if
3019:Algebraic Geometry
2809:
2781:
2752:
2715:
2641:
2619:and is denoted by
2613:category of groups
2589:
2545:
2503:
2458:
2413:
2349:
2313:
2270:
2244:. When a basis on
2234:
2202:
2163:
2131:
2089:
2029:
2001:
1981:
1961:
1939:
1916:
1896:
1876:
1835:
1798:
1764:In particular, if
1751:
1687:
1667:
1640:
1609:
1563:
1536:
1509:
1482:
1452:
1420:
1378:
1346:
1313:
1275:
1209:In category theory
1190:
1128:finitely generated
1105:
1069:
1041:; that is to say,
1023:
985:
953:
917:
875:
804:
760:
716:
615:group homomorphism
587:
559:
519:
485:
465:
376:
313:
291:
209:
108:
57:finite-dimensional
25:automorphism group
3033:978-0-387-90244-9
3014:Hartshorne, Robin
2989:978-0-387-97495-8
2951:978-0-471-43334-7
2812:{\displaystyle G}
2778:
2741:
2482:
2437:
2298:
2187:
2116:
2032:{\displaystyle M}
2004:{\displaystyle G}
1984:{\displaystyle G}
1964:{\displaystyle C}
1919:{\displaystyle G}
1899:{\displaystyle G}
1845:, or the objects
1690:{\displaystyle F}
1341:
1131:projective module
948:
629:-action on a set
488:{\displaystyle G}
475:. In particular,
408:
316:{\displaystyle G}
3146:
3112:
3088:
3044:
3009:
2955:
2940:(3rd ed.).
2938:Abstract Algebra
2922:
2916:
2910:
2904:
2898:
2897:, Exercise 8.28.
2892:
2886:
2885:
2865:
2859:
2853:
2847:
2841:
2824:
2818:
2816:
2815:
2810:
2790:
2788:
2787:
2782:
2780:
2779:
2771:
2761:
2759:
2758:
2753:
2748:
2743:
2742:
2734:
2724:
2722:
2721:
2716:
2714:
2713:
2691:
2650:
2648:
2647:
2642:
2598:
2596:
2595:
2590:
2554:
2552:
2551:
2546:
2512:
2510:
2509:
2504:
2484:
2483:
2480:
2467:
2465:
2464:
2459:
2439:
2438:
2435:
2422:
2420:
2419:
2414:
2372:commutative ring
2358:
2356:
2355:
2350:
2322:
2320:
2319:
2314:
2300:
2299:
2296:
2280:is the space of
2279:
2277:
2276:
2271:
2243:
2241:
2240:
2235:
2211:
2209:
2208:
2203:
2189:
2188:
2185:
2172:
2170:
2169:
2164:
2140:
2138:
2137:
2132:
2118:
2117:
2114:
2098:
2096:
2095:
2090:
2038:
2036:
2035:
2030:
2010:
2008:
2007:
2002:
1990:
1988:
1987:
1982:
1970:
1968:
1967:
1962:
1948:
1946:
1945:
1940:
1938:
1925:
1923:
1922:
1917:
1905:
1903:
1902:
1897:
1885:
1883:
1882:
1877:
1844:
1842:
1841:
1836:
1807:
1805:
1804:
1799:
1760:
1758:
1757:
1752:
1747:
1746:
1722:
1721:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1666:
1665:
1649:
1647:
1646:
1641:
1639:
1638:
1618:
1616:
1615:
1610:
1608:
1607:
1595:
1594:
1572:
1570:
1569:
1564:
1562:
1561:
1545:
1543:
1542:
1537:
1535:
1534:
1518:
1516:
1515:
1510:
1508:
1507:
1491:
1489:
1488:
1483:
1481:
1480:
1461:
1459:
1458:
1453:
1429:
1427:
1426:
1421:
1387:
1385:
1384:
1379:
1355:
1353:
1352:
1347:
1342:
1334:
1322:
1320:
1319:
1314:
1284:
1282:
1281:
1276:
1199:
1197:
1196:
1191:
1177:
1176:
1114:
1112:
1111:
1106:
1078:
1076:
1075:
1070:
1032:
1030:
1029:
1024:
994:
992:
991:
986:
962:
960:
959:
954:
949:
941:
926:
924:
923:
918:
884:
882:
881:
876:
813:
811:
810:
805:
769:
767:
766:
761:
725:
723:
722:
717:
694:
693:
680:
679:
596:
594:
593:
588:
586:
585:
568:
566:
565:
560:
558:
557:
528:
526:
525:
520:
518:
517:
494:
492:
491:
486:
474:
472:
471:
466:
464:
463:
442:
441:
422:
421:
409:
401:
385:
383:
382:
377:
375:
374:
365:
357:
352:
322:
320:
319:
314:
300:
298:
297:
292:
275:
274:
218:
216:
215:
210:
205:
117:
115:
114:
109:
3154:
3153:
3149:
3148:
3147:
3145:
3144:
3143:
3129:
3128:
3120:
3115:
3109:
3069:
3034:
2990:
2960:Fulton, William
2952:
2931:
2926:
2925:
2919:Waterhouse 2012
2917:
2913:
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2844:Hartshorne 1977
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2819:that preserves
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2669:Level structure
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2479:
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2381:, consider the
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2285:
2282:square matrices
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2101:vector subspace
2078:
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2017:
1996:
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1972:
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1215:category theory
1211:
1200:, unique up to
1172:
1168:
1148:
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1144:
1088:
1085:
1084:
1046:
1043:
1042:
1000:
997:
996:
968:
965:
964:
940:
935:
932:
931:
927:the set of all
894:
891:
890:
864:
861:
860:
840:, representing
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772:
771:
731:
728:
727:
689:
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308:
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201:
196:
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190:field extension
170:symmetric group
150:
142:category theory
91:
88:
87:
82:). If instead
74:to itself (the
17:
12:
11:
5:
3152:
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3127:
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3118:External links
3116:
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2876:(2): 209–216.
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2066:Now, consider
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176:. If the set
149:
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131:symmetry group
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35:consisting of
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2601:group functor
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839:
836:of the group
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37:automorphisms
34:
30:
27:of an object
26:
22:
3097:
3053:
3017:
2967:
2937:
2914:
2909:, Lemma 3.2.
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2890:
2873:
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2018:
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1464:
1258:
1249:
1237:
1229:
1221:
1219:
1212:
1137:
1123:
1039:transitively
854:
845:
841:
837:
831:
827:
823:
819:
815:
630:
626:
622:
618:
610:
602:
600:
570:
542:
389:
331:
325:cyclic group
323:of a finite
253:
244:
236:Galois group
227:
220:
181:
177:
173:
165:
161:
153:
151:
139:
134:
128:
123:
83:
79:
71:
63:
60:vector space
52:
40:
28:
24:
18:
2964:Harris, Joe
2907:Milnor 1971
2701:is that of
2248:is chosen,
2072:linear maps
2061:Lie algebra
887:cardinality
633:determines
605:is a group
504:Lie algebra
243:projective
45:composition
21:mathematics
3085:0237.18005
2929:References
2693:First, if
2011:-modules.
1356:is a left
1242:unit group
929:bijections
336:isomorphic
3095:(2012) .
3006:246650103
2831:Citations
2776:~
2739:~
2630:
2581:⊗
2572:
2566:↦
2537:⊗
2528:
2495:⊗
2486:
2450:⊗
2441:
2408:⊗
2402:→
2396:⊗
2338:
2302:
2259:
2223:
2191:
2152:
2120:
2084:→
1862:
1830:∗
1793:→
1733:
1727:→
1708:
1597:→
1573:, and if
1441:
1403:
1367:
1339:∼
1336:→
1296:
1234:morphisms
1179:
1166:↪
1154:
1142:embedding
1094:
1052:
1006:
974:
946:∼
943:→
900:
790:φ
781:⋅
749:
743:→
734:φ
711:⋅
687:σ
673:σ
669:↦
650:
644:→
609:on a set
531:Lie group
435:σ
424:∈
415:σ
411:↦
406:¯
372:×
277:
97:
49:morphisms
3133:Category
3051:(1971).
3016:(1977),
2966:(1991).
2921:, § 7.6.
2658:See also
1770:category
1623:mapping
148:Examples
3077:0349811
3042:0463157
2998:1153249
2882:1990752
2611:to the
2049:algebra
1949:-object
1926:); cf.
1677:, then
1621:functor
1323:of all
1244:of the
1133:over a
963:. Then
256:is the
249:over a
31:is the
3105:
3083:
3075:
3065:
3040:
3030:
3004:
2996:
2986:
2948:
2880:
2762:where
1390:torsor
1226:object
1224:is an
1081:torsor
1035:freely
607:acting
541:). If
537:: see
495:is an
386:, the
247:-space
232:Galois
43:under
23:, the
2942:Wiley
2878:JSTOR
2681:Notes
2607:over
2599:is a
2513:over
2377:over
2363:over
2359:is a
2059:or a
2051:over
1951:. If
1619:is a
1236:from
1126:be a
1115:(cf.
1079:is a
830:is a
617:from
539:below
329:order
251:field
223:that
156:is a
70:from
55:is a
33:group
3103:ISBN
3063:ISBN
3028:ISBN
3002:OCLC
2984:ISBN
2946:ISBN
2555:and
2284:and
2019:Let
1546:and
1492:and
1254:PROP
1135:ring
1122:Let
1083:for
1037:and
889:and
859:Let
3081:Zbl
2976:doi
2627:Aut
2569:Aut
2525:Aut
2481:alg
2477:End
2436:alg
2432:End
2335:Aut
2297:alg
2293:End
2256:End
2220:Aut
2186:alg
2182:End
2149:End
2141:of
2115:alg
2111:End
1859:Obj
1730:Aut
1705:Aut
1650:to
1465:If
1438:Aut
1400:Iso
1364:Aut
1293:Iso
1259:If
1256:.)
1248:of
1220:If
1151:Aut
1091:Aut
1049:Iso
1003:Iso
971:Aut
897:Iso
826:on
746:Aut
647:Aut
601:If
338:to
334:is
327:of
268:PGL
225:fix
172:of
158:set
152:If
126:.
122:of
94:Aut
78:of
47:of
39:of
19:In
3135::
3079:.
3073:MR
3071:.
3061:.
3038:MR
3036:,
3022:,
3000:.
2994:MR
2992:.
2982:.
2970:.
2962:;
2944:.
2874:72
2872:.
2651:.
2367:.
2063:.
1808:,
1217:.
1170:GL
1119:).
852:.
144:.
137:.
3111:.
3087:.
3008:.
2978::
2954:.
2884:.
2821:C
2807:G
2797:G
2793:C
2773:G
2750:C
2746:/
2736:G
2711:g
2699:G
2695:G
2639:)
2636:M
2633:(
2609:k
2587:)
2584:R
2578:M
2575:(
2563:R
2543:)
2540:R
2534:M
2531:(
2515:R
2501:)
2498:R
2492:M
2489:(
2456:)
2453:R
2447:M
2444:(
2411:R
2405:M
2399:R
2393:M
2383:R
2379:k
2375:R
2365:k
2347:)
2344:M
2341:(
2311:)
2308:M
2305:(
2268:)
2265:M
2262:(
2246:M
2232:)
2229:M
2226:(
2200:)
2197:M
2194:(
2161:)
2158:M
2155:(
2129:)
2126:M
2123:(
2087:M
2081:M
2070:-
2068:k
2053:k
2045:M
2041:k
2027:M
1999:G
1979:G
1959:C
1936:S
1914:G
1894:G
1874:)
1871:)
1868:G
1865:(
1856:(
1853:F
1833:)
1827:(
1824:F
1814:G
1810:C
1796:C
1790:G
1787::
1784:F
1774:G
1766:G
1749:)
1744:2
1740:X
1736:(
1724:)
1719:1
1715:X
1711:(
1685:F
1663:2
1659:X
1636:1
1632:X
1605:2
1601:C
1592:1
1588:C
1584::
1581:F
1559:2
1555:C
1532:1
1528:C
1505:2
1501:X
1478:1
1474:X
1450:)
1447:B
1444:(
1418:)
1415:B
1412:,
1409:A
1406:(
1388:-
1376:)
1373:B
1370:(
1344:B
1331:A
1311:)
1308:B
1305:,
1302:A
1299:(
1273:B
1270:,
1267:A
1250:X
1238:X
1230:X
1222:X
1204:.
1188:)
1185:R
1182:(
1174:n
1163:)
1160:P
1157:(
1138:R
1124:P
1103:)
1100:B
1097:(
1067:)
1064:B
1061:,
1058:A
1055:(
1021:)
1018:B
1015:,
1012:A
1009:(
983:)
980:B
977:(
951:B
938:A
915:)
912:B
909:,
906:A
903:(
873:B
870:,
867:A
846:X
842:G
838:G
828:X
824:G
820:X
816:X
802:x
799:)
796:g
793:(
787:=
784:x
778:g
758:)
755:X
752:(
740:G
737::
714:x
708:g
705:=
702:)
699:x
696:(
691:g
682:,
677:g
666:g
662:,
659:)
656:X
653:(
641:G
631:X
627:G
623:X
619:G
611:X
603:G
597:.
583:g
571:G
555:g
543:G
515:g
499:.
483:G
461:a
457:x
453:=
450:)
447:x
444:(
439:a
430:,
427:G
419:a
403:a
390:n
368:)
363:Z
359:n
355:/
350:Z
346:(
332:n
311:G
289:.
286:)
283:k
280:(
272:n
254:k
245:n
228:K
221:L
207:K
203:/
199:L
182:X
178:X
174:X
166:X
162:X
154:X
124:X
106:)
103:X
100:(
84:X
80:X
72:X
64:X
53:X
41:X
29:X
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