41:
2005:
1492:
1390:
1832:
1075:
1786:
1609:
901:
1728:
1135:
2115:
1164:
1107:
1664:
1514:
1412:
1353:
1264:
1223:
1198:
486:
461:
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1700:
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1571:
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951:
1864:
1006:
1032:
1748:
1629:
1542:
1455:
1432:
1331:
1284:
921:
1137:. An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a
788:
1911:. In this sense it has been proved that the classification problem for countable torsion-free abelian groups is as hard as possible.
848:
are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case.
2366:
2222:
346:
296:
1460:
1358:
1287:
1791:
2123:
781:
291:
1040:
2171:
2145:
2101:
2046:
1903:
The hardness of a classification problem for a certain type of structures on a countable set can be quantified using
845:
707:
2718:
2713:
2708:
774:
1753:
1576:
868:
1524:
Torsion-free abelian groups of rank 1 have been completely classified. To do so one associates to a group
391:
205:
2682:
2215:
1705:
1112:
123:
1140:
1083:
2010:
2277:
1929:
Thomas, Simon (2003), "The classification problem for torsion-free abelian groups of finite rank",
1634:
1304:
1077:, as only the integer 0 can be added to itself finitely many times to reach 0. More generally, the
924:
589:
323:
200:
88:
1497:
1395:
1336:
1228:
1206:
1172:
469:
444:
407:
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1669:
1169:
A non-finitely generated countable example is given by the additive group of the polynomial ring
2562:
1908:
739:
529:
1869:
1547:
2548:
2500:
2208:
2063:
Paolini, Gianluca; Shelah, Saharon (2021). "Torsion-Free
Abelian Groups are Borel Complete".
613:
956:
930:
2524:
2518:
2282:
1837:
838:
830:
553:
541:
159:
93:
1952:
1494:. Thus, torsion-free abelian groups of rank 1 are exactly subgroups of the additive group
982:
8:
2257:
2231:
1011:
822:
128:
23:
2457:
2273:
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1733:
1614:
1527:
1440:
1417:
1316:
1269:
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906:
113:
85:
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2190:
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2141:
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818:
518:
361:
255:
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684:
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2575:
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2019:
1948:
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337:
279:
154:
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826:
753:
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732:
689:
577:
500:
330:
244:
184:
64:
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103:
80:
2702:
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2448:
2396:
2348:
2303:
2001:
1392:. Equivalently it is the maximal cardinality of a linearly independent (over
863:
857:
814:
679:
601:
435:
308:
174:
1895:
is a complete isomorphism invariant for rank-1 torsion free abelian groups.
2609:
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144:
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98:
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32:
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802:
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2155:
1203:
More complicated examples are the additive group of the rational field
701:
429:
2510:
522:
2247:
2069:
59:
903:
is said to be torsion-free if no element other than the identity
401:
315:
2194:
2041:. Chicago Lectures in Mathematics. University of Chicago Press.
40:
2200:
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410:
1898:
1487:{\displaystyle \mathbb {Q} \otimes _{\mathbb {Z} }A}
1385:{\displaystyle \mathbb {Q} \otimes _{\mathbb {Z} }A}
2036:
1887:
1858:
1827:{\displaystyle n,m\in \mathbb {Z} \setminus \{0\}}
1826:
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1278:
1266:(rational numbers whose denominator is a power of
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1129:
1101:
1069:
1026:
1000:
971:
945:
915:
895:
480:
455:
418:
16:Abelian group with no non-trivial torsion elements
2006:"Abelian groups without elements of finite order"
2700:
1070:{\displaystyle \langle \mathbb {Z} ,+,0\rangle }
2183:Introduction To Modern Algebra, Revised Edition
2216:
2062:
2000:
1200:(the free abelian group of countable rank).
1037:A natural example of a torsion-free group is
782:
1821:
1815:
1775:
1769:
1598:
1592:
1064:
1044:
890:
872:
1573:of the prime numbers, as follows: pick any
1286:). Yet more involved examples are given by
2223:
2209:
2131:
1988:
789:
775:
2068:
1942:
1808:
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1502:
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1233:
1211:
1177:
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1123:
1089:
1048:
474:
449:
412:
2109:
2087:
1964:
1730:. This does not depend on the choice of
2701:
2367:Classification of finite simple groups
1928:
347:Classification of finite simple groups
2204:
2180:
1927:See for instance the introduction to
1781:{\displaystyle y\in A\setminus \{0\}}
1604:{\displaystyle x\in A\setminus \{0\}}
1457:is torsion-free then it injects into
896:{\displaystyle \langle G,+,0\rangle }
2154:
1976:
1293:
13:
2162:(Revised 3rd ed.), New York:
2090:A First Course In Abstract Algebra
14:
2730:
1899:Classification problem in general
1812:
1766:
1723:{\displaystyle k\in \mathbb {N} }
1589:
1519:
1130:{\displaystyle r\in \mathbb {N} }
846:finitely generated abelian groups
1159:{\displaystyle \mathbb {Z} ^{r}}
1102:{\displaystyle \mathbb {Z} ^{r}}
837:is the only element with finite
39:
2132:Hungerford, Thomas W. (1974),
2056:
2030:
1994:
1982:
1970:
1958:
1921:
1882:
1876:
1653:
1647:
1560:
1554:
1253:
1237:
1187:
1181:
851:
708:Infinite dimensional Lie group
1:
2081:
2039:Infinite Abelian group theory
2024:10.1215/S0012-7094-37-00308-9
1944:10.1090/S0894-0347-02-00409-5
1659:{\displaystyle p\in \tau (A)}
2116:Blaisdell Publishing Company
2037:Phillip A. Griffith (1970).
1509:{\displaystyle \mathbb {Q} }
1407:{\displaystyle \mathbb {Z} }
1348:{\displaystyle \mathbb {Q} }
1259:{\displaystyle \mathbb {Z} }
1218:{\displaystyle \mathbb {Q} }
1193:{\displaystyle \mathbb {Z} }
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
7:
2634:Infinite dimensional groups
1695:{\displaystyle x\in p^{k}A}
1225:, or its subgroups such as
206:List of group theory topics
10:
2735:
2230:
2088:Fraleigh, John B. (1976),
1302:
855:
811:torsion-free abelian group
2673:
2633:
2509:
2357:
2291:
2238:
2092:(2nd ed.), Reading:
2011:Duke Mathematical Journal
817:which has no non-trivial
2537:Special orthogonal group
2110:Herstein, I. N. (1964),
1914:
1888:{\displaystyle \tau (A)}
1566:{\displaystyle \tau (A)}
1333:is the dimension of the
1305:Rank of an abelian group
1109:is torsion-free for any
324:Elementary abelian group
201:Glossary of group theory
2181:McCoy, Neal H. (1968),
1298:
2563:Exceptional Lie groups
1909:descriptive set theory
1889:
1860:
1828:
1782:
1744:
1724:
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1327:
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1260:
1219:
1194:
1160:
1131:
1103:
1071:
1028:
1002:
973:
972:{\displaystyle x\in G}
947:
946:{\displaystyle n>0}
927:. Explicitly, for any
917:
897:
740:Linear algebraic group
482:
457:
420:
2549:Special unitary group
1890:
1861:
1859:{\displaystyle ny=mx}
1829:
1783:
1745:
1725:
1697:
1661:
1626:
1606:
1568:
1539:
1511:
1489:
1452:
1429:
1409:
1387:
1350:
1328:
1288:groups of higher rank
1281:
1261:
1220:
1195:
1161:
1132:
1104:
1072:
1029:
1003:
974:
948:
918:
898:
821:elements; that is, a
483:
458:
421:
2719:Properties of groups
2714:Abelian group theory
2709:Algebraic structures
2646:Diffeomorphism group
2525:Special linear group
2519:General linear group
1870:
1838:
1792:
1754:
1734:
1706:
1670:
1635:
1615:
1577:
1548:
1528:
1498:
1461:
1441:
1418:
1396:
1359:
1337:
1317:
1313:of an abelian group
1270:
1229:
1207:
1173:
1141:
1113:
1084:
1041:
1012:
1001:{\displaystyle nx=0}
983:
957:
931:
907:
869:
470:
445:
408:
2471:Other finite groups
2258:Commutator subgroup
1866:. Baer proved that
1027:{\displaystyle x=0}
953:, the only element
114:Group homomorphisms
24:Algebraic structure
2501:Rubik's Cube group
2458:Baby monster group
2268:Group homomorphism
1885:
1856:
1824:
1778:
1750:since for another
1740:
1720:
1692:
1656:
1621:
1601:
1563:
1534:
1506:
1484:
1447:
1424:
1404:
1382:
1345:
1323:
1276:
1256:
1215:
1190:
1156:
1127:
1099:
1079:free abelian group
1067:
1024:
998:
969:
943:
913:
893:
805:, specifically in
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
2696:
2695:
2381:Alternating group
2112:Topics In Algebra
1931:J. Am. Math. Soc.
1743:{\displaystyle x}
1624:{\displaystyle p}
1537:{\displaystyle A}
1450:{\displaystyle A}
1427:{\displaystyle A}
1326:{\displaystyle A}
1279:{\displaystyle p}
916:{\displaystyle e}
799:
798:
374:
373:
256:Alternating group
213:
212:
2726:
2688:Abstract algebra
2625:Quaternion group
2555:Symplectic group
2531:Orthogonal group
2225:
2218:
2211:
2202:
2201:
2197:
2176:
2150:
2128:
2106:
2075:
2074:
2072:
2060:
2054:
2052:
2034:
2028:
2027:
1998:
1992:
1989:Hungerford (1974
1986:
1980:
1974:
1968:
1962:
1956:
1955:
1946:
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1865:
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1377:
1376:
1366:
1354:
1352:
1351:
1346:
1344:
1332:
1330:
1329:
1324:
1294:Groups of rank 1
1285:
1283:
1282:
1277:
1265:
1263:
1262:
1257:
1252:
1251:
1236:
1224:
1222:
1221:
1216:
1214:
1199:
1197:
1196:
1191:
1180:
1165:
1163:
1162:
1157:
1155:
1154:
1149:
1136:
1134:
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1128:
1126:
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1092:
1076:
1074:
1073:
1068:
1051:
1033:
1031:
1030:
1025:
1007:
1005:
1004:
999:
978:
976:
975:
970:
952:
950:
949:
944:
922:
920:
919:
914:
902:
900:
899:
894:
835:identity element
807:abstract algebra
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
485:
484:
479:
477:
462:
460:
459:
454:
452:
425:
423:
422:
417:
415:
338:Schur multiplier
292:Cauchy's theorem
280:Quaternion group
228:
227:
54:
53:
43:
30:
19:
18:
2734:
2733:
2729:
2728:
2727:
2725:
2724:
2723:
2699:
2698:
2697:
2692:
2669:
2641:Conformal group
2629:
2603:
2595:
2587:
2579:
2571:
2505:
2497:
2484:
2475:Symmetric group
2454:
2444:
2437:
2430:
2423:
2415:
2406:
2402:
2392:Sporadic groups
2386:
2377:
2359:Discrete groups
2353:
2344:Wallpaper group
2324:Solvable groups
2292:Types of groups
2287:
2253:Normal subgroup
2234:
2229:
2187:Allyn and Bacon
2174:
2164:Springer-Verlag
2148:
2138:Springer-Verlag
2126:
2104:
2084:
2079:
2078:
2061:
2057:
2049:
2035:
2031:
1999:
1995:
1987:
1983:
1975:
1971:
1963:
1959:
1926:
1922:
1917:
1901:
1871:
1868:
1867:
1839:
1836:
1835:
1807:
1793:
1790:
1789:
1755:
1752:
1751:
1735:
1732:
1731:
1715:
1707:
1704:
1703:
1683:
1679:
1671:
1668:
1667:
1666:if and only if
1636:
1633:
1632:
1616:
1613:
1612:
1578:
1575:
1574:
1549:
1546:
1545:
1529:
1526:
1525:
1522:
1501:
1499:
1496:
1495:
1474:
1473:
1469:
1464:
1462:
1459:
1458:
1442:
1439:
1438:
1419:
1416:
1415:
1399:
1397:
1394:
1393:
1372:
1371:
1367:
1362:
1360:
1357:
1356:
1340:
1338:
1335:
1334:
1318:
1315:
1314:
1307:
1301:
1296:
1271:
1268:
1267:
1244:
1240:
1232:
1230:
1227:
1226:
1210:
1208:
1205:
1204:
1176:
1174:
1171:
1170:
1150:
1145:
1144:
1142:
1139:
1138:
1122:
1114:
1111:
1110:
1093:
1088:
1087:
1085:
1082:
1081:
1047:
1042:
1039:
1038:
1013:
1010:
1009:
984:
981:
980:
958:
955:
954:
932:
929:
928:
908:
905:
904:
870:
867:
866:
860:
854:
827:group operation
795:
766:
765:
754:Abelian variety
747:Reductive group
735:
725:
724:
723:
722:
673:
665:
657:
649:
641:
614:Special unitary
525:
511:
510:
492:
491:
473:
471:
468:
467:
448:
446:
443:
442:
411:
409:
406:
405:
397:
396:
387:Discrete groups
376:
375:
331:Frobenius group
276:
263:
252:
245:Symmetric group
241:
225:
215:
214:
65:Normal subgroup
51:
31:
22:
17:
12:
11:
5:
2732:
2722:
2721:
2716:
2711:
2694:
2693:
2691:
2690:
2685:
2680:
2674:
2671:
2670:
2668:
2667:
2664:
2661:
2658:
2653:
2648:
2643:
2637:
2635:
2631:
2630:
2628:
2627:
2622:
2620:Poincaré group
2617:
2612:
2606:
2605:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2573:
2569:
2565:
2559:
2558:
2552:
2546:
2540:
2534:
2528:
2522:
2515:
2513:
2507:
2506:
2504:
2503:
2498:
2493:
2488:Dihedral group
2485:
2480:
2472:
2468:
2467:
2461:
2455:
2452:
2446:
2442:
2435:
2428:
2421:
2416:
2413:
2407:
2404:
2400:
2394:
2388:
2387:
2384:
2378:
2375:
2369:
2363:
2361:
2355:
2354:
2352:
2351:
2346:
2341:
2336:
2331:
2329:Symmetry group
2326:
2321:
2316:
2314:Infinite group
2311:
2306:
2304:Abelian groups
2301:
2295:
2293:
2289:
2288:
2286:
2285:
2280:
2278:direct product
2270:
2265:
2263:Quotient group
2260:
2255:
2250:
2244:
2242:
2236:
2235:
2228:
2227:
2220:
2213:
2205:
2199:
2198:
2178:
2172:
2152:
2146:
2129:
2125:978-1114541016
2124:
2107:
2102:
2094:Addison-Wesley
2083:
2080:
2077:
2076:
2055:
2047:
2029:
1993:
1981:
1969:
1965:Fraleigh (1976
1957:
1937:(1): 233–258,
1919:
1918:
1916:
1913:
1900:
1897:
1884:
1881:
1878:
1875:
1855:
1852:
1849:
1846:
1843:
1823:
1820:
1817:
1814:
1810:
1806:
1803:
1800:
1797:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1739:
1718:
1714:
1711:
1691:
1686:
1682:
1678:
1675:
1655:
1652:
1649:
1646:
1643:
1640:
1620:
1611:, for a prime
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1562:
1559:
1556:
1553:
1533:
1521:
1520:Classification
1518:
1504:
1483:
1477:
1472:
1467:
1446:
1423:
1402:
1381:
1375:
1370:
1365:
1355:-vector space
1343:
1322:
1303:Main article:
1300:
1297:
1295:
1292:
1275:
1255:
1250:
1247:
1243:
1239:
1235:
1213:
1189:
1186:
1183:
1179:
1153:
1148:
1125:
1121:
1118:
1096:
1091:
1066:
1063:
1060:
1057:
1054:
1050:
1046:
1023:
1020:
1017:
997:
994:
991:
988:
968:
965:
962:
942:
939:
936:
912:
892:
889:
886:
883:
880:
877:
874:
856:Main article:
853:
850:
797:
796:
794:
793:
786:
779:
771:
768:
767:
764:
763:
761:Elliptic curve
757:
756:
750:
749:
743:
742:
736:
731:
730:
727:
726:
721:
720:
717:
714:
710:
706:
705:
704:
699:
697:Diffeomorphism
693:
692:
687:
682:
676:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
634:
633:
622:
621:
610:
609:
598:
597:
586:
585:
574:
573:
562:
561:
554:Special linear
550:
549:
542:General linear
538:
537:
532:
526:
517:
516:
513:
512:
509:
508:
503:
498:
490:
489:
476:
464:
451:
438:
436:Modular groups
434:
433:
432:
427:
414:
398:
395:
394:
389:
383:
382:
381:
378:
377:
372:
371:
370:
369:
364:
359:
356:
350:
349:
343:
342:
341:
340:
334:
333:
327:
326:
321:
312:
311:
309:Hall's theorem
306:
304:Sylow theorems
300:
299:
294:
286:
285:
284:
283:
277:
272:
269:Dihedral group
265:
264:
259:
253:
248:
242:
237:
226:
221:
220:
217:
216:
211:
210:
209:
208:
203:
195:
194:
193:
192:
187:
182:
177:
172:
167:
162:
160:multiplicative
157:
152:
147:
142:
134:
133:
132:
131:
126:
118:
117:
109:
108:
107:
106:
104:Wreath product
101:
96:
91:
89:direct product
83:
81:Quotient group
75:
74:
73:
72:
67:
62:
52:
49:
48:
45:
44:
36:
35:
15:
9:
6:
4:
3:
2:
2731:
2720:
2717:
2715:
2712:
2710:
2707:
2706:
2704:
2689:
2686:
2684:
2681:
2679:
2676:
2675:
2672:
2665:
2662:
2659:
2657:
2656:Quantum group
2654:
2652:
2649:
2647:
2644:
2642:
2639:
2638:
2636:
2632:
2626:
2623:
2621:
2618:
2616:
2615:Lorentz group
2613:
2611:
2608:
2607:
2604:
2598:
2596:
2590:
2588:
2582:
2580:
2574:
2572:
2566:
2564:
2561:
2560:
2556:
2553:
2550:
2547:
2544:
2543:Unitary group
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2516:
2514:
2512:
2508:
2502:
2499:
2496:
2492:
2489:
2486:
2483:
2479:
2476:
2473:
2470:
2469:
2465:
2464:Monster group
2462:
2459:
2456:
2450:
2449:Fischer group
2447:
2445:
2438:
2431:
2424:
2418:Janko groups
2417:
2411:
2408:
2398:
2397:Mathieu group
2395:
2393:
2390:
2389:
2382:
2379:
2373:
2370:
2368:
2365:
2364:
2362:
2360:
2356:
2350:
2349:Trivial group
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2319:Simple groups
2317:
2315:
2312:
2310:
2309:Cyclic groups
2307:
2305:
2302:
2300:
2299:Finite groups
2297:
2296:
2294:
2290:
2284:
2281:
2279:
2275:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2245:
2243:
2241:
2240:Basic notions
2237:
2233:
2226:
2221:
2219:
2214:
2212:
2207:
2206:
2203:
2196:
2192:
2188:
2184:
2179:
2175:
2173:0-387-95385-X
2169:
2165:
2161:
2157:
2153:
2149:
2147:0-387-90518-9
2143:
2139:
2135:
2130:
2127:
2121:
2117:
2113:
2108:
2105:
2103:0-201-01984-1
2099:
2095:
2091:
2086:
2085:
2071:
2066:
2059:
2050:
2048:0-226-30870-7
2044:
2040:
2033:
2025:
2021:
2018:(1): 68–122.
2017:
2013:
2012:
2007:
2003:
2002:Reinhold Baer
1997:
1991:, p. 78)
1990:
1985:
1979:, p. 42)
1978:
1973:
1967:, p. 78)
1966:
1961:
1954:
1950:
1945:
1940:
1936:
1932:
1924:
1920:
1912:
1910:
1906:
1896:
1879:
1873:
1853:
1850:
1847:
1844:
1841:
1818:
1804:
1801:
1798:
1795:
1788:there exists
1772:
1763:
1760:
1757:
1737:
1712:
1709:
1689:
1684:
1680:
1676:
1673:
1650:
1644:
1641:
1638:
1618:
1595:
1586:
1583:
1580:
1557:
1551:
1531:
1517:
1481:
1470:
1444:
1435:
1421:
1379:
1368:
1320:
1312:
1306:
1291:
1289:
1273:
1248:
1245:
1241:
1201:
1184:
1167:
1151:
1119:
1116:
1094:
1080:
1061:
1058:
1055:
1052:
1035:
1021:
1018:
1015:
995:
992:
989:
986:
966:
963:
960:
940:
937:
934:
926:
923:is of finite
910:
887:
884:
881:
878:
875:
865:
864:abelian group
859:
858:Abelian group
849:
847:
842:
840:
836:
832:
828:
825:in which the
824:
820:
816:
815:abelian group
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
738:
737:
734:
729:
728:
718:
715:
712:
711:
709:
703:
700:
698:
695:
694:
691:
688:
686:
683:
681:
678:
677:
674:
668:
666:
660:
658:
652:
650:
644:
642:
636:
635:
631:
627:
624:
623:
619:
615:
612:
611:
607:
603:
600:
599:
595:
591:
588:
587:
583:
579:
576:
575:
571:
567:
564:
563:
559:
555:
552:
551:
547:
543:
540:
539:
536:
533:
531:
528:
527:
524:
520:
515:
514:
507:
504:
502:
499:
497:
494:
493:
465:
440:
439:
437:
431:
428:
403:
400:
399:
393:
390:
388:
385:
384:
380:
379:
368:
365:
363:
360:
357:
354:
353:
352:
351:
348:
345:
344:
339:
336:
335:
332:
329:
328:
325:
322:
320:
318:
314:
313:
310:
307:
305:
302:
301:
298:
295:
293:
290:
289:
288:
287:
281:
278:
275:
270:
267:
266:
262:
257:
254:
251:
246:
243:
240:
235:
232:
231:
230:
229:
224:
223:Finite groups
219:
218:
207:
204:
202:
199:
198:
197:
196:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
137:
136:
135:
130:
127:
125:
122:
121:
120:
119:
116:
115:
111:
110:
105:
102:
100:
97:
95:
92:
90:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
57:
56:
55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
21:
20:
2683:Applications
2610:Circle group
2494:
2490:
2481:
2477:
2410:Conway group
2372:Cyclic group
2182:
2159:
2136:, New York:
2133:
2111:
2089:
2058:
2053:Chapter VII.
2038:
2032:
2015:
2009:
1996:
1984:
1972:
1960:
1934:
1930:
1923:
1905:model theory
1902:
1631:we say that
1523:
1436:
1414:) subset of
1310:
1308:
1202:
1168:
1036:
861:
843:
810:
800:
629:
617:
605:
593:
581:
569:
557:
545:
316:
273:
260:
249:
238:
234:Cyclic group
112:
99:Free product
70:Group action
33:Group theory
28:Group theory
27:
2339:Point group
2334:Space group
2156:Lang, Serge
2114:, Waltham:
852:Definitions
831:commutative
803:mathematics
519:Topological
358:alternating
2703:Categories
2651:Loop group
2511:Lie groups
2283:direct sum
2185:, Boston:
2082:References
2070:2102.12371
1977:Lang (2002
1953:1021.03043
1834:such that
1702:for every
979:for which
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
1874:τ
1813:∖
1805:∈
1767:∖
1761:∈
1713:∈
1677:∈
1645:τ
1642:∈
1590:∖
1584:∈
1552:τ
1544:a subset
1471:⊗
1369:⊗
1246:−
1120:∈
1065:⟩
1045:⟨
964:∈
891:⟩
873:⟨
690:Conformal
578:Euclidean
185:nilpotent
2248:Subgroup
2195:68-15225
2158:(2002),
2004:(1937).
833:and the
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
2678:History
2160:Algebra
2134:Algebra
819:torsion
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
2453:22..24
2405:22..24
2401:11..12
2232:Groups
2193:
2170:
2144:
2122:
2100:
2045:
1951:
844:While
813:is an
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
2666:Sp(∞)
2663:SU(∞)
2557:Sp(n)
2551:SU(n)
2539:SO(n)
2527:SL(n)
2521:GL(n)
2274:Semi-
2065:arXiv
1915:Notes
925:order
839:order
823:group
719:Sp(∞)
716:SU(∞)
129:image
2660:O(∞)
2545:U(n)
2533:O(n)
2414:1..3
2191:LCCN
2168:ISBN
2142:ISBN
2120:ISBN
2098:ISBN
2043:ISBN
1907:and
1311:rank
1309:The
1299:Rank
938:>
809:, a
713:O(∞)
702:Loop
521:and
2020:doi
1949:Zbl
1939:doi
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1434:.
1166:.
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1008:is
862:An
841:.
829:is
801:In
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616:SU(
592:SO(
556:SL(
544:GL(
2705::
2439:,
2432:,
2425:,
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2166:,
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2096:,
2014:.
2008:.
1947:,
1935:16
1933:,
1516:.
1290:.
604:U(
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568:O(
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2568:G
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2482:n
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2460:B
2451:F
2443:4
2441:J
2436:3
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2429:2
2427:J
2422:1
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2224:e
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2073:.
2067::
2051:.
2026:.
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1816:{
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961:x
941:0
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879:,
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