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