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2400:{\displaystyle {\begin{aligned}&&h(g_{1})&=h(g_{2})\\\Leftrightarrow &&h(g_{1})\cdot h(g_{2})^{-1}&=e_{H}\\\Leftrightarrow &&h\left(g_{1}\circ g_{2}^{-1}\right)&=e_{H},\ \operatorname {ker} (h)=\{e_{G}\}\\\Rightarrow &&g_{1}\circ g_{2}^{-1}&=e_{G}\\\Leftrightarrow &&g_{1}&=g_{2}\end{aligned}}}
2002:
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1997:{\displaystyle {\begin{aligned}h\left(g^{-1}\circ u\circ g\right)&=h(g)^{-1}\cdot h(u)\cdot h(g)\\&=h(g)^{-1}\cdot e_{H}\cdot h(g)\\&=h(g)^{-1}\cdot h(g)=e_{H},\end{aligned}}}
1668:
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1309:; they differ only in the notation of their elements (except of identity element) and are identical for all practical purposes. I.e. we re-label all elements except identity.
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The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function
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2048:}. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:
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sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of
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2571:{\displaystyle G\equiv \left\{{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\bigg |}a>0,b\in \mathbf {R} \right\}}
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The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The
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3466:. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a
1369:, +) contains only two elements, the identity transformation and multiplication with β1; it is isomorphic to (
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The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if
3470:; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an
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1297:; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups
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In areas of mathematics where one considers groups endowed with additional structure, a
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A group endomorphism that is bijective, and hence an isomorphism. The set of all
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Mathematical function between groups that preserves multiplication structure
2662:{\displaystyle {\begin{pmatrix}a&b\\0&1\end{pmatrix}}\mapsto a^{u}}
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The exponential map also yields a group homomorphism from the group of
1329:; the domain and codomain are the same. Also called an endomorphism of
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994:
where the group operation on the left side of the equation is that of
3541:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
2789:* with multiplication. This map is surjective and has the kernel {2Ο
2873:{\displaystyle \Phi :(\mathbb {N} ,+)\rightarrow (\mathbb {R} ,+)}
2581:
forms a group under matrix multiplication. For any complex number
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and its kernel consists of all integers which are divisible by 3.
1723:{\displaystyle g^{-1}\circ u\circ g\in \operatorname {ker} (h)}
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and it also maps inverses to inverses in the sense that
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with addition to the group of non-zero complex numbers
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1285:(or, onto); i.e., reaches every point in the codomain.
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1155:{\displaystyle h\left(u^{-1}\right)=h(u)^{-1}.\,}
1605:states that the image of a group homomorphism,
1273:(or, one-to-one); i.e., preserves distinctness.
2764:yields a group homomorphism from the group of
1365:). As an example, the automorphism group of (
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1663:{\displaystyle u\in \operatorname {ker} (h)}
1169:"is compatible with the group structure".
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1005:From this property, one can deduce that
3579:Rowland, Todd & Weisstein, Eric W.
3515:(3rd ed.). Wiley. pp. 71β72.
412:
178:
88:
1613:) is isomorphic to the quotient group
414:Classification of finite simple groups
1200:is a group homomorphism if whenever
984:{\displaystyle h(u*v)=h(u)\cdot h(v)}
7:
3489:Fundamental theorem on homomorphisms
3210:is itself an abelian group: the sum
3144:are group homomorphisms, then so is
2961:{\displaystyle (\mathbb {R} ^{+},*)}
2917:{\displaystyle \Phi (x)={\sqrt{2}}x}
1407:which are mapped to the identity in
1180:is often required to be continuous.
3220:of two homomorphisms is defined by
2674:Consider a multiplicative group of
34:Depiction of a group homomorphism (
2887:
2827:
25:
3511:Dummit, D. S.; Foote, R. (2004).
2467:3 is a group homomorphism. It is
2023:
3202:of all group homomorphisms from
3096:{\displaystyle f:G\rightarrow H}
3064:{\displaystyle \mathbb {R} ^{+}}
2995:{\displaystyle (\mathbb {R} ,+)}
2559:
106:
3278:is again a group homomorphism.
3174:Homomorphisms of abelian groups
3407:, this shows that the set End(
3368: and
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3002:, represented respectively by
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2745:{\displaystyle f_{u}(a)=a^{u}}
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1079:{\displaystyle h(e_{G})=e_{H}}
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998:and on the right side that of
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775:Infinite dimensional Lie group
1:
3539:Graduate Texts in Mathematics
3444:is isomorphic to the ring of
1403:to be the set of elements in
1293:A group homomorphism that is
1281:A group homomorphism that is
1269:A group homomorphism that is
3252:) for all
2682:, β
) for any complex number
548:{\displaystyle \mathbb {Z} }
523:{\displaystyle \mathbb {Z} }
486:{\displaystyle \mathbb {Z} }
273:List of group theory topics
3626:
1385:
1243:In other words, the group
3403:Since the composition is
1603:first isomorphism theorem
46:(right). The oval inside
3268:is needed to prove that
2927:Consider the two groups
2754:is a group homomorphism.
2671:is a group homomorphism.
1023:to the identity element
391:Elementary abelian group
268:Glossary of group theory
2801:}, as can be seen from
1361:. It is denoted by Aut(
1165:Hence one can say that
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1351:functional composition
1317:A group homomorphism,
1215: we have
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807:Linear algebraic group
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3264:The commutativity of
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2676:positive real numbers
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1624:The kernel of h is a
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1251:and the homomorphism
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3581:"Group Homomorphism"
3468:preadditive category
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2686:. Then the function
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2483:
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2007:The image of h is a
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1749:{\displaystyle u,g}
1388:Image (mathematics)
181:Group homomorphisms
91:Algebraic structure
18:Group homomorphisms
3168:category of groups
3166:(specifically the
3112:Category of groups
3107:is a homomorphism.
3105:logarithm function
3093:
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3032:
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2924:is a homomorphism.
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2815:exponential fields
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2025:group monomorphism
2018:The homomorphism,
1994:
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1355:automorphism group
1178:topological groups
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911:such that for all
886:group homomorphism
657:Special orthogonal
545:
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483:
364:Lagrange's theorem
84:
3548:978-0-387-95385-4
3522:978-0-471-43334-7
3499:Ring homomorphism
3418:endomorphism ring
3035:{\displaystyle H}
3015:{\displaystyle G}
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323:Alternating group
280:
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16:(Redirected from
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3513:Abstract Algebra
3472:abelian category
3455:with entries in
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3305:are elements of
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1382:Image and kernel
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563:Arithmetic group
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2805:. Fields like
2803:Euler's formula
2780:complex numbers
2762:exponential map
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1626:normal subgroup
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1386:Main articles:
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814:Reductive group
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681:Special unitary
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454:Discrete groups
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398:Frobenius group
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312:Symmetric group
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132:Normal subgroup
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3572:External links
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2203:
2199:
2196:
2193:
2190:
2189:
2184:
2180:
2176:
2173:
2171:
2167:
2164:
2160:
2154:
2150:
2146:
2143:
2140:
2137:
2132:
2128:
2124:
2121:
2118:
2115:
2112:
2111:
2108:
2103:
2099:
2095:
2092:
2089:
2086:
2084:
2082:
2077:
2073:
2069:
2066:
2063:
2060:
2042:
2005:
2004:
1989:
1984:
1980:
1976:
1973:
1970:
1967:
1964:
1961:
1956:
1953:
1949:
1945:
1942:
1939:
1936:
1933:
1931:
1929:
1926:
1923:
1920:
1917:
1914:
1909:
1905:
1901:
1896:
1893:
1889:
1885:
1882:
1879:
1876:
1873:
1871:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1834:
1831:
1827:
1823:
1820:
1817:
1814:
1811:
1809:
1806:
1802:
1799:
1796:
1793:
1788:
1785:
1781:
1776:
1772:
1769:
1768:
1745:
1742:
1739:
1730:for arbitrary
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1687:
1684:
1680:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1599:
1598:
1587:
1583:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1554:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1499:
1498:
1487:
1483:
1477:
1473:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1441:
1437:
1434:
1431:
1428:
1425:
1422:
1396:We define the
1383:
1380:
1379:
1378:
1339:
1334:
1315:
1310:
1291:
1286:
1279:
1274:
1267:
1260:
1257:
1241:
1240:
1185:
1182:
1163:
1162:
1150:
1145:
1142:
1138:
1134:
1131:
1128:
1125:
1121:
1116:
1113:
1109:
1105:
1101:
1087:
1086:
1073:
1069:
1065:
1062:
1057:
1053:
1049:
1046:
1026:
1015:
992:
991:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
923:it holds that
864:
863:
861:
860:
853:
846:
838:
835:
834:
831:
830:
828:Elliptic curve
824:
823:
817:
816:
810:
809:
803:
798:
797:
794:
793:
788:
787:
784:
781:
777:
773:
772:
771:
766:
764:Diffeomorphism
760:
759:
754:
749:
743:
742:
738:
734:
730:
726:
722:
718:
714:
710:
706:
701:
700:
689:
688:
677:
676:
665:
664:
653:
652:
641:
640:
629:
628:
621:Special linear
617:
616:
609:General linear
605:
604:
599:
593:
584:
583:
580:
579:
576:
575:
570:
565:
557:
556:
543:
531:
518:
505:
503:Modular groups
501:
500:
499:
494:
481:
465:
462:
461:
456:
450:
449:
448:
445:
444:
439:
438:
437:
436:
431:
426:
423:
417:
416:
410:
409:
408:
407:
401:
400:
394:
393:
388:
379:
378:
376:Hall's theorem
373:
371:Sylow theorems
367:
366:
361:
353:
352:
351:
350:
344:
339:
336:Dihedral group
332:
331:
326:
320:
315:
309:
304:
293:
288:
287:
284:
283:
278:
277:
276:
275:
270:
262:
261:
260:
259:
254:
249:
244:
239:
234:
229:
227:multiplicative
224:
219:
214:
209:
201:
200:
199:
198:
193:
185:
184:
176:
175:
174:
173:
171:Wreath product
168:
163:
158:
156:direct product
150:
148:Quotient group
142:
141:
140:
139:
134:
129:
119:
116:
115:
112:
111:
103:
102:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3622:
3611:
3608:
3606:
3603:
3602:
3600:
3588:
3587:
3582:
3576:
3575:
3571:
3566:
3562:
3558:
3554:
3550:
3544:
3540:
3536:
3532:
3528:
3524:
3518:
3514:
3509:
3508:
3504:
3500:
3497:
3495:
3494:Quasimorphism
3492:
3490:
3487:
3485:
3482:
3481:
3477:
3475:
3473:
3469:
3465:
3462:
3458:
3454:
3451:
3447:
3443:
3440:
3436:
3432:
3428:
3424:
3420:
3419:
3414:
3410:
3406:
3396:
3392:
3388:
3384:
3380:
3376:
3372:
3365:
3361:
3357:
3353:
3349:
3345:
3341:
3336:
3335:
3334:
3330:
3326:
3320:
3314:
3310:
3304:
3300:
3294:
3290:
3284:
3279:
3276:
3272:
3267:
3259:
3255:
3251:
3247:
3243:
3239:
3235:
3231:
3227:
3223:
3222:
3221:
3218:
3214:
3209:
3205:
3199:
3195:
3189:
3185:
3181:
3173:
3171:
3169:
3165:
3160:
3156:
3152:
3148:
3142:
3138:
3134:
3128:
3124:
3120:
3111:
3106:
3090:
3084:
3081:
3078:
3056:
3029:
3009:
2986:
2983:
2952:
2949:
2944:
2926:
2911:
2904:
2899:
2893:
2880:, defined by
2864:
2861:
2844:
2841:
2830:
2820:The function
2819:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2781:
2777:
2774:
2770:
2767:
2763:
2759:
2758:
2737:
2733:
2729:
2723:
2715:
2711:
2703:
2702:
2700:
2696:
2692:
2685:
2681:
2677:
2673:
2654:
2650:
2641:
2635:
2630:
2623:
2618:
2612:
2603:
2602:
2601:
2599:
2595:
2591:
2585:the function
2584:
2564:
2555:
2552:
2549:
2546:
2543:
2540:
2528:
2522:
2517:
2510:
2505:
2499:
2493:
2489:
2486:
2479:
2478:
2476:
2475:
2470:
2466:
2463:
2459:
2455:
2451:
2447:
2443:
2439:
2435:
2431:
2427:
2419:
2416:Consider the
2415:
2414:
2410:
2388:
2384:
2380:
2378:
2371:
2367:
2351:
2347:
2343:
2341:
2334:
2331:
2326:
2322:
2318:
2313:
2309:
2290:
2286:
2279:
2273:
2267:
2264:
2258:
2253:
2249:
2245:
2243:
2237:
2231:
2228:
2223:
2219:
2215:
2210:
2206:
2201:
2197:
2182:
2178:
2174:
2172:
2165:
2162:
2152:
2148:
2141:
2138:
2130:
2126:
2119:
2101:
2097:
2090:
2087:
2085:
2075:
2071:
2064:
2051:
2050:
2049:
2045:
2041:
2037:
2031:
2027:
2026:
2021:
2016:
2014:
2010:
1987:
1982:
1978:
1974:
1968:
1962:
1959:
1954:
1951:
1943:
1937:
1934:
1932:
1921:
1915:
1912:
1907:
1903:
1899:
1894:
1891:
1883:
1877:
1874:
1872:
1861:
1855:
1852:
1846:
1840:
1837:
1832:
1829:
1821:
1815:
1812:
1810:
1804:
1800:
1797:
1794:
1791:
1786:
1783:
1779:
1774:
1770:
1759:
1758:
1757:
1743:
1740:
1737:
1714:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1685:
1682:
1678:
1654:
1648:
1645:
1642:
1639:
1631:
1627:
1622:
1620:
1616:
1612:
1608:
1604:
1585:
1581:
1577:
1574:
1571:
1568:
1562:
1556:
1552:
1548:
1542:
1536:
1533:
1527:
1521:
1518:
1511:
1510:
1509:
1507:
1505:
1485:
1481:
1475:
1471:
1467:
1461:
1455:
1452:
1449:
1446:
1443:
1439:
1435:
1429:
1423:
1420:
1413:
1412:
1411:
1410:
1406:
1402:
1400:
1393:
1389:
1381:
1376:
1372:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1343:automorphisms
1340:
1338:
1335:
1332:
1328:
1324:
1320:
1316:
1314:
1311:
1308:
1304:
1300:
1296:
1292:
1290:
1287:
1284:
1280:
1278:
1275:
1272:
1268:
1266:
1263:
1262:
1258:
1256:
1254:
1250:
1246:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1203:
1202:
1201:
1199:
1195:
1191:
1183:
1181:
1179:
1175:
1170:
1168:
1148:
1143:
1140:
1132:
1126:
1123:
1119:
1114:
1111:
1107:
1103:
1099:
1092:
1091:
1090:
1071:
1067:
1063:
1055:
1051:
1044:
1037:
1036:
1035:
1033:
1029:
1022:
1018:
1012:
1008:
1003:
1001:
997:
975:
969:
966:
960:
954:
951:
945:
942:
939:
933:
926:
925:
924:
922:
918:
914:
910:
906:
902:
899:
895:
891:
887:
883:
879:
875:
871:
859:
854:
852:
847:
845:
840:
839:
837:
836:
829:
826:
825:
822:
819:
818:
815:
812:
811:
808:
805:
804:
801:
796:
795:
785:
782:
779:
778:
776:
770:
767:
765:
762:
761:
758:
755:
753:
750:
748:
745:
744:
741:
735:
733:
727:
725:
719:
717:
711:
709:
703:
702:
698:
694:
691:
690:
686:
682:
679:
678:
674:
670:
667:
666:
662:
658:
655:
654:
650:
646:
643:
642:
638:
634:
631:
630:
626:
622:
619:
618:
614:
610:
607:
606:
603:
600:
598:
595:
594:
591:
587:
582:
581:
574:
571:
569:
566:
564:
561:
560:
532:
507:
506:
504:
498:
495:
470:
467:
466:
460:
457:
455:
452:
451:
447:
446:
435:
432:
430:
427:
424:
421:
420:
419:
418:
415:
411:
406:
403:
402:
399:
396:
395:
392:
389:
387:
385:
381:
380:
377:
374:
372:
369:
368:
365:
362:
360:
357:
356:
355:
354:
348:
345:
342:
337:
334:
333:
329:
324:
321:
318:
313:
310:
307:
302:
299:
298:
297:
296:
291:
290:Finite groups
286:
285:
274:
271:
269:
266:
265:
264:
263:
258:
255:
253:
250:
248:
245:
243:
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
213:
210:
208:
205:
204:
203:
202:
197:
194:
192:
189:
188:
187:
186:
183:
182:
177:
172:
169:
167:
164:
162:
159:
157:
154:
151:
149:
146:
145:
144:
143:
138:
135:
133:
130:
128:
125:
124:
123:
122:
117:Basic notions
114:
113:
109:
105:
104:
101:
96:
92:
87:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
32:
19:
3605:Group theory
3584:
3534:
3512:
3484:Homomorphism
3463:
3460:
3456:
3449:
3445:
3441:
3438:
3434:
3430:
3422:
3416:
3408:
3402:
3394:
3390:
3386:
3382:
3378:
3374:
3370:
3363:
3359:
3355:
3351:
3347:
3343:
3339:
3328:
3324:
3318:
3312:
3308:
3302:
3298:
3292:
3288:
3282:
3280:
3274:
3270:
3265:
3263:
3257:
3253:
3249:
3245:
3241:
3237:
3233:
3229:
3225:
3216:
3212:
3207:
3203:
3197:
3193:
3183:
3179:
3177:
3158:
3154:
3150:
3146:
3140:
3136:
3132:
3126:
3122:
3118:
3115:
2810:
2806:
2798:
2794:
2790:
2786:
2782:
2772:
2768:
2766:real numbers
2698:
2694:
2687:
2683:
2679:
2597:
2593:
2586:
2582:
2580:
2461:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2429:
2425:
2418:cyclic group
2043:
2039:
2035:
2029:
2024:
2019:
2017:
2012:
2006:
1629:
1623:
1618:
1614:
1610:
1606:
1600:
1502:
1500:
1408:
1404:
1397:
1395:
1374:
1370:
1366:
1362:
1358:
1354:
1346:
1337:Automorphism
1330:
1326:
1322:
1318:
1313:Endomorphism
1306:
1302:
1298:
1265:Monomorphism
1252:
1248:
1244:
1242:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1197:
1193:
1189:
1187:
1174:homomorphism
1173:
1171:
1166:
1164:
1088:
1031:
1024:
1020:
1013:
1006:
1004:
999:
995:
993:
920:
916:
912:
908:
904:
900:
893:
889:
885:
881:
877:
872:, given two
867:
696:
684:
672:
660:
648:
636:
624:
612:
383:
340:
327:
316:
305:
301:Cyclic group
180:
179:
166:Free product
137:Group action
100:Group theory
95:Group theory
94:
79:
71:
67:
59:
55:
47:
43:
39:
35:
3531:Lang, Serge
3405:associative
2701:defined by
2600:defined by
1345:of a group
1305:are called
1289:Isomorphism
1277:Epimorphism
870:mathematics
586:Topological
425:alternating
3599:Categories
3565:0984.00001
3505:References
3433:copies of
3427:direct sum
2469:surjective
1307:isomorphic
1283:surjective
896:, Β·) is a
693:Symplectic
633:Orthogonal
590:Lie groups
497:Free group
222:continuous
161:Direct sum
42:(left) to
3610:Morphisms
3586:MathWorld
3088:→
2953:∗
2888:Φ
2851:→
2828:Φ
2647:↦
2556:∈
2490:≡
2361:⇔
2332:−
2319:∘
2303:⇒
2268:
2229:−
2216:∘
2192:⇔
2163:−
2139:⋅
2114:⇔
1960:⋅
1952:−
1913:⋅
1900:⋅
1892:−
1853:⋅
1838:⋅
1830:−
1798:∘
1792:∘
1784:−
1709:
1703:∈
1697:∘
1691:∘
1683:−
1670:and show
1649:
1643:∈
1632:. Assume
1575:∈
1569::
1549:≡
1522:
1453::
1447:∈
1424:
1295:bijective
1271:injective
1184:Intuition
1141:−
1112:−
1009:maps the
967:⋅
943:∗
880:,β) and (
757:Conformal
645:Euclidean
252:nilpotent
3533:(2002),
3478:See also
3453:matrices
3333:, then
3164:category
3153: :
3135: :
3121: :
3042:, where
2793: :
2693: :
2592: :
2477:The set
2440: :
2411:Examples
2028:; i.e.,
2009:subgroup
1501:and the
1192: :
903: :
898:function
892:,β) to (
884:, Β·), a
752:PoincarΓ©
597:Solenoid
469:Integers
459:Lattices
434:sporadic
429:Lie type
257:solvable
247:dihedral
232:additive
217:infinite
127:Subgroup
3557:1878556
3535:Algebra
3188:abelian
2022:, is a
1349:, with
747:Lorentz
669:Unitary
568:Lattice
508:PSL(2,
242:abelian
153:(Semi-)
62:is the
50:is the
38:) from
3563:
3555:
3545:
3519:
3415:, the
3381:) = (
3321:is in
3317:, and
3285:is in
2262:
1508:to be
1399:kernel
888:from (
874:groups
602:Circle
533:SL(2,
422:cyclic
386:-group
237:cyclic
212:finite
207:simple
191:kernel
64:kernel
3389:) + (
3358:) + (
2452:with
2038:) = {
1617:/ker
1504:image
1377:, +).
1259:Types
786:Sp(β)
783:SU(β)
196:image
76:coset
74:is a
52:image
3543:ISBN
3517:ISBN
3448:-by-
3413:ring
3350:= (
3346:) β
3323:Hom(
3307:Hom(
3287:Hom(
3244:) +
3236:) =
3192:Hom(
3186:are
3182:and
3130:and
3022:and
2968:and
2809:and
2760:The
2544:>
2460:) =
2034:ker(
1506:of h
1401:of h
1390:and
1301:and
1231:) =
1223:) β
915:and
780:O(β)
769:Loop
588:and
70:and
3561:Zbl
3429:of
3421:of
3373:β (
3256:in
3206:to
3178:If
3170:).
3116:If
2465:mod
2424:= (
2265:ker
2011:of
1706:ker
1646:ker
1628:of
1421:ker
1357:of
1030:of
1019:of
919:in
876:, (
868:In
695:Sp(
683:SU(
659:SO(
623:SL(
611:GL(
78:of
66:of
54:of
3601::
3583:.
3559:,
3553:MR
3551:,
3537:,
3474:.
3393:β
3385:β
3377:+
3362:β
3354:β
3342:+
3327:,
3311:,
3301:,
3297:,
3291:,
3273:+
3232:)(
3228:+
3215:+
3196:,
3157:β
3149:β
3139:β
3125:β
2797:β
2791:ki
2697:β
2596:β
2448:/3
2444:β
2428:/3
2015:.
1756::
1621:.
1534::=
1519:im
1436::=
1373:/2
1325:β
1321::
1239:).
1211:=
1207:β
1196:β
1034:,
1002:.
907:β
671:U(
647:E(
635:O(
93:β
72:aN
58:.
3589:.
3525:.
3464:Z
3461:n
3459:/
3457:Z
3450:m
3446:m
3442:Z
3439:n
3437:/
3435:Z
3431:m
3423:G
3409:G
3399:.
3397:)
3395:k
3391:g
3387:h
3383:g
3379:k
3375:h
3371:g
3366:)
3364:f
3360:k
3356:f
3352:h
3348:f
3344:k
3340:h
3338:(
3331:)
3329:L
3325:H
3319:g
3315:)
3313:H
3309:G
3303:k
3299:h
3295:)
3293:G
3289:K
3283:f
3275:k
3271:h
3266:H
3260:.
3258:G
3254:u
3250:u
3248:(
3246:k
3242:u
3240:(
3238:h
3234:u
3230:k
3226:h
3224:(
3217:k
3213:h
3208:H
3204:G
3200:)
3198:H
3194:G
3184:H
3180:G
3159:K
3155:G
3151:h
3147:k
3141:K
3137:H
3133:k
3127:H
3123:G
3119:h
3091:H
3085:G
3082::
3079:f
3057:+
3052:R
3030:H
3010:G
2990:)
2987:+
2984:,
2980:R
2976:(
2956:)
2950:,
2945:+
2940:R
2935:(
2912:x
2905:2
2900:=
2897:)
2894:x
2891:(
2868:)
2865:+
2862:,
2858:R
2854:(
2848:)
2845:+
2842:,
2838:N
2834:(
2831::
2817:.
2811:C
2807:R
2799:Z
2795:k
2787:C
2783:C
2773:R
2769:R
2738:u
2734:a
2730:=
2727:)
2724:a
2721:(
2716:u
2712:f
2699:C
2695:R
2690:u
2688:f
2684:u
2680:R
2678:(
2655:u
2651:a
2642:)
2636:1
2631:0
2624:b
2619:a
2613:(
2598:C
2594:G
2589:u
2587:f
2583:u
2565:}
2560:R
2553:b
2550:,
2547:0
2541:a
2536:|
2529:)
2523:1
2518:0
2511:b
2506:a
2500:(
2494:{
2487:G
2462:u
2458:u
2456:(
2454:h
2450:Z
2446:Z
2442:Z
2438:h
2434:Z
2430:Z
2426:Z
2422:3
2420:Z
2389:2
2385:g
2381:=
2372:1
2368:g
2352:G
2348:e
2344:=
2335:1
2327:2
2323:g
2314:1
2310:g
2296:}
2291:G
2287:e
2283:{
2280:=
2277:)
2274:h
2271:(
2259:,
2254:H
2250:e
2246:=
2238:)
2232:1
2224:2
2220:g
2211:1
2207:g
2202:(
2198:h
2183:H
2179:e
2175:=
2166:1
2159:)
2153:2
2149:g
2145:(
2142:h
2136:)
2131:1
2127:g
2123:(
2120:h
2107:)
2102:2
2098:g
2094:(
2091:h
2088:=
2081:)
2076:1
2072:g
2068:(
2065:h
2044:G
2040:e
2036:h
2030:h
2020:h
2013:H
1988:,
1983:H
1979:e
1975:=
1972:)
1969:g
1966:(
1963:h
1955:1
1948:)
1944:g
1941:(
1938:h
1935:=
1925:)
1922:g
1919:(
1916:h
1908:H
1904:e
1895:1
1888:)
1884:g
1881:(
1878:h
1875:=
1865:)
1862:g
1859:(
1856:h
1850:)
1847:u
1844:(
1841:h
1833:1
1826:)
1822:g
1819:(
1816:h
1813:=
1805:)
1801:g
1795:u
1787:1
1780:g
1775:(
1771:h
1744:g
1741:,
1738:u
1718:)
1715:h
1712:(
1700:g
1694:u
1686:1
1679:g
1658:)
1655:h
1652:(
1640:u
1630:G
1619:h
1615:G
1611:G
1609:(
1607:h
1586:.
1582:}
1578:G
1572:u
1566:)
1563:u
1560:(
1557:h
1553:{
1546:)
1543:G
1540:(
1537:h
1531:)
1528:h
1525:(
1486:.
1482:}
1476:H
1472:e
1468:=
1465:)
1462:u
1459:(
1456:h
1450:G
1444:u
1440:{
1433:)
1430:h
1427:(
1409:H
1405:G
1375:Z
1371:Z
1367:Z
1363:G
1359:G
1347:G
1333:.
1331:G
1327:G
1323:G
1319:h
1303:H
1299:G
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1249:G
1245:H
1237:c
1235:(
1233:h
1229:b
1227:(
1225:h
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1219:(
1217:h
1213:c
1209:b
1205:a
1198:H
1194:G
1190:h
1167:h
1149:.
1144:1
1137:)
1133:u
1130:(
1127:h
1124:=
1120:)
1115:1
1108:u
1104:(
1100:h
1072:H
1068:e
1064:=
1061:)
1056:G
1052:e
1048:(
1045:h
1032:H
1027:H
1025:e
1021:G
1016:G
1014:e
1007:h
1000:H
996:G
979:)
976:v
973:(
970:h
964:)
961:u
958:(
955:h
952:=
949:)
946:v
940:u
937:(
934:h
921:G
917:v
913:u
909:H
905:G
901:h
894:H
890:G
882:H
878:G
857:e
850:t
843:v
739:8
737:E
731:7
729:E
723:6
721:E
715:4
713:F
707:2
705:G
699:)
697:n
687:)
685:n
675:)
673:n
663:)
661:n
651:)
649:n
639:)
637:n
627:)
625:n
615:)
613:n
555:)
542:Z
530:)
517:Z
493:)
480:Z
471:(
384:p
349:Q
341:n
338:D
328:n
325:A
317:n
314:S
306:n
303:Z
82:.
80:N
68:h
60:N
56:h
48:H
44:H
40:G
36:h
20:)
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