1125:
31:
1351:
A cyclic permutation of three (other one remains unchanged). Cycle type = . Order = 3. Members = { (1, 3, 4, 2), (1, 4, 2, 3), (3, 2, 4, 1), (4, 2, 1, 3), (4, 1, 3, 2), (2, 4, 3, 1), (3, 1, 2, 4), (2, 3, 1, 4) }). The 8 rows containing this conjugacy class are shown with normal print (no boldface
835:
Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity
2341:
th powers gives a map on conjugacy classes, and one may consider which conjugacy classes are in its preimage. For example, in the symmetric group, the square of an element of type (3)(2) (a 3-cycle and a 2-cycle) is an element of type (3), therefore one of the power-up classes of (3) is the class
3195:
1347:
Interchanging two (other two remain unchanged). Cycle type = . Order = 2. Members = { (1, 2, 4, 3), (1, 4, 3, 2), (1, 3, 2, 4), (4, 2, 3, 1), (3, 2, 1, 4), (2, 1, 3, 4) }). The 6 rows containing this conjugacy class are highlighted in green in the adjacent
2335:
1355:
A cyclic permutation of all four. Cycle type = . Order = 4. Members = { (2, 3, 4, 1), (2, 4, 1, 3), (3, 1, 4, 2), (3, 4, 2, 1), (4, 1, 2, 3), (4, 3, 1, 2) }). The 6 rows containing this conjugacy class are highlighted in orange in the adjacent
3914:
1359:
Interchanging two, and also the other two. Cycle type = . Order = 2. Members = { (2, 1, 4, 3), (4, 3, 2, 1), (3, 4, 1, 2) }). The 3 rows containing this conjugacy class are shown with boldface entries in the adjacent
3028:
771:
4861:
5721:
4412:
3003:
1032:
6449:
5432:
5345:
5291:
4976:
4225:
1668:
1812:
6299:
4928:
3595:
5633:
5534:
4038:
3723:
2801:
2530:
1107:
6516:
6224:
6012:
622:
590:
527:
6148:
4287:
2451:
2013:
4179:
3920:, when considering the group as acting on itself through conjugation, so that orbits are conjugacy classes and stabilizer subgroups are centralizers. The converse holds as well.
3449:
3289:
1550:
6395:
2188:
1499:
6477:
4755:
2742:
2939:
1344:
No change. Cycle type = . Order = 1. Members = { (1, 2, 3, 4) }. The single row containing this conjugacy class is shown as a row of black circles in the adjacent table.
6112:
6038:
5889:
5843:
1233:
934:
5980:
1962:
403:
165:
4713:
2183:
1736:
219:
6337:
3238:
6553:
1893:
1768:
1162:
328:
3023:
2895:
2821:
2565:
244:
Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of
6585:, but isomorphic subgroups need not be conjugate. For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate.
5935:
5375:
5197:
5147:
5077:
4575:
4525:
4255:
3748:
2624:
2416:
2141:
1838:
1396:
1332:
1296:
881:
695:
358:
6581:
The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate. Conjugate subgroups are
4676:
4088:
3999:
2875:
2848:
2682:
2387:
2111:
1595:
1427:
1194:
5007:
3664:
4445:
6579:
6081:
5807:
5764:
5656:
5495:
5240:
4649:
3526:
3472:
3398:
3336:
1455:
1265:
798:
490:
6192:
6172:
6058:
5909:
5863:
5784:
5741:
5594:
5574:
5554:
5472:
5452:
5217:
5167:
5117:
5097:
5047:
5027:
4881:
4626:
4597:
4545:
4495:
4474:
4058:
3941:
3743:
3684:
3635:
3615:
3546:
3499:
3371:
3313:
2702:
2589:
2491:
2471:
2360:
2084:
2064:
2037:
1917:
1858:
1690:
826:
665:
645:
558:
467:
447:
423:
296:
239:
120:
92:
72:
1561:
1237:
700:
4296:
6734:
6771:
4760:
3190:{\displaystyle {\frac {n!}{\left(k_{1}!m_{1}^{k_{1}}\right)\left(k_{2}!m_{2}^{k_{2}}\right)\cdots \left(k_{s}!m_{s}^{k_{s}}\right)}}.}
6718:
6682:
5661:
6710:
560:
into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes
828:
is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same
6746:
6798:
6706:
4093:
3403:
3243:
2944:
945:
6400:
6793:
6618:
5380:
5296:
5245:
4933:
4184:
3343:
3339:
3292:
1626:
1773:
6646:
853:
6232:
4886:
3558:
5599:
5500:
4004:
3917:
3689:
2747:
2705:
2496:
1336:
consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their description,
1044:
6486:
6197:
5985:
595:
563:
500:
257:
6121:
4260:
2424:
836:
which has order 1. In some cases, conjugacy classes can be described in a uniform way; for example, in the
2330:{\displaystyle a^{k}=\left(gbg^{-1}\right)\left(gbg^{-1}\right)\cdots \left(gbg^{-1}\right)=gb^{k}g^{-1}.}
1967:
263:
1368:, which can be characterized by permutations of the body diagonals, are also described by conjugation in
6815:
6788:
6634:
4600:
1508:
1124:
6342:
1460:
6454:
4718:
2714:
6674:
2900:
1896:
1118:
829:
495:
168:
4414:
where the sum is over a representative element from each conjugacy class that is not in the center.
6115:
6088:
3944:
2627:
1365:
530:
95:
6017:
5868:
5822:
1199:
895:
6555:
this formula generalizes the one given earlier for the number of elements in a conjugacy class.
5940:
4447:
can often be used to gain information about the order of the center or of the conjugacy classes.
2040:
1922:
1038:
939:
363:
125:
4681:
2146:
1699:
182:
6307:
3208:
6767:
6714:
6678:
6594:
6582:
6523:
2419:
2016:
1866:
1741:
1431:
1304:
1131:
534:
301:
253:
245:
172:
3008:
2880:
2806:
2541:
1895:
belong to the same conjugacy class (that is, if they are conjugate), then they have the same
5914:
5350:
5172:
5122:
5052:
4554:
4500:
4230:
2594:
2395:
2116:
1817:
1371:
1307:
1271:
856:
674:
337:
30:
5786:
is abelian and in fact isomorphic to the direct product of two cyclic groups each of order
4654:
4066:
3949:
2853:
2826:
2632:
2365:
2089:
1573:
1405:
1167:
4983:
4090:
from every conjugacy class, we infer from the disjointness of the conjugacy classes that
3640:
1556:
837:
540:
It can be easily shown that conjugacy is an equivalence relation and therefore partitions
17:
4420:
6561:
6063:
5789:
5746:
5638:
5477:
5222:
4631:
3508:
3454:
3380:
3318:
1437:
1247:
780:
472:
6622:
6598:
6177:
6157:
6043:
5894:
5848:
5769:
5726:
5579:
5559:
5539:
5457:
5437:
5202:
5152:
5102:
5082:
5032:
5012:
4866:
4611:
4582:
4530:
4480:
4459:
4043:
3926:
3728:
3669:
3620:
3600:
3531:
3484:
3356:
3298:
2687:
2574:
2476:
2456:
2345:
2069:
2049:
2022:
1902:
1843:
1675:
1502:
1114:
811:
650:
630:
625:
543:
452:
432:
408:
281:
267:
224:
105:
77:
57:
39:
6809:
1693:
668:
249:
6614:
4548:
3502:
51:
35:
3553:
3347:
2803:
be the distinct integers which appear as lengths of cycles in the cycle type of
2534:
1337:
885:
841:
47:
3909:{\displaystyle bab^{-1}=cza(cz)^{-1}=czaz^{-1}c^{-1}=cazz^{-1}c^{-1}=cac^{-1}.}
6698:
6151:
1840:(and the converse is also true: if all conjugacy classes are singletons then
1457:
This is because each conjugacy class corresponds to exactly one partition of
6602:
6640:
4060:; hence the size of each conjugacy class divides the order of the group.
2568:
6766:. Graduate texts in mathematics. Vol. 242 (2 ed.). Springer.
4456:
1623:
The identity element is always the only element in its class, that is
5817:
3374:
766:{\displaystyle \operatorname {Cl} (a)=\left\{gag^{-1}:g\in G\right\}}
266:
that are constant for members of the same conjugacy class are called
5029:
is an abelian group since any non-trivial group element is of order
4856:{\textstyle |G|=p^{n}=|{\operatorname {Z} (G)}|+\sum _{i}p^{k_{i}}.}
27:
In group theory, equivalence class under the relation of conjugation
6480:
5199:
hence abelian. On the other hand, if every non-trivial element in
3549:
6558:
The above is particularly useful when talking about subgroups of
4289:
forms a conjugacy class containing just itself gives rise to the
1113:
These three classes also correspond to the classification of the
6609:
Conjugacy class and irreducible representations in finite group
6601:
topological space can be thought of as equivalence classes of
5716:{\displaystyle \left|\operatorname {C} _{G}(b)\right|=p^{2},}
4407:{\displaystyle |G|=|{\operatorname {Z} (G)}|+\sum _{i}\left,}
2684:
is equal to the number of elements in the conjugacy class of
671:
otherwise.) The equivalence class that contains the element
4063:
Furthermore, if we choose a single representative element
2473:
if and only if its conjugacy class has only one element,
1242:. Each row contains all elements of the conjugacy class
248:
is fundamental for the study of their structure. For an
179:. In other words, each conjugacy class is closed under
4678:
that is not in the center also has order some power of
4763:
3923:
Thus the number of elements in the conjugacy class of
6564:
6526:
6489:
6457:
6403:
6345:
6310:
6235:
6200:
6180:
6160:
6124:
6091:
6066:
6046:
6020:
5988:
5943:
5917:
5897:
5871:
5851:
5825:
5792:
5772:
5749:
5729:
5664:
5641:
5602:
5582:
5562:
5542:
5503:
5480:
5460:
5440:
5383:
5353:
5299:
5248:
5225:
5205:
5175:
5155:
5125:
5105:
5085:
5055:
5035:
5015:
4986:
4936:
4889:
4869:
4721:
4684:
4657:
4634:
4614:
4585:
4557:
4533:
4503:
4483:
4462:
4423:
4299:
4263:
4233:
4187:
4096:
4069:
4046:
4007:
3952:
3929:
3751:
3731:
3692:
3672:
3643:
3623:
3603:
3561:
3534:
3511:
3487:
3457:
3406:
3383:
3359:
3321:
3301:
3246:
3211:
3031:
3011:
2947:
2903:
2883:
2856:
2829:
2809:
2750:
2717:
2690:
2635:
2597:
2577:
2544:
2499:
2479:
2459:
2427:
2398:
2368:
2348:
2191:
2149:
2119:
2092:
2072:
2052:
2025:
1970:
1925:
1905:
1869:
1846:
1820:
1776:
1744:
1702:
1678:
1629:
1576:
1511:
1463:
1440:
1408:
1374:
1310:
1274:
1250:
1202:
1170:
1134:
1047:
948:
898:
859:
814:
783:
703:
677:
653:
633:
598:
566:
546:
503:
475:
455:
435:
411:
366:
340:
304:
284:
227:
185:
128:
108:
80:
60:
6085:
A frequently used theorem is that, given any subset
3597:
This can be seen by observing that any two elements
2998:{\displaystyle \sum \limits _{i=1}^{s}k_{i}m_{i}=n}
1400:In general, the number of conjugacy classes in the
6573:
6547:
6510:
6471:
6443:
6389:
6331:
6293:
6218:
6186:
6166:
6142:
6106:
6075:
6052:
6032:
6006:
5974:
5929:
5903:
5883:
5857:
5837:
5801:
5778:
5758:
5735:
5715:
5650:
5627:
5588:
5568:
5548:
5528:
5489:
5466:
5446:
5426:
5369:
5339:
5285:
5234:
5211:
5191:
5161:
5141:
5111:
5091:
5071:
5041:
5021:
5001:
4970:
4922:
4875:
4855:
4749:
4707:
4670:
4643:
4620:
4591:
4569:
4539:
4519:
4489:
4468:
4439:
4406:
4281:
4249:
4219:
4173:
4082:
4052:
4032:
3993:
3935:
3908:
3737:
3717:
3678:
3658:
3629:
3609:
3589:
3540:
3520:
3493:
3466:
3443:
3392:
3365:
3342:of this action are the conjugacy classes, and the
3330:
3307:
3283:
3232:
3189:
3017:
2997:
2933:
2889:
2869:
2842:
2815:
2795:
2736:
2696:
2676:
2618:
2583:
2559:
2524:
2485:
2465:
2445:
2410:
2381:
2354:
2329:
2177:
2135:
2105:
2078:
2058:
2031:
2007:
1956:
1911:
1887:
1852:
1832:
1806:
1762:
1730:
1684:
1662:
1589:
1544:
1493:
1449:
1421:
1390:
1326:
1290:
1259:
1227:
1188:
1156:
1101:
1027:{\displaystyle (abc\to acb,abc\to bac,abc\to cba)}
1026:
928:
875:
820:
792:
765:
689:
659:
639:
616:
584:
552:
521:
484:
461:
441:
417:
397:
352:
322:
290:
233:
213:
159:
114:
86:
66:
6444:{\displaystyle g^{-1}h\in \operatorname {N} (S),}
3725:) give rise to the same element when conjugating
888:of three elements, has three conjugacy classes:
5427:{\displaystyle |\operatorname {Z} (G)|=p>1,}
6625:is precisely the number of conjugacy classes.
5340:{\displaystyle |\operatorname {Z} (G)|=p>1}
42:with conjugacy classes distinguished by color.
5865:not necessarily a subgroup), define a subset
5286:{\displaystyle |\operatorname {Z} (G)|>1,}
4971:{\displaystyle |\operatorname {Z} (G)|>1.}
4417:Knowledge of the divisors of the group order
4220:{\displaystyle \operatorname {C} _{G}(x_{i})}
1663:{\displaystyle \operatorname {Cl} (e)=\{e\}.}
1352:or color highlighting) in the adjacent table.
8:
6669:Dummit, David S.; Foote, Richard M. (2004).
6539:
6533:
1807:{\displaystyle \operatorname {Cl} (a)=\{a\}}
1801:
1795:
1654:
1648:
1562:conjugation of isometries in Euclidean space
1536:
1512:
1488:
1464:
5169:is isomorphic to the cyclic group of order
3353:Similarly, we can define a group action of
6294:{\displaystyle |\operatorname {Cl} (S)|=.}
5812:Conjugacy of subgroups and general subsets
4923:{\displaystyle |{\operatorname {Z} (G)}|,}
4757:But then the class equation requires that
4608:Since the order of any conjugacy class of
4257:Observing that each element of the center
3590:{\displaystyle \operatorname {C} _{G}(a).}
6643: – Group in group theory mathematics
6563:
6525:
6488:
6461:
6456:
6408:
6402:
6378:
6356:
6344:
6309:
6256:
6236:
6234:
6199:
6179:
6159:
6123:
6090:
6065:
6045:
6019:
5987:
5960:
5942:
5916:
5896:
5870:
5850:
5824:
5791:
5771:
5748:
5728:
5704:
5674:
5663:
5640:
5628:{\displaystyle \operatorname {C} _{G}(b)}
5607:
5601:
5581:
5561:
5541:
5529:{\displaystyle \operatorname {C} _{G}(b)}
5508:
5502:
5479:
5459:
5439:
5404:
5384:
5382:
5358:
5352:
5320:
5300:
5298:
5269:
5249:
5247:
5224:
5204:
5180:
5174:
5154:
5130:
5124:
5104:
5084:
5060:
5054:
5034:
5014:
4985:
4957:
4937:
4935:
4912:
4895:
4890:
4888:
4868:
4842:
4837:
4827:
4815:
4798:
4793:
4784:
4772:
4764:
4762:
4732:
4720:
4694:
4689:
4683:
4662:
4656:
4633:
4613:
4584:
4556:
4532:
4508:
4502:
4482:
4461:
4432:
4424:
4422:
4387:
4371:
4350:
4338:
4321:
4316:
4308:
4300:
4298:
4262:
4238:
4232:
4208:
4192:
4186:
4154:
4138:
4117:
4105:
4097:
4095:
4074:
4068:
4045:
4033:{\displaystyle \operatorname {C} _{G}(a)}
4012:
4006:
3968:
3951:
3928:
3894:
3872:
3859:
3834:
3821:
3796:
3762:
3750:
3730:
3718:{\displaystyle \operatorname {C} _{G}(a)}
3697:
3691:
3671:
3642:
3622:
3602:
3566:
3560:
3533:
3510:
3486:
3456:
3429:
3405:
3382:
3358:
3320:
3300:
3269:
3245:
3210:
3168:
3163:
3158:
3145:
3120:
3115:
3110:
3097:
3075:
3070:
3065:
3052:
3032:
3030:
3010:
2983:
2973:
2963:
2952:
2946:
2902:
2882:
2861:
2855:
2834:
2828:
2808:
2796:{\displaystyle m_{1},m_{2},\ldots ,m_{s}}
2787:
2768:
2755:
2749:
2728:
2716:
2689:
2651:
2634:
2596:
2576:
2543:
2525:{\displaystyle \operatorname {C} _{G}(a)}
2504:
2498:
2478:
2458:
2426:
2397:
2373:
2367:
2347:
2315:
2305:
2281:
2249:
2220:
2196:
2190:
2166:
2148:
2124:
2118:
2097:
2091:
2071:
2051:
2024:
1996:
1969:
1942:
1924:
1919:can be translated into a statement about
1904:
1868:
1845:
1819:
1775:
1743:
1713:
1701:
1677:
1628:
1611:= ( 1 2 3 ) ( 2 3 ) ( 3 2 1 ) = ( 3 1 )
1581:
1575:
1510:
1462:
1439:
1413:
1407:
1379:
1373:
1315:
1309:
1279:
1273:
1268:and each column contains all elements of
1249:
1219:
1201:
1169:
1145:
1133:
1046:
947:
897:
864:
858:
813:
782:
737:
702:
676:
652:
632:
597:
565:
545:
502:
474:
454:
434:
410:
377:
365:
339:
303:
283:
226:
202:
184:
145:
127:
107:
79:
59:
6664:
6662:
3637:belonging to the same coset (and hence,
2086:are conjugate, then so are their powers
1899:. More generally, every statement about
1123:
29:
6658:
5377:We only need to consider the case when
3528:the elements in the conjugacy class of
2043:. See the next property for an example.
1505:, up to permutation of the elements of
1102:{\displaystyle (abc\to bca,abc\to cab)}
6511:{\displaystyle \operatorname {N} (S).}
6219:{\displaystyle \operatorname {Cl} (S)}
6007:{\displaystyle \operatorname {Cl} (S)}
5556:and the center which does not contain
3548:are in one-to-one correspondence with
617:{\displaystyle \operatorname {Cl} (b)}
585:{\displaystyle \operatorname {Cl} (a)}
522:{\displaystyle \operatorname {GL} (n)}
6143:{\displaystyle \operatorname {N} (S)}
4651:it follows that each conjugacy class
4282:{\displaystyle \operatorname {Z} (G)}
2446:{\displaystyle \operatorname {Z} (G)}
7:
3346:of a given element is the element's
3005:). Then the number of conjugates of
2008:{\displaystyle \varphi (x)=gxg^{-1}}
533:, the conjugacy relation is called
4174:{\displaystyle |G|=\sum _{i}\left,}
3444:{\displaystyle g\cdot S:=gSg^{-1},}
3284:{\displaystyle g\cdot x:=gxg^{-1}.}
2949:
1109:. The 2 members both have order 3.
6490:
6423:
6125:
5671:
5604:
5505:
5389:
5305:
5254:
4942:
4896:
4799:
4368:
4322:
4264:
4227:is the centralizer of the element
4189:
4135:
4009:
3965:
3694:
3563:
3451:or on the set of the subgroups of
2850:be the number of cycles of length
2648:
2501:
2428:
1614:= ( 3 1 ) is Conjugate of ( 2 3 )
1545:{\displaystyle \{1,2,\ldots ,n\}.}
1034:. The 3 members all have order 2.
25:
6390:{\displaystyle gSg^{-1}=hSh^{-1}}
1494:{\displaystyle \{1,2,\ldots ,n\}}
936:. The single member has order 1.
6762:Grillet, Pierre Antoine (2007).
6472:{\displaystyle g{\text{ and }}h}
4750:{\displaystyle 0<k_{i}<n.}
2737:{\displaystyle \sigma \in S_{n}}
6451:in other words, if and only if
5743:is an element of the center of
3916:That can also be seen from the
2934:{\displaystyle i=1,2,\ldots ,s}
6617:, the number of nonisomorphic
6502:
6496:
6435:
6429:
6285:
6282:
6276:
6264:
6257:
6253:
6247:
6237:
6213:
6207:
6137:
6131:
6001:
5995:
5689:
5683:
5622:
5616:
5596:elements. Hence the order of
5523:
5517:
5474:which is not in the center of
5405:
5401:
5395:
5385:
5321:
5317:
5311:
5301:
5270:
5266:
5260:
5250:
5242:hence by the conclusion above
4958:
4954:
4948:
4938:
4913:
4908:
4902:
4891:
4816:
4811:
4805:
4794:
4773:
4765:
4577:). We are going to prove that
4433:
4425:
4393:
4380:
4339:
4334:
4328:
4317:
4309:
4301:
4276:
4270:
4214:
4201:
4160:
4147:
4106:
4098:
4027:
4021:
3983:
3977:
3793:
3783:
3712:
3706:
3581:
3575:
2666:
2660:
2519:
2513:
2440:
2434:
1980:
1974:
1789:
1783:
1642:
1636:
1183:
1171:
1096:
1084:
1060:
1048:
1021:
1009:
985:
961:
949:
923:
911:
899:
716:
710:
611:
605:
579:
573:
516:
510:
1:
6707:Graduate Texts in Mathematics
6107:{\displaystyle S\subseteq G,}
4497:(that is, a group with order
3505:, then for any group element
1340:, member order, and members:
6033:{\displaystyle T\subseteq G}
5884:{\displaystyle T\subseteq G}
5838:{\displaystyle S\subseteq G}
1366:proper rotations of the cube
1228:{\displaystyle a,b\in S_{4}}
929:{\displaystyle (abc\to abc)}
252:, each conjugacy class is a
6794:Encyclopedia of Mathematics
6637: – Concept in topology
6619:irreducible representations
5975:{\displaystyle T=gSg^{-1}.}
2571:consisting of all elements
2493:itself. More generally, if
1957:{\displaystyle b=gag^{-1},}
398:{\displaystyle gag^{-1}=b,}
334:if there exists an element
160:{\displaystyle b=gag^{-1}.}
6832:
6194:equals the cardinality of
6014:be the set of all subsets
5816:More generally, given any
4708:{\displaystyle p^{k_{i}},}
2823:(including 1-cycles). Let
2178:{\displaystyle a=gbg^{-1}}
1731:{\displaystyle gag^{-1}=a}
1430:is equal to the number of
214:{\displaystyle b=gag^{-1}}
18:Conjugation (group theory)
6647:Conjugacy-closed subgroup
6593:Conjugacy classes in the
6332:{\displaystyle g,h\in G,}
5434:then there is an element
4628:must divide the order of
3233:{\displaystyle g,x\in G,}
3201:Conjugacy as group action
840:they can be described by
469:is called a conjugate of
298:be a group. Two elements
6589:Geometric interpretation
6548:{\displaystyle S=\{a\},}
5635:is strictly larger than
3918:orbit-stabilizer theorem
3477:Conjugacy class equation
2706:orbit-stabilizer theorem
1888:{\displaystyle a,b\in G}
1763:{\displaystyle a,g\in G}
1157:{\displaystyle bab^{-1}}
323:{\displaystyle a,b\in G}
256:containing one element (
6304:This follows since, if
5766:a contradiction. Hence
3018:{\displaystyle \sigma }
2890:{\displaystyle \sigma }
2816:{\displaystyle \sigma }
2560:{\displaystyle a\in G,}
2362:is a power-up class of
122:in the group such that
102:if there is an element
6575:
6549:
6512:
6473:
6445:
6391:
6333:
6295:
6220:
6188:
6168:
6144:
6108:
6077:
6054:
6034:
6008:
5976:
5931:
5930:{\displaystyle g\in G}
5905:
5885:
5859:
5839:
5803:
5780:
5760:
5737:
5717:
5652:
5629:
5590:
5570:
5550:
5530:
5491:
5468:
5448:
5428:
5371:
5370:{\displaystyle p^{2}.}
5341:
5287:
5236:
5213:
5193:
5192:{\displaystyle p^{2},}
5163:
5143:
5142:{\displaystyle p^{2},}
5113:
5093:
5073:
5072:{\displaystyle p^{2}.}
5043:
5023:
5003:
4972:
4924:
4877:
4863:From this we see that
4857:
4751:
4709:
4672:
4645:
4622:
4593:
4571:
4570:{\displaystyle n>0}
4541:
4521:
4520:{\displaystyle p^{n},}
4491:
4470:
4441:
4408:
4283:
4251:
4250:{\displaystyle x_{i}.}
4221:
4175:
4084:
4054:
4034:
3995:
3937:
3910:
3739:
3719:
3680:
3660:
3631:
3611:
3591:
3542:
3522:
3495:
3468:
3445:
3394:
3367:
3332:
3309:
3285:
3234:
3191:
3019:
2999:
2968:
2935:
2891:
2871:
2844:
2817:
2797:
2738:
2698:
2678:
2620:
2619:{\displaystyle ga=ag,}
2585:
2561:
2526:
2487:
2467:
2447:
2412:
2411:{\displaystyle a\in G}
2383:
2356:
2331:
2179:
2137:
2136:{\displaystyle b^{k}.}
2107:
2080:
2060:
2033:
2009:
1958:
1913:
1889:
1854:
1834:
1833:{\displaystyle a\in G}
1808:
1764:
1732:
1686:
1664:
1591:
1546:
1495:
1451:
1423:
1392:
1391:{\displaystyle S_{4}.}
1328:
1327:{\displaystyle S_{4},}
1298:
1292:
1291:{\displaystyle S_{4}.}
1261:
1229:
1190:
1158:
1103:
1028:
930:
877:
876:{\displaystyle S_{3},}
822:
794:
767:
691:
690:{\displaystyle a\in G}
661:
641:
618:
586:
554:
523:
486:
463:
443:
419:
399:
354:
353:{\displaystyle g\in G}
324:
292:
235:
215:
161:
116:
88:
68:
43:
6675:John Wiley & Sons
6635:Topological conjugacy
6605:under free homotopy.
6576:
6550:
6513:
6474:
6446:
6392:
6334:
6296:
6221:
6189:
6169:
6145:
6109:
6078:
6055:
6035:
6009:
5977:
5932:
5911:if there exists some
5906:
5886:
5860:
5840:
5804:
5781:
5761:
5738:
5718:
5653:
5630:
5591:
5571:
5551:
5531:
5492:
5469:
5449:
5429:
5372:
5342:
5288:
5237:
5214:
5194:
5164:
5144:
5114:
5094:
5074:
5044:
5024:
5004:
4973:
4925:
4878:
4858:
4752:
4710:
4673:
4671:{\displaystyle H_{i}}
4646:
4623:
4594:
4572:
4542:
4522:
4492:
4471:
4442:
4409:
4284:
4252:
4222:
4176:
4085:
4083:{\displaystyle x_{i}}
4055:
4035:
3996:
3994:{\displaystyle \left}
3938:
3911:
3740:
3720:
3681:
3661:
3632:
3612:
3592:
3543:
3523:
3496:
3469:
3446:
3395:
3368:
3333:
3310:
3286:
3235:
3205:For any two elements
3192:
3020:
3000:
2948:
2936:
2892:
2872:
2870:{\displaystyle m_{i}}
2845:
2843:{\displaystyle k_{i}}
2818:
2798:
2739:
2699:
2679:
2677:{\displaystyle \left}
2621:
2586:
2562:
2527:
2488:
2468:
2448:
2413:
2384:
2382:{\displaystyle a^{k}}
2357:
2332:
2180:
2138:
2108:
2106:{\displaystyle a^{k}}
2081:
2061:
2034:
2010:
1959:
1914:
1890:
1855:
1835:
1809:
1765:
1733:
1687:
1665:
1592:
1590:{\displaystyle S_{3}}
1547:
1496:
1452:
1424:
1422:{\displaystyle S_{n}}
1393:
1329:
1293:
1262:
1230:
1191:
1189:{\displaystyle (a,b)}
1159:
1127:
1104:
1029:
931:
878:
823:
795:
768:
692:
662:
642:
619:
587:
555:
524:
487:
464:
444:
420:
400:
355:
325:
293:
236:
216:
162:
117:
89:
69:
33:
6789:"Conjugate elements"
6562:
6524:
6487:
6455:
6401:
6343:
6308:
6233:
6198:
6178:
6158:
6122:
6089:
6064:
6044:
6018:
5986:
5941:
5915:
5895:
5869:
5849:
5823:
5790:
5770:
5747:
5727:
5662:
5639:
5600:
5580:
5560:
5540:
5501:
5478:
5458:
5438:
5381:
5351:
5297:
5246:
5223:
5203:
5173:
5153:
5123:
5103:
5083:
5053:
5033:
5013:
5002:{\displaystyle n=2,}
4984:
4980:In particular, when
4934:
4887:
4867:
4761:
4719:
4682:
4655:
4632:
4612:
4583:
4555:
4531:
4501:
4481:
4460:
4421:
4297:
4261:
4231:
4185:
4094:
4067:
4044:
4005:
3950:
3927:
3749:
3729:
3690:
3670:
3659:{\displaystyle b=cz}
3641:
3621:
3601:
3559:
3532:
3509:
3485:
3455:
3404:
3381:
3357:
3319:
3299:
3244:
3209:
3029:
3009:
2945:
2901:
2881:
2854:
2827:
2807:
2748:
2715:
2688:
2633:
2595:
2575:
2542:
2497:
2477:
2457:
2425:
2396:
2366:
2346:
2189:
2147:
2117:
2090:
2070:
2050:
2023:
1968:
1923:
1903:
1867:
1844:
1818:
1774:
1742:
1700:
1676:
1627:
1574:
1509:
1461:
1438:
1406:
1372:
1308:
1272:
1248:
1200:
1168:
1132:
1119:equilateral triangle
1045:
946:
896:
884:consisting of the 6
857:
852:The symmetric group
812:
781:
701:
675:
651:
631:
596:
564:
544:
501:
496:general linear group
473:
453:
433:
409:
364:
338:
302:
282:
225:
183:
169:equivalence relation
126:
106:
78:
58:
5891:to be conjugate to
4440:{\displaystyle |G|}
4001:of the centralizer
3686:in the centralizer
3175:
3127:
3082:
667:are conjugate, and
531:invertible matrices
494:In the case of the
173:equivalence classes
6574:{\displaystyle G.}
6571:
6545:
6508:
6469:
6441:
6387:
6329:
6291:
6216:
6184:
6164:
6140:
6104:
6076:{\displaystyle S.}
6073:
6050:
6030:
6004:
5972:
5927:
5901:
5881:
5855:
5835:
5802:{\displaystyle p.}
5799:
5776:
5759:{\displaystyle G,}
5756:
5733:
5713:
5651:{\displaystyle p,}
5648:
5625:
5586:
5566:
5546:
5526:
5490:{\displaystyle G.}
5487:
5464:
5444:
5424:
5367:
5337:
5283:
5235:{\displaystyle p,}
5232:
5209:
5189:
5159:
5139:
5109:
5089:
5069:
5039:
5019:
4999:
4968:
4920:
4873:
4853:
4832:
4747:
4705:
4668:
4644:{\displaystyle G,}
4641:
4618:
4589:
4567:
4537:
4517:
4487:
4466:
4455:Consider a finite
4437:
4404:
4355:
4279:
4247:
4217:
4171:
4122:
4080:
4050:
4030:
3991:
3933:
3906:
3735:
3715:
3676:
3656:
3627:
3607:
3587:
3538:
3521:{\displaystyle a,}
3518:
3491:
3467:{\displaystyle G.}
3464:
3441:
3393:{\displaystyle G,}
3390:
3373:on the set of all
3363:
3331:{\displaystyle G.}
3328:
3305:
3281:
3230:
3187:
3154:
3106:
3061:
3015:
2995:
2931:
2887:
2867:
2840:
2813:
2793:
2734:
2694:
2674:
2616:
2581:
2557:
2522:
2483:
2463:
2443:
2408:
2379:
2352:
2327:
2175:
2133:
2103:
2076:
2056:
2041:inner automorphism
2029:
2005:
1954:
1909:
1885:
1850:
1830:
1804:
1760:
1728:
1682:
1660:
1587:
1560:can be studied by
1542:
1491:
1450:{\displaystyle n.}
1447:
1432:integer partitions
1419:
1388:
1324:
1299:
1288:
1260:{\displaystyle a,}
1257:
1225:
1186:
1154:
1099:
1039:cyclic permutation
1024:
926:
873:
818:
793:{\displaystyle a.}
790:
773:and is called the
763:
687:
657:
637:
614:
582:
550:
519:
485:{\displaystyle b.}
482:
459:
439:
415:
395:
350:
320:
288:
246:non-abelian groups
231:
211:
157:
112:
84:
64:
44:
6773:978-0-387-71567-4
6595:fundamental group
6464:
6187:{\displaystyle G}
6167:{\displaystyle S}
6053:{\displaystyle T}
5904:{\displaystyle S}
5858:{\displaystyle S}
5779:{\displaystyle G}
5736:{\displaystyle b}
5589:{\displaystyle p}
5569:{\displaystyle b}
5549:{\displaystyle b}
5467:{\displaystyle G}
5447:{\displaystyle b}
5212:{\displaystyle G}
5162:{\displaystyle G}
5112:{\displaystyle G}
5092:{\displaystyle a}
5042:{\displaystyle p}
5022:{\displaystyle G}
4876:{\displaystyle p}
4823:
4621:{\displaystyle G}
4599:-group has a non-
4592:{\displaystyle p}
4540:{\displaystyle p}
4490:{\displaystyle G}
4469:{\displaystyle p}
4346:
4113:
4053:{\displaystyle G}
3936:{\displaystyle a}
3738:{\displaystyle a}
3679:{\displaystyle z}
3630:{\displaystyle c}
3610:{\displaystyle b}
3541:{\displaystyle a}
3494:{\displaystyle G}
3366:{\displaystyle G}
3308:{\displaystyle G}
3182:
2697:{\displaystyle a}
2584:{\displaystyle g}
2486:{\displaystyle a}
2466:{\displaystyle G}
2355:{\displaystyle a}
2079:{\displaystyle b}
2059:{\displaystyle a}
2032:{\displaystyle G}
1912:{\displaystyle a}
1853:{\displaystyle G}
1685:{\displaystyle G}
1241:
821:{\displaystyle G}
660:{\displaystyle b}
640:{\displaystyle a}
553:{\displaystyle G}
535:matrix similarity
462:{\displaystyle a}
442:{\displaystyle a}
418:{\displaystyle b}
291:{\displaystyle G}
234:{\displaystyle g}
221:for all elements
177:conjugacy classes
115:{\displaystyle g}
87:{\displaystyle b}
67:{\displaystyle a}
16:(Redirected from
6823:
6802:
6777:
6764:Abstract algebra
6749:
6745:Grillet (2007),
6743:
6737:
6733:Grillet (2007),
6731:
6725:
6724:
6695:
6689:
6688:
6673:(3rd ed.).
6671:Abstract Algebra
6666:
6580:
6578:
6577:
6572:
6554:
6552:
6551:
6546:
6517:
6515:
6514:
6509:
6479:are in the same
6478:
6476:
6475:
6470:
6465:
6462:
6450:
6448:
6447:
6442:
6416:
6415:
6396:
6394:
6393:
6388:
6386:
6385:
6364:
6363:
6338:
6336:
6335:
6330:
6300:
6298:
6297:
6292:
6260:
6240:
6225:
6223:
6222:
6217:
6193:
6191:
6190:
6185:
6173:
6171:
6170:
6165:
6149:
6147:
6146:
6141:
6113:
6111:
6110:
6105:
6082:
6080:
6079:
6074:
6060:is conjugate to
6059:
6057:
6056:
6051:
6039:
6037:
6036:
6031:
6013:
6011:
6010:
6005:
5981:
5979:
5978:
5973:
5968:
5967:
5936:
5934:
5933:
5928:
5910:
5908:
5907:
5902:
5890:
5888:
5887:
5882:
5864:
5862:
5861:
5856:
5844:
5842:
5841:
5836:
5808:
5806:
5805:
5800:
5785:
5783:
5782:
5777:
5765:
5763:
5762:
5757:
5742:
5740:
5739:
5734:
5722:
5720:
5719:
5714:
5709:
5708:
5696:
5692:
5679:
5678:
5657:
5655:
5654:
5649:
5634:
5632:
5631:
5626:
5612:
5611:
5595:
5593:
5592:
5587:
5575:
5573:
5572:
5567:
5555:
5553:
5552:
5547:
5535:
5533:
5532:
5527:
5513:
5512:
5496:
5494:
5493:
5488:
5473:
5471:
5470:
5465:
5453:
5451:
5450:
5445:
5433:
5431:
5430:
5425:
5408:
5388:
5376:
5374:
5373:
5368:
5363:
5362:
5346:
5344:
5343:
5338:
5324:
5304:
5292:
5290:
5289:
5284:
5273:
5253:
5241:
5239:
5238:
5233:
5218:
5216:
5215:
5210:
5198:
5196:
5195:
5190:
5185:
5184:
5168:
5166:
5165:
5160:
5148:
5146:
5145:
5140:
5135:
5134:
5118:
5116:
5115:
5110:
5098:
5096:
5095:
5090:
5079:If some element
5078:
5076:
5075:
5070:
5065:
5064:
5048:
5046:
5045:
5040:
5028:
5026:
5025:
5020:
5008:
5006:
5005:
5000:
4977:
4975:
4974:
4969:
4961:
4941:
4929:
4927:
4926:
4921:
4916:
4911:
4894:
4882:
4880:
4879:
4874:
4862:
4860:
4859:
4854:
4849:
4848:
4847:
4846:
4831:
4819:
4814:
4797:
4789:
4788:
4776:
4768:
4756:
4754:
4753:
4748:
4737:
4736:
4714:
4712:
4711:
4706:
4701:
4700:
4699:
4698:
4677:
4675:
4674:
4669:
4667:
4666:
4650:
4648:
4647:
4642:
4627:
4625:
4624:
4619:
4598:
4596:
4595:
4590:
4576:
4574:
4573:
4568:
4546:
4544:
4543:
4538:
4526:
4524:
4523:
4518:
4513:
4512:
4496:
4494:
4493:
4488:
4475:
4473:
4472:
4467:
4446:
4444:
4443:
4438:
4436:
4428:
4413:
4411:
4410:
4405:
4400:
4396:
4392:
4391:
4376:
4375:
4354:
4342:
4337:
4320:
4312:
4304:
4288:
4286:
4285:
4280:
4256:
4254:
4253:
4248:
4243:
4242:
4226:
4224:
4223:
4218:
4213:
4212:
4197:
4196:
4180:
4178:
4177:
4172:
4167:
4163:
4159:
4158:
4143:
4142:
4121:
4109:
4101:
4089:
4087:
4086:
4081:
4079:
4078:
4059:
4057:
4056:
4051:
4039:
4037:
4036:
4031:
4017:
4016:
4000:
3998:
3997:
3992:
3990:
3986:
3973:
3972:
3942:
3940:
3939:
3934:
3915:
3913:
3912:
3907:
3902:
3901:
3880:
3879:
3867:
3866:
3842:
3841:
3829:
3828:
3804:
3803:
3770:
3769:
3744:
3742:
3741:
3736:
3724:
3722:
3721:
3716:
3702:
3701:
3685:
3683:
3682:
3677:
3665:
3663:
3662:
3657:
3636:
3634:
3633:
3628:
3616:
3614:
3613:
3608:
3596:
3594:
3593:
3588:
3571:
3570:
3547:
3545:
3544:
3539:
3527:
3525:
3524:
3519:
3500:
3498:
3497:
3492:
3473:
3471:
3470:
3465:
3450:
3448:
3447:
3442:
3437:
3436:
3399:
3397:
3396:
3391:
3372:
3370:
3369:
3364:
3337:
3335:
3334:
3329:
3314:
3312:
3311:
3306:
3290:
3288:
3287:
3282:
3277:
3276:
3239:
3237:
3236:
3231:
3196:
3194:
3193:
3188:
3183:
3181:
3180:
3176:
3174:
3173:
3172:
3162:
3150:
3149:
3132:
3128:
3126:
3125:
3124:
3114:
3102:
3101:
3087:
3083:
3081:
3080:
3079:
3069:
3057:
3056:
3041:
3033:
3024:
3022:
3021:
3016:
3004:
3002:
3001:
2996:
2988:
2987:
2978:
2977:
2967:
2962:
2940:
2938:
2937:
2932:
2896:
2894:
2893:
2888:
2876:
2874:
2873:
2868:
2866:
2865:
2849:
2847:
2846:
2841:
2839:
2838:
2822:
2820:
2819:
2814:
2802:
2800:
2799:
2794:
2792:
2791:
2773:
2772:
2760:
2759:
2743:
2741:
2740:
2735:
2733:
2732:
2703:
2701:
2700:
2695:
2683:
2681:
2680:
2675:
2673:
2669:
2656:
2655:
2625:
2623:
2622:
2617:
2590:
2588:
2587:
2582:
2566:
2564:
2563:
2558:
2531:
2529:
2528:
2523:
2509:
2508:
2492:
2490:
2489:
2484:
2472:
2470:
2469:
2464:
2452:
2450:
2449:
2444:
2417:
2415:
2414:
2409:
2388:
2386:
2385:
2380:
2378:
2377:
2361:
2359:
2358:
2353:
2340:
2336:
2334:
2333:
2328:
2323:
2322:
2310:
2309:
2294:
2290:
2289:
2288:
2262:
2258:
2257:
2256:
2233:
2229:
2228:
2227:
2201:
2200:
2184:
2182:
2181:
2176:
2174:
2173:
2142:
2140:
2139:
2134:
2129:
2128:
2112:
2110:
2109:
2104:
2102:
2101:
2085:
2083:
2082:
2077:
2065:
2063:
2062:
2057:
2038:
2036:
2035:
2030:
2014:
2012:
2011:
2006:
2004:
2003:
1964:because the map
1963:
1961:
1960:
1955:
1950:
1949:
1918:
1916:
1915:
1910:
1894:
1892:
1891:
1886:
1863:If two elements
1859:
1857:
1856:
1851:
1839:
1837:
1836:
1831:
1813:
1811:
1810:
1805:
1769:
1767:
1766:
1761:
1737:
1735:
1734:
1729:
1721:
1720:
1691:
1689:
1688:
1683:
1669:
1667:
1666:
1661:
1596:
1594:
1593:
1588:
1586:
1585:
1554:In general, the
1551:
1549:
1548:
1543:
1500:
1498:
1497:
1492:
1456:
1454:
1453:
1448:
1428:
1426:
1425:
1420:
1418:
1417:
1402:symmetric group
1397:
1395:
1394:
1389:
1384:
1383:
1333:
1331:
1330:
1325:
1320:
1319:
1303:symmetric group
1297:
1295:
1294:
1289:
1284:
1283:
1267:
1266:
1264:
1263:
1258:
1235:
1234:
1232:
1231:
1226:
1224:
1223:
1195:
1193:
1192:
1187:
1163:
1161:
1160:
1155:
1153:
1152:
1108:
1106:
1105:
1100:
1033:
1031:
1030:
1025:
935:
933:
932:
927:
882:
880:
879:
874:
869:
868:
827:
825:
824:
819:
806:
805:
799:
797:
796:
791:
772:
770:
769:
764:
762:
758:
745:
744:
696:
694:
693:
688:
666:
664:
663:
658:
646:
644:
643:
638:
623:
621:
620:
615:
591:
589:
588:
583:
559:
557:
556:
551:
528:
526:
525:
520:
491:
489:
488:
483:
468:
466:
465:
460:
448:
446:
445:
440:
424:
422:
421:
416:
404:
402:
401:
396:
385:
384:
359:
357:
356:
351:
329:
327:
326:
321:
297:
295:
294:
289:
240:
238:
237:
232:
220:
218:
217:
212:
210:
209:
166:
164:
163:
158:
153:
152:
121:
119:
118:
113:
93:
91:
90:
85:
73:
71:
70:
65:
21:
6831:
6830:
6826:
6825:
6824:
6822:
6821:
6820:
6806:
6805:
6787:
6784:
6774:
6761:
6758:
6753:
6752:
6744:
6740:
6732:
6728:
6721:
6697:
6696:
6692:
6685:
6668:
6667:
6660:
6655:
6631:
6623:complex numbers
6611:
6591:
6560:
6559:
6522:
6521:
6485:
6484:
6463: and
6453:
6452:
6404:
6399:
6398:
6397:if and only if
6374:
6352:
6341:
6340:
6306:
6305:
6231:
6230:
6196:
6195:
6176:
6175:
6156:
6155:
6120:
6119:
6087:
6086:
6062:
6061:
6042:
6041:
6016:
6015:
5984:
5983:
5956:
5939:
5938:
5913:
5912:
5893:
5892:
5867:
5866:
5847:
5846:
5821:
5820:
5814:
5788:
5787:
5768:
5767:
5745:
5744:
5725:
5724:
5700:
5670:
5669:
5665:
5660:
5659:
5637:
5636:
5603:
5598:
5597:
5578:
5577:
5558:
5557:
5538:
5537:
5504:
5499:
5498:
5476:
5475:
5456:
5455:
5436:
5435:
5379:
5378:
5354:
5349:
5348:
5295:
5294:
5244:
5243:
5221:
5220:
5201:
5200:
5176:
5171:
5170:
5151:
5150:
5126:
5121:
5120:
5101:
5100:
5081:
5080:
5056:
5051:
5050:
5031:
5030:
5011:
5010:
4982:
4981:
4932:
4931:
4885:
4884:
4865:
4864:
4838:
4833:
4780:
4759:
4758:
4728:
4717:
4716:
4690:
4685:
4680:
4679:
4658:
4653:
4652:
4630:
4629:
4610:
4609:
4581:
4580:
4553:
4552:
4529:
4528:
4504:
4499:
4498:
4479:
4478:
4458:
4457:
4453:
4419:
4418:
4383:
4367:
4360:
4356:
4295:
4294:
4259:
4258:
4234:
4229:
4228:
4204:
4188:
4183:
4182:
4150:
4134:
4127:
4123:
4092:
4091:
4070:
4065:
4064:
4042:
4041:
4008:
4003:
4002:
3964:
3957:
3953:
3948:
3947:
3925:
3924:
3890:
3868:
3855:
3830:
3817:
3792:
3758:
3747:
3746:
3727:
3726:
3693:
3688:
3687:
3668:
3667:
3639:
3638:
3619:
3618:
3599:
3598:
3562:
3557:
3556:
3530:
3529:
3507:
3506:
3483:
3482:
3479:
3453:
3452:
3425:
3402:
3401:
3379:
3378:
3355:
3354:
3317:
3316:
3297:
3296:
3291:This defines a
3265:
3242:
3241:
3207:
3206:
3203:
3164:
3141:
3140:
3136:
3116:
3093:
3092:
3088:
3071:
3048:
3047:
3043:
3042:
3034:
3027:
3026:
3007:
3006:
2979:
2969:
2943:
2942:
2899:
2898:
2879:
2878:
2857:
2852:
2851:
2830:
2825:
2824:
2805:
2804:
2783:
2764:
2751:
2746:
2745:
2724:
2713:
2712:
2686:
2685:
2647:
2640:
2636:
2631:
2630:
2593:
2592:
2573:
2572:
2540:
2539:
2500:
2495:
2494:
2475:
2474:
2455:
2454:
2423:
2422:
2394:
2393:
2369:
2364:
2363:
2344:
2343:
2338:
2311:
2301:
2277:
2270:
2266:
2245:
2238:
2234:
2216:
2209:
2205:
2192:
2187:
2186:
2162:
2145:
2144:
2120:
2115:
2114:
2093:
2088:
2087:
2068:
2067:
2048:
2047:
2021:
2020:
1992:
1966:
1965:
1938:
1921:
1920:
1901:
1900:
1865:
1864:
1842:
1841:
1816:
1815:
1772:
1771:
1740:
1739:
1709:
1698:
1697:
1674:
1673:
1625:
1624:
1620:
1577:
1572:
1571:
1557:Euclidean group
1507:
1506:
1459:
1458:
1436:
1435:
1409:
1404:
1403:
1375:
1370:
1369:
1311:
1306:
1305:
1275:
1270:
1269:
1246:
1245:
1243:
1215:
1198:
1197:
1166:
1165:
1141:
1130:
1129:
1043:
1042:
944:
943:
894:
893:
860:
855:
854:
850:
838:symmetric group
810:
809:
803:
802:
779:
778:
775:conjugacy class
733:
726:
722:
699:
698:
673:
672:
649:
648:
629:
628:
594:
593:
562:
561:
542:
541:
499:
498:
471:
470:
451:
450:
431:
430:
407:
406:
373:
362:
361:
336:
335:
300:
299:
280:
279:
276:
268:class functions
223:
222:
198:
181:
180:
141:
124:
123:
104:
103:
76:
75:
56:
55:
54:, two elements
40:dihedral groups
28:
23:
22:
15:
12:
11:
5:
6829:
6827:
6819:
6818:
6808:
6807:
6804:
6803:
6783:
6782:External links
6780:
6779:
6778:
6772:
6757:
6754:
6751:
6750:
6738:
6726:
6719:
6690:
6683:
6657:
6656:
6654:
6651:
6650:
6649:
6644:
6638:
6630:
6627:
6610:
6607:
6599:path-connected
6590:
6587:
6570:
6567:
6544:
6541:
6538:
6535:
6532:
6529:
6507:
6504:
6501:
6498:
6495:
6492:
6468:
6460:
6440:
6437:
6434:
6431:
6428:
6425:
6422:
6419:
6414:
6411:
6407:
6384:
6381:
6377:
6373:
6370:
6367:
6362:
6359:
6355:
6351:
6348:
6328:
6325:
6322:
6319:
6316:
6313:
6302:
6301:
6290:
6287:
6284:
6281:
6278:
6275:
6272:
6269:
6266:
6263:
6259:
6255:
6252:
6249:
6246:
6243:
6239:
6215:
6212:
6209:
6206:
6203:
6183:
6163:
6139:
6136:
6133:
6130:
6127:
6103:
6100:
6097:
6094:
6072:
6069:
6049:
6029:
6026:
6023:
6003:
6000:
5997:
5994:
5991:
5971:
5966:
5963:
5959:
5955:
5952:
5949:
5946:
5926:
5923:
5920:
5900:
5880:
5877:
5874:
5854:
5834:
5831:
5828:
5813:
5810:
5798:
5795:
5775:
5755:
5752:
5732:
5712:
5707:
5703:
5699:
5695:
5691:
5688:
5685:
5682:
5677:
5673:
5668:
5647:
5644:
5624:
5621:
5618:
5615:
5610:
5606:
5585:
5565:
5545:
5525:
5522:
5519:
5516:
5511:
5507:
5486:
5483:
5463:
5443:
5423:
5420:
5417:
5414:
5411:
5407:
5403:
5400:
5397:
5394:
5391:
5387:
5366:
5361:
5357:
5336:
5333:
5330:
5327:
5323:
5319:
5316:
5313:
5310:
5307:
5303:
5282:
5279:
5276:
5272:
5268:
5265:
5262:
5259:
5256:
5252:
5231:
5228:
5208:
5188:
5183:
5179:
5158:
5138:
5133:
5129:
5108:
5088:
5068:
5063:
5059:
5038:
5018:
4998:
4995:
4992:
4989:
4967:
4964:
4960:
4956:
4953:
4950:
4947:
4944:
4940:
4919:
4915:
4910:
4907:
4904:
4901:
4898:
4893:
4872:
4852:
4845:
4841:
4836:
4830:
4826:
4822:
4818:
4813:
4810:
4807:
4804:
4801:
4796:
4792:
4787:
4783:
4779:
4775:
4771:
4767:
4746:
4743:
4740:
4735:
4731:
4727:
4724:
4704:
4697:
4693:
4688:
4665:
4661:
4640:
4637:
4617:
4604:
4588:
4566:
4563:
4560:
4536:
4516:
4511:
4507:
4486:
4465:
4452:
4449:
4435:
4431:
4427:
4403:
4399:
4395:
4390:
4386:
4382:
4379:
4374:
4370:
4366:
4363:
4359:
4353:
4349:
4345:
4341:
4336:
4333:
4330:
4327:
4324:
4319:
4315:
4311:
4307:
4303:
4291:class equation
4278:
4275:
4272:
4269:
4266:
4246:
4241:
4237:
4216:
4211:
4207:
4203:
4200:
4195:
4191:
4170:
4166:
4162:
4157:
4153:
4149:
4146:
4141:
4137:
4133:
4130:
4126:
4120:
4116:
4112:
4108:
4104:
4100:
4077:
4073:
4049:
4029:
4026:
4023:
4020:
4015:
4011:
3989:
3985:
3982:
3979:
3976:
3971:
3967:
3963:
3960:
3956:
3932:
3905:
3900:
3897:
3893:
3889:
3886:
3883:
3878:
3875:
3871:
3865:
3862:
3858:
3854:
3851:
3848:
3845:
3840:
3837:
3833:
3827:
3824:
3820:
3816:
3813:
3810:
3807:
3802:
3799:
3795:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3768:
3765:
3761:
3757:
3754:
3734:
3714:
3711:
3708:
3705:
3700:
3696:
3675:
3655:
3652:
3649:
3646:
3626:
3606:
3586:
3583:
3580:
3577:
3574:
3569:
3565:
3537:
3517:
3514:
3490:
3478:
3475:
3463:
3460:
3440:
3435:
3432:
3428:
3424:
3421:
3418:
3415:
3412:
3409:
3389:
3386:
3362:
3327:
3324:
3304:
3280:
3275:
3272:
3268:
3264:
3261:
3258:
3255:
3252:
3249:
3229:
3226:
3223:
3220:
3217:
3214:
3202:
3199:
3198:
3197:
3186:
3179:
3171:
3167:
3161:
3157:
3153:
3148:
3144:
3139:
3135:
3131:
3123:
3119:
3113:
3109:
3105:
3100:
3096:
3091:
3086:
3078:
3074:
3068:
3064:
3060:
3055:
3051:
3046:
3040:
3037:
3014:
2994:
2991:
2986:
2982:
2976:
2972:
2966:
2961:
2958:
2955:
2951:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2886:
2864:
2860:
2837:
2833:
2812:
2790:
2786:
2782:
2779:
2776:
2771:
2767:
2763:
2758:
2754:
2731:
2727:
2723:
2720:
2709:
2693:
2672:
2668:
2665:
2662:
2659:
2654:
2650:
2646:
2643:
2639:
2615:
2612:
2609:
2606:
2603:
2600:
2580:
2556:
2553:
2550:
2547:
2537:
2521:
2518:
2515:
2512:
2507:
2503:
2482:
2462:
2442:
2439:
2436:
2433:
2430:
2407:
2404:
2401:
2390:
2376:
2372:
2351:
2342:(3)(2) (where
2337:) Thus taking
2326:
2321:
2318:
2314:
2308:
2304:
2300:
2297:
2293:
2287:
2284:
2280:
2276:
2273:
2269:
2265:
2261:
2255:
2252:
2248:
2244:
2241:
2237:
2232:
2226:
2223:
2219:
2215:
2212:
2208:
2204:
2199:
2195:
2172:
2169:
2165:
2161:
2158:
2155:
2152:
2132:
2127:
2123:
2100:
2096:
2075:
2055:
2044:
2028:
2002:
1999:
1995:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1953:
1948:
1945:
1941:
1937:
1934:
1931:
1928:
1908:
1884:
1881:
1878:
1875:
1872:
1861:
1849:
1829:
1826:
1823:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1759:
1756:
1753:
1750:
1747:
1727:
1724:
1719:
1716:
1712:
1708:
1705:
1681:
1670:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1619:
1616:
1605:x = ( 3 2 1 )
1602:x = ( 1 2 3 )
1584:
1580:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1446:
1443:
1416:
1412:
1387:
1382:
1378:
1362:
1361:
1357:
1353:
1349:
1345:
1323:
1318:
1314:
1287:
1282:
1278:
1256:
1253:
1222:
1218:
1214:
1211:
1208:
1205:
1185:
1182:
1179:
1176:
1173:
1164:for all pairs
1151:
1148:
1144:
1140:
1137:
1128:Table showing
1111:
1110:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1035:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
937:
925:
922:
919:
916:
913:
910:
907:
904:
901:
872:
867:
863:
849:
846:
817:
789:
786:
761:
757:
754:
751:
748:
743:
740:
736:
732:
729:
725:
721:
718:
715:
712:
709:
706:
686:
683:
680:
656:
636:
626:if and only if
613:
610:
607:
604:
601:
581:
578:
575:
572:
569:
549:
518:
515:
512:
509:
506:
481:
478:
458:
438:
428:
414:
405:in which case
394:
391:
388:
383:
380:
376:
372:
369:
349:
346:
343:
319:
316:
313:
310:
307:
287:
275:
272:
241:in the group.
230:
208:
205:
201:
197:
194:
191:
188:
156:
151:
148:
144:
140:
137:
134:
131:
111:
83:
63:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6828:
6817:
6814:
6813:
6811:
6800:
6796:
6795:
6790:
6786:
6785:
6781:
6775:
6769:
6765:
6760:
6759:
6755:
6748:
6742:
6739:
6736:
6730:
6727:
6722:
6720:0-387-95385-X
6716:
6712:
6708:
6704:
6700:
6694:
6691:
6686:
6684:0-471-43334-9
6680:
6676:
6672:
6665:
6663:
6659:
6652:
6648:
6645:
6642:
6639:
6636:
6633:
6632:
6628:
6626:
6624:
6620:
6616:
6608:
6606:
6604:
6600:
6596:
6588:
6586:
6584:
6568:
6565:
6556:
6542:
6536:
6530:
6527:
6518:
6505:
6499:
6493:
6482:
6466:
6458:
6438:
6432:
6426:
6420:
6417:
6412:
6409:
6405:
6382:
6379:
6375:
6371:
6368:
6365:
6360:
6357:
6353:
6349:
6346:
6326:
6323:
6320:
6317:
6314:
6311:
6288:
6279:
6273:
6270:
6267:
6261:
6250:
6244:
6241:
6229:
6228:
6227:
6210:
6204:
6201:
6181:
6161:
6153:
6134:
6128:
6117:
6101:
6098:
6095:
6092:
6083:
6070:
6067:
6047:
6027:
6024:
6021:
5998:
5992:
5989:
5969:
5964:
5961:
5957:
5953:
5950:
5947:
5944:
5924:
5921:
5918:
5898:
5878:
5875:
5872:
5852:
5832:
5829:
5826:
5819:
5811:
5809:
5796:
5793:
5773:
5753:
5750:
5730:
5710:
5705:
5701:
5697:
5693:
5686:
5680:
5675:
5666:
5645:
5642:
5619:
5613:
5608:
5583:
5576:but at least
5563:
5543:
5520:
5514:
5509:
5484:
5481:
5461:
5441:
5421:
5418:
5415:
5412:
5409:
5398:
5392:
5364:
5359:
5355:
5334:
5331:
5328:
5325:
5314:
5308:
5280:
5277:
5274:
5263:
5257:
5229:
5226:
5206:
5186:
5181:
5177:
5156:
5136:
5131:
5127:
5106:
5086:
5066:
5061:
5057:
5036:
5016:
4996:
4993:
4990:
4987:
4978:
4965:
4962:
4951:
4945:
4917:
4905:
4899:
4870:
4850:
4843:
4839:
4834:
4828:
4824:
4820:
4808:
4802:
4790:
4785:
4781:
4777:
4769:
4744:
4741:
4738:
4733:
4729:
4725:
4722:
4702:
4695:
4691:
4686:
4663:
4659:
4638:
4635:
4615:
4606:
4602:
4586:
4579:every finite
4578:
4564:
4561:
4558:
4550:
4534:
4514:
4509:
4505:
4484:
4477:
4463:
4450:
4448:
4429:
4415:
4401:
4397:
4388:
4384:
4377:
4372:
4364:
4361:
4357:
4351:
4347:
4343:
4331:
4325:
4313:
4305:
4292:
4273:
4267:
4244:
4239:
4235:
4209:
4205:
4198:
4193:
4168:
4164:
4155:
4151:
4144:
4139:
4131:
4128:
4124:
4118:
4114:
4110:
4102:
4075:
4071:
4061:
4047:
4024:
4018:
4013:
3987:
3980:
3974:
3969:
3961:
3958:
3954:
3946:
3930:
3921:
3919:
3903:
3898:
3895:
3891:
3887:
3884:
3881:
3876:
3873:
3869:
3863:
3860:
3856:
3852:
3849:
3846:
3843:
3838:
3835:
3831:
3825:
3822:
3818:
3814:
3811:
3808:
3805:
3800:
3797:
3789:
3786:
3780:
3777:
3774:
3771:
3766:
3763:
3759:
3755:
3752:
3732:
3709:
3703:
3698:
3673:
3653:
3650:
3647:
3644:
3624:
3604:
3584:
3578:
3572:
3567:
3555:
3551:
3535:
3515:
3512:
3504:
3488:
3476:
3474:
3461:
3458:
3438:
3433:
3430:
3426:
3422:
3419:
3416:
3413:
3410:
3407:
3387:
3384:
3376:
3360:
3351:
3349:
3345:
3341:
3325:
3322:
3302:
3294:
3278:
3273:
3270:
3266:
3262:
3259:
3256:
3253:
3250:
3247:
3227:
3224:
3221:
3218:
3215:
3212:
3200:
3184:
3177:
3169:
3165:
3159:
3155:
3151:
3146:
3142:
3137:
3133:
3129:
3121:
3117:
3111:
3107:
3103:
3098:
3094:
3089:
3084:
3076:
3072:
3066:
3062:
3058:
3053:
3049:
3044:
3038:
3035:
3012:
2992:
2989:
2984:
2980:
2974:
2970:
2964:
2959:
2956:
2953:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2884:
2862:
2858:
2835:
2831:
2810:
2788:
2784:
2780:
2777:
2774:
2769:
2765:
2761:
2756:
2752:
2729:
2725:
2721:
2718:
2710:
2707:
2691:
2670:
2663:
2657:
2652:
2644:
2641:
2637:
2629:
2613:
2610:
2607:
2604:
2601:
2598:
2578:
2570:
2554:
2551:
2548:
2545:
2536:
2533:
2516:
2510:
2505:
2480:
2460:
2437:
2431:
2421:
2405:
2402:
2399:
2391:
2374:
2370:
2349:
2324:
2319:
2316:
2312:
2306:
2302:
2298:
2295:
2291:
2285:
2282:
2278:
2274:
2271:
2267:
2263:
2259:
2253:
2250:
2246:
2242:
2239:
2235:
2230:
2224:
2221:
2217:
2213:
2210:
2206:
2202:
2197:
2193:
2170:
2167:
2163:
2159:
2156:
2153:
2150:
2130:
2125:
2121:
2098:
2094:
2073:
2053:
2045:
2042:
2026:
2018:
2000:
1997:
1993:
1989:
1986:
1983:
1977:
1971:
1951:
1946:
1943:
1939:
1935:
1932:
1929:
1926:
1906:
1898:
1882:
1879:
1876:
1873:
1870:
1862:
1847:
1827:
1824:
1821:
1798:
1792:
1786:
1780:
1777:
1757:
1754:
1751:
1748:
1745:
1725:
1722:
1717:
1714:
1710:
1706:
1703:
1695:
1679:
1671:
1657:
1651:
1645:
1639:
1633:
1630:
1622:
1621:
1617:
1615:
1612:
1609:
1606:
1603:
1600:
1599:a = ( 2 3 )
1597:
1582:
1578:
1568:
1565:
1563:
1559:
1558:
1552:
1539:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1504:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1444:
1441:
1433:
1429:
1414:
1410:
1398:
1385:
1380:
1376:
1367:
1358:
1354:
1350:
1346:
1343:
1342:
1341:
1339:
1335:
1334:
1321:
1316:
1312:
1285:
1280:
1276:
1254:
1251:
1239:
1238:numbered list
1220:
1216:
1212:
1209:
1206:
1203:
1180:
1177:
1174:
1149:
1146:
1142:
1138:
1135:
1126:
1122:
1120:
1116:
1093:
1090:
1087:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1057:
1054:
1051:
1041:of all three
1040:
1036:
1018:
1015:
1012:
1006:
1003:
1000:
997:
994:
991:
988:
982:
979:
976:
973:
970:
967:
964:
958:
955:
952:
941:
938:
920:
917:
914:
908:
905:
902:
891:
890:
889:
887:
883:
870:
865:
861:
847:
845:
843:
839:
833:
831:
815:
807:
787:
784:
776:
759:
755:
752:
749:
746:
741:
738:
734:
730:
727:
723:
719:
713:
707:
704:
684:
681:
678:
670:
654:
634:
627:
608:
602:
599:
576:
570:
567:
547:
538:
536:
532:
513:
507:
504:
497:
492:
479:
476:
456:
436:
426:
412:
392:
389:
386:
381:
378:
374:
370:
367:
347:
344:
341:
333:
317:
314:
311:
308:
305:
285:
273:
271:
269:
265:
261:
259:
258:singleton set
255:
251:
250:abelian group
247:
242:
228:
206:
203:
199:
195:
192:
189:
186:
178:
174:
170:
154:
149:
146:
142:
138:
135:
132:
129:
109:
101:
97:
81:
61:
53:
50:, especially
49:
41:
37:
36:Cayley graphs
32:
19:
6816:Group theory
6792:
6763:
6741:
6729:
6702:
6693:
6670:
6615:finite group
6612:
6592:
6557:
6519:
6303:
6084:
5815:
5219:is of order
5119:is of order
4979:
4883:must divide
4607:
4549:prime number
4454:
4416:
4290:
4062:
3922:
3503:finite group
3480:
3352:
3293:group action
3204:
2532:denotes the
2418:lies in the
2017:automorphism
1860:is abelian).
1613:
1610:
1607:
1604:
1601:
1598:
1569:
1566:
1555:
1553:
1401:
1399:
1363:
1302:
1300:
1112:
886:permutations
851:
834:
804:class number
801:
774:
539:
493:
331:
277:
262:
243:
176:
99:
52:group theory
45:
6699:Lang, Serge
3554:centralizer
3400:by writing
3348:centralizer
2535:centralizer
2392:An element
2143:(Proof: if
940:Transposing
427:a conjugate
175:are called
167:This is an
48:mathematics
6756:References
6603:free loops
6583:isomorphic
6152:normalizer
6040:such that
5937:such that
5723:therefore
5658:therefore
5497:Note that
3344:stabilizer
2591:such that
2567:i.e., the
2039:called an
1618:Properties
1338:cycle type
1115:isometries
892:No change
842:cycle type
624:are equal
425:is called
360:such that
274:Definition
6799:EMS Press
6621:over the
6520:By using
6494:
6427:
6421:∈
6410:−
6380:−
6358:−
6321:∈
6245:
6205:
6129:
6096:⊆
6025:⊆
5993:
5962:−
5922:∈
5876:⊆
5830:⊆
5681:
5614:
5536:includes
5515:
5393:
5309:
5258:
4946:
4900:
4825:∑
4803:
4378:
4348:∑
4326:
4268:
4199:
4145:
4115:∑
4019:
3975:
3896:−
3874:−
3861:−
3836:−
3823:−
3798:−
3764:−
3704:
3666:for some
3573:
3431:−
3411:⋅
3271:−
3251:⋅
3222:∈
3134:⋯
3013:σ
2950:∑
2941:(so that
2923:…
2897:for each
2885:σ
2811:σ
2778:…
2722:∈
2719:σ
2658:
2626:then the
2549:∈
2511:
2432:
2403:∈
2317:−
2283:−
2264:⋯
2251:−
2222:−
2168:−
1998:−
1972:φ
1944:−
1880:∈
1825:∈
1781:
1755:∈
1715:−
1634:
1608:Then xax
1528:…
1480:…
1236:(compare
1213:∈
1147:−
1085:→
1061:→
1010:→
986:→
962:→
912:→
753:∈
739:−
708:
682:∈
603:
571:
508:
379:−
345:∈
332:conjugate
315:∈
264:Functions
204:−
147:−
100:conjugate
6810:Category
6711:Springer
6701:(2002).
6641:FC-group
6629:See also
2744:and let
2704:(by the
2569:subgroup
1814:for all
1738:for all
1570:Let G =
1567:Example
848:Examples
669:disjoint
6801:, 2001
6703:Algebra
6613:In any
4601:trivial
4451:Example
3943:is the
3552:of the
3375:subsets
1770:, i.e.
1694:abelian
6770:
6717:
6681:
5818:subset
4715:where
4603:center
4527:where
4476:-group
4181:where
3550:cosets
3340:orbits
2420:center
2015:is an
1503:cycles
1360:table.
1356:table.
1348:table.
1117:of an
171:whose
6747:p. 57
6735:p. 56
6653:Notes
6597:of a
6481:coset
6339:then
6174:) in
6150:(the
6116:index
5293:then
5149:then
5009:then
4547:is a
3945:index
3501:is a
2711:Take
2628:index
2185:then
1897:order
1696:then
1501:into
1196:with
830:order
96:group
94:of a
6768:ISBN
6715:ISBN
6679:ISBN
6114:the
5982:Let
5416:>
5332:>
5275:>
4963:>
4739:<
4726:<
4562:>
4551:and
3617:and
3338:The
3240:let
2113:and
2066:and
1364:The
1301:The
942:two
800:The
647:and
592:and
449:and
330:are
278:Let
98:are
74:and
34:Two
6483:of
6154:of
6118:of
5454:of
5347:or
5099:of
5049:or
4930:so
4040:in
3745::
3481:If
3377:of
3315:on
3295:of
3025:is:
2877:in
2538:of
2453:of
2046:If
2019:of
1692:is
1672:If
1434:of
1244:of
808:of
777:of
697:is
529:of
429:of
260:).
254:set
46:In
38:of
6812::
6797:,
6791:,
6713:.
6709:.
6705:.
6677:.
6661:^
6242:Cl
6226::
6202:Cl
5990:Cl
4966:1.
4605:.
4293::
3417::=
3350:.
3257::=
2708:).
2389:).
1778:Cl
1631:Cl
1564:.
1121:.
1037:A
844:.
832:.
705:Cl
600:Cl
568:Cl
537:.
505:GL
270:.
6776:.
6723:.
6687:.
6569:.
6566:G
6543:,
6540:}
6537:a
6534:{
6531:=
6528:S
6506:.
6503:)
6500:S
6497:(
6491:N
6467:h
6459:g
6439:,
6436:)
6433:S
6430:(
6424:N
6418:h
6413:1
6406:g
6383:1
6376:h
6372:S
6369:h
6366:=
6361:1
6354:g
6350:S
6347:g
6327:,
6324:G
6318:h
6315:,
6312:g
6289:.
6286:]
6283:)
6280:S
6277:(
6274:N
6271::
6268:G
6265:[
6262:=
6258:|
6254:)
6251:S
6248:(
6238:|
6214:)
6211:S
6208:(
6182:G
6162:S
6138:)
6135:S
6132:(
6126:N
6102:,
6099:G
6093:S
6071:.
6068:S
6048:T
6028:G
6022:T
6002:)
5999:S
5996:(
5970:.
5965:1
5958:g
5954:S
5951:g
5948:=
5945:T
5925:G
5919:g
5899:S
5879:G
5873:T
5853:S
5845:(
5833:G
5827:S
5797:.
5794:p
5774:G
5754:,
5751:G
5731:b
5711:,
5706:2
5702:p
5698:=
5694:|
5690:)
5687:b
5684:(
5676:G
5672:C
5667:|
5646:,
5643:p
5623:)
5620:b
5617:(
5609:G
5605:C
5584:p
5564:b
5544:b
5524:)
5521:b
5518:(
5510:G
5506:C
5485:.
5482:G
5462:G
5442:b
5422:,
5419:1
5413:p
5410:=
5406:|
5402:)
5399:G
5396:(
5390:Z
5386:|
5365:.
5360:2
5356:p
5335:1
5329:p
5326:=
5322:|
5318:)
5315:G
5312:(
5306:Z
5302:|
5281:,
5278:1
5271:|
5267:)
5264:G
5261:(
5255:Z
5251:|
5230:,
5227:p
5207:G
5187:,
5182:2
5178:p
5157:G
5137:,
5132:2
5128:p
5107:G
5087:a
5067:.
5062:2
5058:p
5037:p
5017:G
4997:,
4994:2
4991:=
4988:n
4959:|
4955:)
4952:G
4949:(
4943:Z
4939:|
4918:,
4914:|
4909:)
4906:G
4903:(
4897:Z
4892:|
4871:p
4851:.
4844:i
4840:k
4835:p
4829:i
4821:+
4817:|
4812:)
4809:G
4806:(
4800:Z
4795:|
4791:=
4786:n
4782:p
4778:=
4774:|
4770:G
4766:|
4745:.
4742:n
4734:i
4730:k
4723:0
4703:,
4696:i
4692:k
4687:p
4664:i
4660:H
4639:,
4636:G
4616:G
4587:p
4565:0
4559:n
4535:p
4515:,
4510:n
4506:p
4485:G
4464:p
4434:|
4430:G
4426:|
4402:,
4398:]
4394:)
4389:i
4385:x
4381:(
4373:G
4369:C
4365::
4362:G
4358:[
4352:i
4344:+
4340:|
4335:)
4332:G
4329:(
4323:Z
4318:|
4314:=
4310:|
4306:G
4302:|
4277:)
4274:G
4271:(
4265:Z
4245:.
4240:i
4236:x
4215:)
4210:i
4206:x
4202:(
4194:G
4190:C
4169:,
4165:]
4161:)
4156:i
4152:x
4148:(
4140:G
4136:C
4132::
4129:G
4125:[
4119:i
4111:=
4107:|
4103:G
4099:|
4076:i
4072:x
4048:G
4028:)
4025:a
4022:(
4014:G
4010:C
3988:]
3984:)
3981:a
3978:(
3970:G
3966:C
3962::
3959:G
3955:[
3931:a
3904:.
3899:1
3892:c
3888:a
3885:c
3882:=
3877:1
3870:c
3864:1
3857:z
3853:z
3850:a
3847:c
3844:=
3839:1
3832:c
3826:1
3819:z
3815:a
3812:z
3809:c
3806:=
3801:1
3794:)
3790:z
3787:c
3784:(
3781:a
3778:z
3775:c
3772:=
3767:1
3760:b
3756:a
3753:b
3733:a
3713:)
3710:a
3707:(
3699:G
3695:C
3674:z
3654:z
3651:c
3648:=
3645:b
3625:c
3605:b
3585:.
3582:)
3579:a
3576:(
3568:G
3564:C
3536:a
3516:,
3513:a
3489:G
3462:.
3459:G
3439:,
3434:1
3427:g
3423:S
3420:g
3414:S
3408:g
3388:,
3385:G
3361:G
3326:.
3323:G
3303:G
3279:.
3274:1
3267:g
3263:x
3260:g
3254:x
3248:g
3228:,
3225:G
3219:x
3216:,
3213:g
3185:.
3178:)
3170:s
3166:k
3160:s
3156:m
3152:!
3147:s
3143:k
3138:(
3130:)
3122:2
3118:k
3112:2
3108:m
3104:!
3099:2
3095:k
3090:(
3085:)
3077:1
3073:k
3067:1
3063:m
3059:!
3054:1
3050:k
3045:(
3039:!
3036:n
2993:n
2990:=
2985:i
2981:m
2975:i
2971:k
2965:s
2960:1
2957:=
2954:i
2929:s
2926:,
2920:,
2917:2
2914:,
2911:1
2908:=
2905:i
2863:i
2859:m
2836:i
2832:k
2789:s
2785:m
2781:,
2775:,
2770:2
2766:m
2762:,
2757:1
2753:m
2730:n
2726:S
2692:a
2671:]
2667:)
2664:a
2661:(
2653:G
2649:C
2645::
2642:G
2638:[
2614:,
2611:g
2608:a
2605:=
2602:a
2599:g
2579:g
2555:,
2552:G
2546:a
2520:)
2517:a
2514:(
2506:G
2502:C
2481:a
2461:G
2441:)
2438:G
2435:(
2429:Z
2406:G
2400:a
2375:k
2371:a
2350:a
2339:k
2325:.
2320:1
2313:g
2307:k
2303:b
2299:g
2296:=
2292:)
2286:1
2279:g
2275:b
2272:g
2268:(
2260:)
2254:1
2247:g
2243:b
2240:g
2236:(
2231:)
2225:1
2218:g
2214:b
2211:g
2207:(
2203:=
2198:k
2194:a
2171:1
2164:g
2160:b
2157:g
2154:=
2151:a
2131:.
2126:k
2122:b
2099:k
2095:a
2074:b
2054:a
2027:G
2001:1
1994:g
1990:x
1987:g
1984:=
1981:)
1978:x
1975:(
1952:,
1947:1
1940:g
1936:a
1933:g
1930:=
1927:b
1907:a
1883:G
1877:b
1874:,
1871:a
1848:G
1828:G
1822:a
1802:}
1799:a
1796:{
1793:=
1790:)
1787:a
1784:(
1758:G
1752:g
1749:,
1746:a
1726:a
1723:=
1718:1
1711:g
1707:a
1704:g
1680:G
1658:.
1655:}
1652:e
1649:{
1646:=
1643:)
1640:e
1637:(
1583:3
1579:S
1540:.
1537:}
1534:n
1531:,
1525:,
1522:2
1519:,
1516:1
1513:{
1489:}
1486:n
1483:,
1477:,
1474:2
1471:,
1468:1
1465:{
1445:.
1442:n
1415:n
1411:S
1386:.
1381:4
1377:S
1322:,
1317:4
1313:S
1286:.
1281:4
1277:S
1255:,
1252:a
1240:)
1221:4
1217:S
1210:b
1207:,
1204:a
1184:)
1181:b
1178:,
1175:a
1172:(
1150:1
1143:b
1139:a
1136:b
1097:)
1094:b
1091:a
1088:c
1082:c
1079:b
1076:a
1073:,
1070:a
1067:c
1064:b
1058:c
1055:b
1052:a
1049:(
1022:)
1019:a
1016:b
1013:c
1007:c
1004:b
1001:a
998:,
995:c
992:a
989:b
983:c
980:b
977:a
974:,
971:b
968:c
965:a
959:c
956:b
953:a
950:(
924:)
921:c
918:b
915:a
909:c
906:b
903:a
900:(
871:,
866:3
862:S
816:G
788:.
785:a
760:}
756:G
750:g
747::
742:1
735:g
731:a
728:g
724:{
720:=
717:)
714:a
711:(
685:G
679:a
655:b
635:a
612:)
609:b
606:(
580:)
577:a
574:(
548:G
517:)
514:n
511:(
480:.
477:b
457:a
437:a
413:b
393:,
390:b
387:=
382:1
375:g
371:a
368:g
348:G
342:g
318:G
312:b
309:,
306:a
286:G
229:g
207:1
200:g
196:a
193:g
190:=
187:b
155:.
150:1
143:g
139:a
136:g
133:=
130:b
110:g
82:b
62:a
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
