1326:
1322:; or more practically, by expressing all such extensions in terms of mathematical objects that are easier to understand and compute. In general, this problem is very hard, and all the most useful results classify extensions that satisfy some additional condition.
2682:
1511:
220:
1412:
856:. Several other general classes of extensions are known but no theory exists that treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the
847:
3451:
1682:
2102:
1908:
2794:
2174:
1749:
2252:
2631:
2986:
2346:
1996:
1852:
1631:
1306:
gives us a complete list of finite simple groups; so the solution to the extension problem would give us enough information to construct and classify all finite groups in general.
2733:
2709:
2623:
2034:
1555:
2899:
2846:
1051:
1263:
894:
396:
2574:
2534:
2444:
2488:
1937:
1296:
1224:
332:
3442:
2382:
630:
1789:
1769:
1710:
1575:
1183:
1163:
1143:
1111:
1091:
1071:
1019:
999:
975:
955:
935:
915:
785:
765:
741:
721:
689:
665:
596:
570:
544:
524:
504:
484:
464:
444:
420:
356:
303:
283:
263:
243:
152:
132:
108:
88:
68:
1557:
making commutative the diagram of Figure 1. In fact it is sufficient to have a group homomorphism; due to the assumed commutativity of the diagram, the map
163:
1423:
1339:
1303:
637:
2592:
3086:
occurring in a central extension are Lie groups, and the maps between them are homomorphisms of Lie groups, then if the Lie algebra of
3400:
3432:
797:
3277:
2138:. Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from
3329:
2739:
2270:
3188:
1636:
2063:
1863:
2761:
2141:
2577:
3332:; Porter, Timothy (1996). "On the Schreier theory of non-abelian extensions: generalisations and computations".
1719:
3184:
2677:{\displaystyle 0\rightarrow {\mathfrak {a}}\rightarrow {\mathfrak {e}}\rightarrow {\mathfrak {g}}\rightarrow 0}
2219:
700:
2952:
2307:
1957:
1813:
1592:
1189:, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the
3468:
3257:
3129:
2581:
1333:
It is important to know when two extensions are equivalent or congruent. We say that the extensions
3165:
2714:
2690:
2604:
2491:
155:
2007:
1913:
where the left and right arrows are respectively the inclusion and the projection of each factor of
3233:
3145:
2853:
2800:
39:
1024:
2915:
2548:
2216:. However, in group theory the opposite terminology has crept in, partly because of the notation
2189:
2127:
1229:
1190:
978:
867:
364:
2553:
2497:
2407:
2467:
1916:
1523:
1268:
1196:
308:
3428:
3416:
3272:
3245:
3237:
3195:
3024:
2385:
2053:
668:
3262:
3199:
3169:
2401:
1792:
1713:
1578:
573:
3394:
3380:
3345:
3296:
3424:
3404:
3376:
3341:
3221:
3010:
2923:
2537:
2358:
2213:
576:
335:
43:
607:
2918:. Roughly speaking, central extensions of Lie groups by discrete groups are the same as
3449:
R. Brown and O. Mucuk, Covering groups of non-connected topological groups revisited,
3364:
3155:
2926:
2919:
2298:
2119:
2112:
1774:
1754:
1695:
1560:
1168:
1148:
1128:
1114:
1096:
1076:
1056:
1004:
984:
960:
940:
920:
900:
770:
750:
726:
706:
674:
650:
581:
555:
529:
509:
489:
469:
449:
429:
405:
359:
341:
288:
268:
248:
228:
137:
117:
93:
73:
53:
47:
215:{\displaystyle 1\to N\;{\overset {\iota }{\to }}\;G\;{\overset {\pi }{\to }}\;Q\to 1.}
3462:
3267:
2588:
2454:
2274:
1506:{\displaystyle 1\to K{\stackrel {i'}{{}\to {}}}G'{\stackrel {\pi '}{{}\to {}}}H\to 1}
744:
3225:
3168:, generators of symmetry groups correspond to conserved quantities, referred to as
2181:
2002:
1299:
633:
602:
599:
550:
1407:{\displaystyle 1\to K{\stackrel {i}{{}\to {}}}G{\stackrel {\pi }{{}\to {}}}H\to 1}
17:
2599:
1325:
853:
31:
2580:, in cases where the projective representation cannot be lifted to an ordinary
3180:
788:
399:
2277:'s theory of nonabelian extensions uses the terminology that an extension of
632:, all finite groups may be constructed as a series of extensions with finite
3360:
3241:
2911:
2118:
Split extensions are very easy to classify, because an extension is split
2997:
depends on the choice of an identity element mapping to the identity in
3440:
R.L. Taylor, Covering groups of non connected topological groups,
3034:
1314:
Solving the extension problem amounts to classifying all extensions of
2449:
Examples of central extensions can be constructed by taking any group
3208:
2536:
under the above correspondence. Another split example is given for a
1751:, but there are, up to group isomorphism, only four groups of order
2796:, a tedious but explicitly checkable existence condition involving
1324:
3319:(Third edition), John Wiley & Sons, Inc., Hoboken, NJ (2004).
3175:
The basic examples of central extensions as covering groups are:
2991:
is a group homomorphism, and surjective. (The group structure on
3300:
3452:
Mathematical
Proceedings of the Cambridge Philosophical Society
842:{\displaystyle \operatorname {Ext} _{\mathbb {Z} }^{1}(Q,N);}
2052:
by the quotient map of the short exact sequence induces the
1657:
2392:. The set of isomorphism classes of central extensions of
3224:. Here the central extension involved is well known in
2750:
There is a similar classification of all extensions of
2200:
In general in mathematics, an extension of a structure
747:, then the set of isomorphism classes of extensions of
3232:. A projective representation that corresponds is the
2955:
2856:
2803:
2764:
2717:
2693:
2634:
2607:
2556:
2500:
2470:
2410:
2361:
2310:
2222:
2144:
2066:
2010:
1960:
1919:
1866:
1816:
1777:
1757:
1722:
1698:
1639:
1595:
1563:
1526:
1426:
1342:
1271:
1232:
1199:
1171:
1151:
1131:
1099:
1079:
1059:
1027:
1007:
987:
963:
943:
923:
903:
870:
800:
773:
753:
729:
709:
677:
653:
610:
584:
558:
532:
512:
492:
472:
452:
432:
408:
367:
344:
311:
291:
271:
251:
231:
166:
140:
120:
96:
76:
56:
2914:
theory, central extensions arise in connection with
1193:
of a finite group is a finite sequence of subgroups
3315:page no. 830, Dummit, David S., Foote, Richard M.,
2738:There is a general theory of central extensions in
2576:. More serious examples are found in the theory of
1520:(or congruent) if there exists a group isomorphism
3407:. From his collection of short mathematical notes.
2980:
2893:
2840:
2788:
2727:
2703:
2676:
2617:
2568:
2528:
2482:
2438:
2376:
2340:
2246:
2188:. For a full discussion of why this is true, see
2168:
2096:
2028:
1990:
1931:
1902:
1846:
1783:
1763:
1743:
1704:
1692:are isomorphic as groups. For instance, there are
1676:
1625:
1569:
1549:
1505:
1406:
1290:
1257:
1218:
1177:
1157:
1137:
1105:
1085:
1065:
1045:
1013:
993:
969:
949:
929:
909:
888:
841:
779:
759:
735:
715:
683:
659:
624:
590:
564:
538:
518:
498:
478:
458:
438:
414:
390:
350:
326:
297:
277:
257:
237:
214:
146:
126:
102:
82:
62:
27:Group for which a given group is a normal subgroup
3443:Proceedings of the American Mathematical Society
636:. This fact was a motivation for completing the
422:. Group extensions arise in the context of the
1677:{\displaystyle 1\to K\to G^{\prime }\to H\to 1}
2104:. In this situation, it is usually said that
486:are to be determined. Note that the phrasing "
3334:Proceedings of the Royal Irish Academy Sect A
3187:, which (in even dimension) doubly cover the
2097:{\displaystyle \pi \circ s=\mathrm {id} _{H}}
1903:{\displaystyle 1\to K\to K\times H\to H\to 1}
8:
2789:{\displaystyle G\to \operatorname {Out} (A)}
2169:{\displaystyle H\to \operatorname {Aut} (K)}
1285:
1272:
1252:
1233:
1213:
1200:
703:, is immediately obvious. If one requires
3365:"Central extensions in Malt'sev varieties"
1744:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
202:
191:
187:
176:
3158:. These generators are in the center of
2966:
2954:
2861:
2855:
2808:
2802:
2763:
2719:
2718:
2716:
2695:
2694:
2692:
2662:
2661:
2652:
2651:
2642:
2641:
2633:
2609:
2608:
2606:
2555:
2505:
2499:
2469:
2415:
2409:
2400:is in one-to-one correspondence with the
2360:
2309:
2247:{\displaystyle \operatorname {Ext} (Q,N)}
2221:
2143:
2088:
2080:
2065:
2009:
1959:
1918:
1865:
1815:
1776:
1756:
1737:
1736:
1728:
1724:
1723:
1721:
1697:
1656:
1638:
1594:
1562:
1525:
1481:
1478:
1473:
1472:
1470:
1469:
1448:
1445:
1440:
1439:
1437:
1436:
1425:
1387:
1384:
1379:
1378:
1376:
1375:
1364:
1361:
1356:
1355:
1353:
1352:
1341:
1279:
1270:
1240:
1231:
1207:
1198:
1170:
1150:
1130:
1098:
1078:
1058:
1026:
1006:
986:
962:
942:
922:
902:
869:
812:
807:
806:
805:
799:
772:
752:
728:
708:
676:
652:
614:
609:
583:
557:
531:
511:
491:
471:
451:
431:
407:
371:
366:
343:
310:
290:
270:
250:
230:
192:
177:
165:
139:
119:
95:
75:
55:
2494:example corresponds to the element 0 in
1117:provide further examples of extensions.
3289:
3228:theory, in the case of forms of weight
2598:Similarly, the central extension of a
2254:, which reads easily as extensions of
1791:with quotient group isomorphic to the
1771:containing a normal subgroup of order
1577:is forced to be an isomorphism by the
1304:classification of finite simple groups
638:classification of finite simple groups
3369:Theory and Applications of Categories
3220:involves a fundamental group that is
2981:{\displaystyle \pi \colon G^{*}\to G}
2341:{\displaystyle 1\to A\to E\to G\to 1}
1991:{\displaystyle 1\to K\to G\to H\to 1}
1847:{\displaystyle 1\to K\to G\to H\to 1}
1626:{\displaystyle 1\to K\to G\to H\to 1}
7:
3033:, which is known to be abelian (see
2946:, in such a way that the projection
2940:is naturally a central extension of
2746:Generalization to general extensions
1857:that is equivalent to the extension
3244:. Metaplectic groups also occur in
2720:
2696:
2663:
2653:
2643:
2610:
2593:universal perfect central extension
2212:is a substructure. See for example
2204:is usually regarded as a structure
38:is a general means of describing a
2084:
2081:
1589:It may happen that the extensions
25:
3037:). Conversely, given a Lie group
3278:Extension of a topological group
3068:More generally, when the groups
3043:and a discrete central subgroup
2262:, and the focus is on the group
466:are known and the properties of
2758:in terms of homomorphisms from
2728:{\displaystyle {\mathfrak {e}}}
2704:{\displaystyle {\mathfrak {a}}}
2618:{\displaystyle {\mathfrak {g}}}
1712:inequivalent extensions of the
2972:
2888:
2885:
2879:
2867:
2835:
2832:
2826:
2814:
2783:
2777:
2768:
2668:
2658:
2648:
2638:
2523:
2511:
2433:
2421:
2371:
2365:
2332:
2326:
2320:
2314:
2241:
2229:
2163:
2157:
2148:
2029:{\displaystyle s\colon H\to G}
2020:
1982:
1976:
1970:
1964:
1894:
1888:
1876:
1870:
1838:
1832:
1826:
1820:
1668:
1662:
1649:
1643:
1617:
1611:
1605:
1599:
1536:
1497:
1475:
1442:
1430:
1398:
1381:
1358:
1346:
863:To consider some examples, if
833:
821:
385:
379:
321:
315:
206:
194:
179:
170:
1:
3130:central Lie algebra extension
2894:{\displaystyle H^{2}(G,Z(A))}
2841:{\displaystyle H^{3}(G,Z(A))}
787:is in fact a group, which is
1943:Classifying split extensions
1125:The question of what groups
1046:{\displaystyle G=K\rtimes H}
3423:, Classics in Mathematics,
3189:projective orthogonal group
3065:is a covering space of it.
1258:{\displaystyle \{A_{i+1}\}}
889:{\displaystyle G=K\times H}
767:by a given (abelian) group
391:{\displaystyle G/\iota (N)}
3485:
3455:, vol. 115 (1994), 97–110.
3297:group+extension#Definition
3144:. In the terminology of
2578:projective representations
2569:{\displaystyle A\rtimes G}
2529:{\displaystyle H^{2}(G,A)}
2439:{\displaystyle H^{2}(G,A)}
2281:gives a larger structure.
1113:, so such products as the
3446:, vol. 5 (1954), 753–768.
3198:, which double cover the
3185:special orthogonal groups
3183:, which double cover the
2934:of a connected Lie group
2849:and the cohomology group
2483:{\displaystyle A\times G}
1932:{\displaystyle K\times H}
1550:{\displaystyle T:G\to G'}
1291:{\displaystyle \{A_{i}\}}
1219:{\displaystyle \{A_{i}\}}
643:An extension is called a
327:{\displaystyle \iota (N)}
42:in terms of a particular
917:is an extension of both
546:" is also used by some.
3236:, constructed from the
3361:Kelly, Gregory Maxwell
3240:, in this case on the
2982:
2895:
2842:
2790:
2729:
2705:
2678:
2619:
2587:In the case of finite
2570:
2530:
2484:
2440:
2378:
2342:
2273:and Timothy Porter on
2248:
2196:Warning on terminology
2170:
2098:
2030:
1992:
1933:
1904:
1848:
1785:
1765:
1745:
1706:
1678:
1627:
1571:
1551:
1507:
1408:
1330:
1310:Classifying extensions
1292:
1259:
1220:
1179:
1159:
1139:
1107:
1087:
1067:
1047:
1015:
995:
971:
951:
931:
911:
890:
843:
781:
761:
737:
717:
685:
661:
626:
592:
566:
540:
520:
500:
480:
460:
440:
416:
392:
352:
328:
299:
279:
259:
239:
216:
148:
128:
104:
84:
64:
3395:Group Extensions and
3258:Lie algebra extension
3003:.) For example, when
2983:
2896:
2843:
2791:
2730:
2706:
2679:
2625:is an exact sequence
2620:
2582:linear representation
2571:
2531:
2485:
2441:
2379:
2343:
2249:
2171:
2099:
2036:such that going from
2031:
1993:
1934:
1905:
1849:
1786:
1766:
1746:
1707:
1684:are inequivalent but
1679:
1628:
1572:
1552:
1508:
1409:
1328:
1293:
1260:
1221:
1180:
1160:
1140:
1108:
1088:
1068:
1048:
1016:
996:
972:
957:. More generally, if
952:
932:
912:
891:
844:
782:
762:
738:
718:
695:Extensions in general
686:
662:
627:
593:
567:
541:
521:
501:
481:
461:
441:
417:
393:
353:
329:
300:
280:
260:
240:
217:
149:
129:
105:
90:are two groups, then
85:
65:
2953:
2922:. More precisely, a
2854:
2801:
2762:
2715:
2711:is in the center of
2691:
2632:
2605:
2554:
2498:
2468:
2408:
2377:{\displaystyle Z(E)}
2359:
2308:
2220:
2142:
2064:
2008:
1958:
1917:
1864:
1814:
1775:
1755:
1720:
1696:
1637:
1593:
1561:
1524:
1424:
1340:
1269:
1230:
1197:
1169:
1149:
1129:
1097:
1077:
1057:
1025:
1005:
985:
961:
941:
921:
901:
868:
798:
771:
751:
727:
707:
675:
651:
608:
582:
556:
530:
510:
490:
470:
450:
430:
406:
365:
342:
309:
289:
269:
249:
229:
164:
156:short exact sequence
138:
118:
94:
74:
54:
3359:Janelidze, George;
3234:Weil representation
3146:theoretical physics
3059:is a Lie group and
1265:is an extension of
1073:is an extension of
817:
699:One extension, the
625:{\displaystyle G/N}
506:is an extension of
426:, where the groups
245:is an extension of
3417:Mac Lane, Saunders
3403:2018-05-17 at the
3196:metaplectic groups
2978:
2916:algebraic topology
2891:
2838:
2786:
2725:
2701:
2674:
2615:
2566:
2549:semidirect product
2526:
2480:
2436:
2374:
2338:
2244:
2190:semidirect product
2166:
2128:semidirect product
2094:
2026:
1988:
1929:
1900:
1844:
1799:Trivial extensions
1781:
1761:
1741:
1702:
1674:
1623:
1567:
1547:
1503:
1404:
1331:
1288:
1255:
1216:
1191:composition series
1175:
1155:
1145:are extensions of
1135:
1103:
1083:
1063:
1043:
1011:
991:
979:semidirect product
967:
947:
927:
907:
886:
839:
801:
777:
757:
733:
713:
681:
657:
622:
588:
562:
536:
516:
496:
476:
456:
436:
412:
388:
348:
324:
295:
275:
255:
235:
212:
144:
124:
100:
80:
60:
3273:Group contraction
3246:quantum mechanics
3238:Fourier transform
3200:symplectic groups
3166:Noether's theorem
3025:fundamental group
2740:Maltsev varieties
2291:central extension
2285:Central extension
2048:and then back to
1805:trivial extension
1784:{\displaystyle 2}
1764:{\displaystyle 8}
1705:{\displaystyle 8}
1570:{\displaystyle T}
1491:
1458:
1392:
1369:
1187:extension problem
1178:{\displaystyle N}
1158:{\displaystyle H}
1138:{\displaystyle G}
1121:Extension problem
1106:{\displaystyle K}
1086:{\displaystyle H}
1066:{\displaystyle G}
1014:{\displaystyle H}
994:{\displaystyle K}
970:{\displaystyle G}
950:{\displaystyle K}
930:{\displaystyle H}
910:{\displaystyle G}
858:extension problem
780:{\displaystyle N}
760:{\displaystyle Q}
736:{\displaystyle Q}
716:{\displaystyle G}
684:{\displaystyle G}
660:{\displaystyle N}
645:central extension
591:{\displaystyle N}
565:{\displaystyle G}
539:{\displaystyle Q}
519:{\displaystyle N}
499:{\displaystyle G}
479:{\displaystyle G}
459:{\displaystyle N}
439:{\displaystyle Q}
424:extension problem
415:{\displaystyle Q}
351:{\displaystyle G}
298:{\displaystyle G}
278:{\displaystyle N}
258:{\displaystyle Q}
238:{\displaystyle G}
200:
185:
147:{\displaystyle N}
127:{\displaystyle Q}
103:{\displaystyle G}
83:{\displaystyle N}
63:{\displaystyle Q}
18:Extension problem
16:(Redirected from
3476:
3437:
3408:
3391:
3385:
3384:
3356:
3350:
3349:
3326:
3320:
3317:Abstract algebra
3313:
3307:
3294:
3263:Virasoro algebra
3231:
3219:
3163:
3153:
3148:, generators of
3143:
3137:
3127:
3121:
3115:
3109:
3103:
3097:
3091:
3085:
3079:
3073:
3064:
3058:
3048:
3042:
3032:
3019:, the kernel of
3018:
3008:
3002:
2996:
2987:
2985:
2984:
2979:
2971:
2970:
2945:
2939:
2933:
2901:
2900:
2898:
2897:
2892:
2866:
2865:
2848:
2847:
2845:
2844:
2839:
2813:
2812:
2795:
2793:
2792:
2787:
2734:
2732:
2731:
2726:
2724:
2723:
2710:
2708:
2707:
2702:
2700:
2699:
2683:
2681:
2680:
2675:
2667:
2666:
2657:
2656:
2647:
2646:
2624:
2622:
2621:
2616:
2614:
2613:
2575:
2573:
2572:
2567:
2535:
2533:
2532:
2527:
2510:
2509:
2489:
2487:
2486:
2481:
2445:
2443:
2442:
2437:
2420:
2419:
2383:
2381:
2380:
2375:
2347:
2345:
2344:
2339:
2253:
2251:
2250:
2245:
2175:
2173:
2172:
2167:
2103:
2101:
2100:
2095:
2093:
2092:
2087:
2035:
2033:
2032:
2027:
1997:
1995:
1994:
1989:
1951:is an extension
1938:
1936:
1935:
1930:
1909:
1907:
1906:
1901:
1853:
1851:
1850:
1845:
1807:is an extension
1793:Klein four-group
1790:
1788:
1787:
1782:
1770:
1768:
1767:
1762:
1750:
1748:
1747:
1742:
1740:
1732:
1727:
1714:Klein four-group
1711:
1709:
1708:
1703:
1683:
1681:
1680:
1675:
1661:
1660:
1632:
1630:
1629:
1624:
1579:short five lemma
1576:
1574:
1573:
1568:
1556:
1554:
1553:
1548:
1546:
1512:
1510:
1509:
1504:
1493:
1492:
1490:
1489:
1480:
1479:
1474:
1471:
1468:
1460:
1459:
1457:
1456:
1447:
1446:
1441:
1438:
1413:
1411:
1410:
1405:
1394:
1393:
1391:
1386:
1385:
1380:
1377:
1371:
1370:
1368:
1363:
1362:
1357:
1354:
1297:
1295:
1294:
1289:
1284:
1283:
1264:
1262:
1261:
1256:
1251:
1250:
1225:
1223:
1222:
1217:
1212:
1211:
1184:
1182:
1181:
1176:
1164:
1162:
1161:
1156:
1144:
1142:
1141:
1136:
1112:
1110:
1109:
1104:
1092:
1090:
1089:
1084:
1072:
1070:
1069:
1064:
1052:
1050:
1049:
1044:
1020:
1018:
1017:
1012:
1000:
998:
997:
992:
976:
974:
973:
968:
956:
954:
953:
948:
936:
934:
933:
928:
916:
914:
913:
908:
896:
895:
893:
892:
887:
848:
846:
845:
840:
816:
811:
810:
786:
784:
783:
778:
766:
764:
763:
758:
742:
740:
739:
734:
722:
720:
719:
714:
690:
688:
687:
682:
666:
664:
663:
658:
647:if the subgroup
631:
629:
628:
623:
618:
597:
595:
594:
589:
571:
569:
568:
563:
545:
543:
542:
537:
525:
523:
522:
517:
505:
503:
502:
497:
485:
483:
482:
477:
465:
463:
462:
457:
445:
443:
442:
437:
421:
419:
418:
413:
397:
395:
394:
389:
375:
357:
355:
354:
349:
333:
331:
330:
325:
304:
302:
301:
296:
284:
282:
281:
276:
264:
262:
261:
256:
244:
242:
241:
236:
221:
219:
218:
213:
201:
193:
186:
178:
153:
151:
150:
145:
133:
131:
130:
125:
109:
107:
106:
101:
89:
87:
86:
81:
69:
67:
66:
61:
21:
3484:
3483:
3479:
3478:
3477:
3475:
3474:
3473:
3459:
3458:
3435:
3425:Springer Verlag
3415:
3412:
3411:
3405:Wayback Machine
3393:P. J. Morandi,
3392:
3388:
3375:(10): 219–226.
3358:
3357:
3353:
3328:
3327:
3323:
3314:
3310:
3295:
3291:
3286:
3254:
3229:
3222:infinite cyclic
3212:
3207:
3159:
3156:central charges
3149:
3139:
3133:
3123:
3117:
3111:
3105:
3099:
3093:
3087:
3081:
3075:
3069:
3060:
3050:
3049:, the quotient
3044:
3038:
3028:
3014:
3011:universal cover
3004:
2998:
2992:
2962:
2951:
2950:
2941:
2935:
2929:
2920:covering groups
2908:
2857:
2852:
2851:
2850:
2804:
2799:
2798:
2797:
2760:
2759:
2748:
2713:
2712:
2689:
2688:
2630:
2629:
2603:
2602:
2552:
2551:
2538:normal subgroup
2501:
2496:
2495:
2490:. This kind of
2466:
2465:
2411:
2406:
2405:
2357:
2356:
2355:is included in
2306:
2305:
2287:
2218:
2217:
2214:field extension
2198:
2140:
2139:
2079:
2062:
2061:
2006:
2005:
1956:
1955:
1949:split extension
1945:
1915:
1914:
1862:
1861:
1812:
1811:
1801:
1773:
1772:
1753:
1752:
1718:
1717:
1694:
1693:
1652:
1635:
1634:
1591:
1590:
1587:
1559:
1558:
1539:
1522:
1521:
1482:
1461:
1449:
1422:
1421:
1338:
1337:
1312:
1275:
1267:
1266:
1236:
1228:
1227:
1203:
1195:
1194:
1167:
1166:
1147:
1146:
1127:
1126:
1123:
1095:
1094:
1075:
1074:
1055:
1054:
1023:
1022:
1003:
1002:
983:
982:
959:
958:
939:
938:
919:
918:
899:
898:
866:
865:
864:
796:
795:
769:
768:
749:
748:
725:
724:
705:
704:
697:
673:
672:
649:
648:
606:
605:
580:
579:
577:normal subgroup
554:
553:
528:
527:
508:
507:
488:
487:
468:
467:
448:
447:
428:
427:
404:
403:
363:
362:
340:
339:
336:normal subgroup
307:
306:
287:
286:
267:
266:
247:
246:
227:
226:
162:
161:
136:
135:
116:
115:
92:
91:
72:
71:
52:
51:
44:normal subgroup
36:group extension
28:
23:
22:
15:
12:
11:
5:
3482:
3480:
3472:
3471:
3461:
3460:
3457:
3456:
3447:
3438:
3433:
3410:
3409:
3386:
3351:
3340:(2): 213–227.
3321:
3308:
3288:
3287:
3285:
3282:
3281:
3280:
3275:
3270:
3265:
3260:
3253:
3250:
3210:
3204:
3203:
3192:
3110:, and that of
2989:
2988:
2977:
2974:
2969:
2965:
2961:
2958:
2927:covering space
2907:
2904:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2864:
2860:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2811:
2807:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2747:
2744:
2722:
2698:
2685:
2684:
2673:
2670:
2665:
2660:
2655:
2650:
2645:
2640:
2637:
2612:
2589:perfect groups
2565:
2562:
2559:
2525:
2522:
2519:
2516:
2513:
2508:
2504:
2479:
2476:
2473:
2460:, and setting
2435:
2432:
2429:
2426:
2423:
2418:
2414:
2373:
2370:
2367:
2364:
2349:
2348:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2299:exact sequence
2286:
2283:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2197:
2194:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2120:if and only if
2113:exact sequence
2091:
2086:
2083:
2078:
2075:
2072:
2069:
2025:
2022:
2019:
2016:
2013:
1999:
1998:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1944:
1941:
1928:
1925:
1922:
1911:
1910:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1855:
1854:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1800:
1797:
1780:
1760:
1739:
1735:
1731:
1726:
1701:
1673:
1670:
1667:
1664:
1659:
1655:
1651:
1648:
1645:
1642:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1586:
1583:
1566:
1545:
1542:
1538:
1535:
1532:
1529:
1514:
1513:
1502:
1499:
1496:
1488:
1485:
1477:
1467:
1464:
1455:
1452:
1444:
1435:
1432:
1429:
1415:
1414:
1403:
1400:
1397:
1390:
1383:
1374:
1367:
1360:
1351:
1348:
1345:
1311:
1308:
1287:
1282:
1278:
1274:
1254:
1249:
1246:
1243:
1239:
1235:
1215:
1210:
1206:
1202:
1185:is called the
1174:
1154:
1134:
1122:
1119:
1115:wreath product
1102:
1082:
1062:
1042:
1039:
1036:
1033:
1030:
1010:
990:
966:
946:
926:
906:
885:
882:
879:
876:
873:
850:
849:
838:
835:
832:
829:
826:
823:
820:
815:
809:
804:
776:
756:
745:abelian groups
732:
712:
701:direct product
696:
693:
680:
656:
621:
617:
613:
587:
561:
535:
515:
495:
475:
455:
435:
411:
387:
384:
381:
378:
374:
370:
360:quotient group
347:
323:
320:
317:
314:
294:
274:
254:
234:
223:
222:
211:
208:
205:
199:
196:
190:
184:
181:
175:
172:
169:
154:if there is a
143:
123:
99:
79:
59:
48:quotient group
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3481:
3470:
3467:
3466:
3464:
3454:
3453:
3448:
3445:
3444:
3439:
3436:
3434:3-540-58662-8
3430:
3426:
3422:
3418:
3414:
3413:
3406:
3402:
3399:
3398:
3390:
3387:
3382:
3378:
3374:
3370:
3366:
3362:
3355:
3352:
3347:
3343:
3339:
3335:
3331:
3330:Brown, Ronald
3325:
3322:
3318:
3312:
3309:
3305:
3303:
3298:
3293:
3290:
3283:
3279:
3276:
3274:
3271:
3269:
3268:HNN extension
3266:
3264:
3261:
3259:
3256:
3255:
3251:
3249:
3247:
3243:
3239:
3235:
3227:
3223:
3218:
3216:
3201:
3197:
3193:
3190:
3186:
3182:
3178:
3177:
3176:
3173:
3171:
3167:
3162:
3157:
3152:
3147:
3142:
3136:
3131:
3126:
3120:
3114:
3108:
3102:
3096:
3090:
3084:
3078:
3072:
3066:
3063:
3057:
3053:
3047:
3041:
3036:
3031:
3026:
3022:
3017:
3012:
3007:
3001:
2995:
2975:
2967:
2963:
2959:
2956:
2949:
2948:
2947:
2944:
2938:
2932:
2928:
2925:
2921:
2917:
2913:
2905:
2903:
2882:
2876:
2873:
2870:
2862:
2858:
2829:
2823:
2820:
2817:
2809:
2805:
2780:
2774:
2771:
2765:
2757:
2753:
2745:
2743:
2741:
2736:
2671:
2635:
2628:
2627:
2626:
2601:
2596:
2594:
2591:, there is a
2590:
2585:
2583:
2579:
2563:
2560:
2557:
2550:
2546:
2542:
2539:
2520:
2517:
2514:
2506:
2502:
2493:
2477:
2474:
2471:
2463:
2459:
2456:
2455:abelian group
2452:
2447:
2430:
2427:
2424:
2416:
2412:
2403:
2399:
2395:
2391:
2388:of the group
2387:
2368:
2362:
2354:
2335:
2329:
2323:
2317:
2311:
2304:
2303:
2302:
2300:
2296:
2292:
2284:
2282:
2280:
2276:
2275:Otto Schreier
2272:
2267:
2265:
2261:
2257:
2238:
2235:
2232:
2226:
2223:
2215:
2211:
2207:
2203:
2195:
2193:
2191:
2187:
2183:
2179:
2160:
2154:
2151:
2145:
2137:
2133:
2129:
2125:
2121:
2116:
2114:
2110:
2107:
2089:
2076:
2073:
2070:
2067:
2059:
2055:
2051:
2047:
2043:
2039:
2023:
2017:
2014:
2011:
2004:
1985:
1979:
1973:
1967:
1961:
1954:
1953:
1952:
1950:
1942:
1940:
1926:
1923:
1920:
1897:
1891:
1885:
1882:
1879:
1873:
1867:
1860:
1859:
1858:
1841:
1835:
1829:
1823:
1817:
1810:
1809:
1808:
1806:
1798:
1796:
1794:
1778:
1758:
1733:
1729:
1715:
1699:
1691:
1687:
1671:
1665:
1653:
1646:
1640:
1620:
1614:
1608:
1602:
1596:
1584:
1582:
1580:
1564:
1543:
1540:
1533:
1530:
1527:
1519:
1500:
1494:
1486:
1483:
1465:
1462:
1453:
1450:
1433:
1427:
1420:
1419:
1418:
1401:
1395:
1388:
1372:
1365:
1349:
1343:
1336:
1335:
1334:
1327:
1323:
1321:
1317:
1309:
1307:
1305:
1301:
1280:
1276:
1247:
1244:
1241:
1237:
1226:, where each
1208:
1204:
1192:
1188:
1172:
1152:
1132:
1120:
1118:
1116:
1100:
1080:
1060:
1040:
1037:
1034:
1031:
1028:
1021:, written as
1008:
988:
980:
964:
944:
924:
904:
883:
880:
877:
874:
871:
861:
859:
855:
836:
830:
827:
824:
818:
813:
802:
794:
793:
792:
790:
774:
754:
746:
730:
710:
702:
694:
692:
678:
670:
654:
646:
641:
639:
635:
634:simple groups
619:
615:
611:
604:
601:
585:
578:
575:
559:
552:
547:
533:
513:
493:
473:
453:
433:
425:
409:
402:to the group
401:
382:
376:
372:
368:
361:
345:
337:
318:
312:
292:
272:
252:
232:
209:
203:
197:
188:
182:
173:
167:
160:
159:
158:
157:
141:
121:
113:
97:
77:
57:
49:
45:
41:
37:
33:
19:
3469:Group theory
3450:
3441:
3420:
3396:
3389:
3372:
3368:
3354:
3337:
3333:
3324:
3316:
3311:
3301:
3292:
3226:modular form
3214:
3206:The case of
3205:
3174:
3160:
3150:
3140:
3134:
3124:
3118:
3112:
3106:
3100:
3094:
3088:
3082:
3076:
3070:
3067:
3061:
3055:
3051:
3045:
3039:
3029:
3020:
3015:
3005:
2999:
2993:
2990:
2942:
2936:
2930:
2909:
2755:
2751:
2749:
2737:
2686:
2597:
2586:
2544:
2540:
2461:
2457:
2450:
2448:
2397:
2393:
2389:
2352:
2350:
2294:
2290:
2288:
2278:
2271:Ronald Brown
2268:
2263:
2259:
2255:
2209:
2205:
2201:
2199:
2185:
2182:automorphism
2177:
2176:, where Aut(
2135:
2131:
2123:
2117:
2108:
2105:
2057:
2054:identity map
2049:
2045:
2041:
2037:
2003:homomorphism
2000:
1948:
1946:
1912:
1856:
1804:
1802:
1689:
1685:
1588:
1517:
1515:
1416:
1332:
1319:
1315:
1313:
1300:simple group
1186:
1124:
862:
857:
851:
698:
667:lies in the
644:
642:
603:factor group
572:possesses a
551:finite group
548:
423:
305:is a group,
224:
111:
35:
29:
3306:Remark 2.2.
3181:spin groups
3154:are called
2600:Lie algebra
2547:set to the
2297:is a short
2293:of a group
2269:A paper of
854:Ext functor
32:mathematics
3284:References
3098:, that of
2906:Lie groups
2687:such that
2402:cohomology
2351:such that
2301:of groups
2122:the group
2111:the above
1518:equivalent
789:isomorphic
549:Since any
400:isomorphic
3242:real line
2973:→
2968:∗
2960::
2957:π
2924:connected
2912:Lie group
2775:
2769:→
2669:→
2659:→
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2227:
2208:of which
2184:group of
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2155:
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2071:∘
2068:π
2021:→
2015::
1983:→
1977:→
1971:→
1965:→
1924:×
1895:→
1889:→
1883:×
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881:×
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377:ι
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198:π
195:→
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180:→
171:→
112:extension
3463:Category
3421:Homology
3419:(1975),
3401:Archived
3363:(2000).
3252:See also
2453:and any
1544:′
1487:′
1466:′
1454:′
1329:Figure 1
1298:by some
852:cf. the
358:and the
3381:1774075
3346:1641218
3299:at the
3170:charges
3122:, then
3035:H-space
3023:is the
3009:is the
2001:with a
1585:Warning
1053:, then
897:, then
574:maximal
285:, then
3431:
3379:
3344:
2464:to be
2404:group
2386:center
2384:, the
2109:splits
2060:i.e.,
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743:to be
669:center
600:simple
110:is an
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