31:
429:, considered as a group under addition, has a unique nontrivial automorphism: negation. Considered as a ring, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any
268:
713:
295:
638:
has no automorphisms other than the identity. Indeed, the rational numbers must be fixed by every automorphism, per above; an automorphism must preserve inequalities since
742:
632:
581:
1042:
214:
182:
662:
1022:
796:
770:
887:
is a permutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by an edge, so are their images under the permutation.
992:
One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician
1423:
440:
from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group
609:); finally, since every rational number is the quotient of two integers, all rational numbers must be fixed by any automorphism.
219:
133:
of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example,
17:
1369:
1302:
715:
and the latter property is preserved by every automorphism; finally every real number must be fixed since it is the
845:
1319:
305:. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the
1229:
966:
667:
273:
1270:
1406:
152:, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism
1265:
831:
1484:
810:
1075:—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.
1058:
605:
must be fixed, as well as the additive inverses of these sums (that is, the automorphism fixes all
933:
725:
615:
564:
1235:
864:
595:
1414:
1402:
1245:
993:
149:
1290:
1027:
187:
155:
499:
469:
418:
641:
1442:
1200:
526:
518:
335:
1007:
8:
1064:
974:
775:
749:
555:
453:
331:
111:
107:
87:
71:
1479:
1342:
1255:
1208:
1196:
1163:
1156:
1079:
1068:
1054:
884:
755:
461:
344:
321:
134:
115:
91:
1489:
1452:
1365:
1298:
1240:
895:
716:
588:
551:
437:
306:
298:
138:
83:
79:
1334:
1224:
997:
870:
465:
39:
1455:
1144:
917:
902:
799:
584:
142:
587:
has no other automorphism than the identity, since an automorphism must fix the
1193:
1167:
947:
929:
910:
745:
537:
488:
95:
47:
1473:
970:
925:
814:
481:
430:
748:
has a unique nontrivial automorphism that fixes the real numbers. It is the
1259:
1082:
are the conjugations by the elements of the group itself. For each element
921:
880:
492:
130:
119:
51:
43:
1212:
1072:
803:
635:
513:. When the vector space is finite-dimensional, the automorphism group of
399:
67:
59:
1346:
823:
395:
1460:
867:
856:
548:
127:
30:
1338:
352:. This results straightforwardly from the definition of a category.
1388:
1250:
1024:
is a new fifth root of unity, connected with the former fifth root
962:
940:
928:), from a surface to itself. For example, the automorphisms of the
906:
891:
606:
339:
75:
422:
82:
the object to itself while preserving all of its structure. The
1186:
849:
1364:(2nd ed.), Cambridge University Press, pp. 22–23,
954:
to itself. The automorphism group is sometimes denoted Diff(
1401:
1297:(Felix Pahl translation ed.). Springer. p. 376.
1139:; usage varies). One can easily check that conjugation by
1000:, where he discovered an order two automorphism, writing:
484:
center it can be embedded into its own automorphism group.
263:{\displaystyle g\circ f=f\circ g=\operatorname {id} _{X},}
1143:
is a group automorphism. The inner automorphisms form a
1407:"Memorandum respecting a new System of Roots of Unity"
509:. An automorphism is an invertible linear operator on
1450:
1295:
Mathematical foundations of computational engineering
1030:
1010:
778:
758:
728:
670:
644:
618:
567:
276:
222:
190:
158:
981:
for a morphism to be bijective to be an isomorphism.
898:
of the space. Specialized terminology is also used:
973:of the space to itself, or self-homeomorphism (see
1288:
1036:
1016:
969:, and an automorphism of a topological space is a
965:, morphisms between topological spaces are called
790:
764:
736:
707:
656:
626:
575:
289:
262:
208:
176:
532:is itself an algebra over the same base field as
1471:
1048:
852:, the group of rotations in 3-dimensional space.
830:) as a ring are the inner automorphisms, by the
806:automorphisms that do not fix the real numbers.
406:is an automorphism. The automorphism group of
813:is the starting point and the main object of
909:. The automorphism group is also called the
184:is an automorphism if there is a morphism
86:of all automorphisms of an object forms a
730:
620:
569:
385:) if the category is clear from context.
1359:
708:{\displaystyle \exists z\mid y-x=z^{2},}
29:
290:{\displaystyle \operatorname {id} _{X}}
14:
1472:
1320:"Automorphisms of the Complex Numbers"
1215:the definition is slightly different.
1189:that contain the outer automorphisms.
444:there is a natural group homomorphism
410:is also called the symmetric group on
148:More generally, for an object in some
1451:
315:
1317:
1185:); the non-trivial elements are the
1044:by relations of perfect reciprocity.
939:An automorphism of a differentiable
355:The automorphism group of an object
1282:
1162:The other automorphisms are called
74:to itself. It is, in some sense, a
24:
894:, an automorphism may be called a
719:of a sequence of rational numbers.
671:
27:Isomorphism of an object to itself
25:
1501:
1436:
1192:The same definition holds in any
1429:from the original on 2022-10-09.
1266:Relation-preserving automorphism
601:; the sum of a finite number of
525:). (The algebraic structure of
1446:at Encyclopaedia of Mathematics
326:The automorphisms of an object
94:. It is, loosely speaking, the
42:shown as a mapping between two
1395:
1377:
1353:
1311:
855:The automorphism group of the
822:The automorphism group of the
809:The study of automorphisms of
200:
168:
13:
1:
1362:Clifford Algebras and Spinors
1276:
1049:Inner and outer automorphisms
433:, but not of a ring or field.
101:
1271:Fractional Fourier transform
737:{\displaystyle \mathbb {C} }
627:{\displaystyle \mathbb {R} }
576:{\displaystyle \mathbb {Q} }
78:of the object, and a way of
50:, and a mapping between two
7:
1289:PJ Pahl, R Damrath (2001).
1218:
1078:In the case of groups, the
1063:In some categories—notably
388:
10:
1506:
1403:Sir William Rowan Hamilton
1318:Yale, Paul B. (May 1966).
1181:is usually denoted by Out(
1052:
987:
905:an automorphism is a self-
811:algebraic field extensions
547:A field automorphism is a
436:A group automorphism is a
319:
309:, and is often called the
1360:Lounesto, Pertti (2001),
977:). In this example it is
802:implies the existence of
402:of the elements of a set
1059:Outer automorphism group
1037:{\displaystyle \lambda }
540:precisely consist of GL(
209:{\displaystyle g:X\to X}
177:{\displaystyle f:X\to X}
1236:Characteristic subgroup
920:, an automorphism is a
885:automorphism of a graph
596:multiplicative identity
491:, an endomorphism of a
1415:Philosophical Magazine
1291:"§7.5.5 Automorphisms"
1246:Frobenius automorphism
1046:
1038:
1018:
994:William Rowan Hamilton
934:Möbius transformations
832:Skolem–Noether theorem
792:
766:
738:
709:
658:
657:{\displaystyle x<y}
628:
577:
342:, which is called the
291:
264:
210:
178:
55:
1039:
1019:
1002:
793:
767:
739:
710:
659:
629:
578:
527:all endomorphisms of
419:elementary arithmetic
292:
265:
211:
179:
33:
1327:Mathematics Magazine
1028:
1017:{\displaystyle \mu }
1008:
776:
756:
726:
668:
642:
616:
565:
519:general linear group
311:trivial automorphism
274:
220:
188:
156:
1391:, 2003, p. 453
1385:Handbook of algebra
1232:(in Sudoku puzzles)
1164:outer automorphisms
1080:inner automorphisms
975:homeomorphism group
924:map (also called a
916:In the category of
834:: maps of the form
791:{\displaystyle -i.}
750:complex conjugation
538:invertible elements
517:is the same as the
462:inner automorphisms
108:algebraic structure
72:mathematical object
46:, a permutation in
1453:Weisstein, Eric W.
1256:Order automorphism
1209:invertible element
1155:); this is called
1151:), denoted by Inn(
1055:Inner automorphism
1034:
1014:
788:
762:
734:
705:
654:
624:
573:
345:automorphism group
322:Automorphism group
316:Automorphism group
287:
260:
206:
174:
135:group homomorphism
92:automorphism group
56:
18:Field automorphism
1241:Endomorphism ring
1094:is the operation
1090:, conjugation by
765:{\displaystyle i}
717:least upper bound
664:is equivalent to
589:additive identity
552:ring homomorphism
456:is the group Inn(
438:group isomorphism
367:is often denoted
307:identity function
299:identity morphism
139:ring homomorphism
16:(Redirected from
1497:
1485:Abstract algebra
1466:
1465:
1431:
1430:
1428:
1411:
1399:
1393:
1392:
1381:
1375:
1374:
1357:
1351:
1350:
1324:
1315:
1309:
1308:
1286:
1225:Antiautomorphism
1180:
1131:
1112:
1043:
1041:
1040:
1035:
1023:
1021:
1020:
1015:
998:icosian calculus
996:in 1856, in his
918:Riemann surfaces
844:. This group is
843:
804:uncountably many
797:
795:
794:
789:
771:
769:
768:
763:
743:
741:
740:
735:
733:
714:
712:
711:
706:
701:
700:
663:
661:
660:
655:
633:
631:
630:
625:
623:
604:
600:
593:
585:rational numbers
582:
580:
579:
574:
572:
381:, or simply Aut(
380:
366:
360:
351:
329:
304:
296:
294:
293:
288:
286:
285:
269:
267:
266:
261:
256:
255:
215:
213:
212:
207:
183:
181:
180:
175:
40:Klein four-group
21:
1505:
1504:
1500:
1499:
1498:
1496:
1495:
1494:
1470:
1469:
1439:
1434:
1426:
1409:
1400:
1396:
1387:, vol. 3,
1383:
1382:
1378:
1372:
1358:
1354:
1339:10.2307/2689301
1322:
1316:
1312:
1305:
1287:
1283:
1279:
1221:
1170:
1157:Goursat's lemma
1145:normal subgroup
1122:
1114:
1103:
1095:
1061:
1053:Main articles:
1051:
1029:
1026:
1025:
1009:
1006:
1005:
990:
967:continuous maps
903:metric geometry
874:
835:
800:axiom of choice
777:
774:
773:
757:
754:
753:
746:complex numbers
729:
727:
724:
723:
696:
692:
669:
666:
665:
643:
640:
639:
619:
617:
614:
613:
602:
598:
591:
568:
566:
563:
562:
500:linear operator
398:, an arbitrary
391:
374:
368:
362:
356:
349:
327:
324:
318:
302:
281:
277:
275:
272:
271:
251:
247:
221:
218:
217:
189:
186:
185:
157:
154:
153:
143:linear operator
104:
98:of the object.
28:
23:
22:
15:
12:
11:
5:
1503:
1493:
1492:
1487:
1482:
1468:
1467:
1456:"Automorphism"
1448:
1438:
1437:External links
1435:
1433:
1432:
1394:
1376:
1370:
1352:
1333:(3): 135–141.
1310:
1303:
1280:
1278:
1275:
1274:
1273:
1268:
1263:
1253:
1248:
1243:
1238:
1233:
1227:
1220:
1217:
1168:quotient group
1118:
1099:
1050:
1047:
1033:
1013:
989:
986:
985:
984:
983:
982:
979:not sufficient
959:
948:diffeomorphism
937:
930:Riemann sphere
914:
911:isometry group
888:
877:
872:
853:
820:
819:
818:
807:
787:
784:
781:
761:
732:
720:
704:
699:
695:
691:
688:
685:
682:
679:
676:
673:
653:
650:
647:
622:
610:
571:
545:
489:linear algebra
485:
434:
415:
390:
387:
370:
361:in a category
320:Main article:
317:
314:
284:
280:
259:
254:
250:
246:
243:
240:
237:
234:
231:
228:
225:
205:
202:
199:
196:
193:
173:
170:
167:
164:
161:
103:
100:
96:symmetry group
48:cycle notation
26:
9:
6:
4:
3:
2:
1502:
1491:
1488:
1486:
1483:
1481:
1478:
1477:
1475:
1463:
1462:
1457:
1454:
1449:
1447:
1445:
1441:
1440:
1425:
1421:
1417:
1416:
1408:
1404:
1398:
1390:
1386:
1380:
1373:
1371:0-521-00551-5
1367:
1363:
1356:
1348:
1344:
1340:
1336:
1332:
1328:
1321:
1314:
1306:
1304:3-540-67995-2
1300:
1296:
1292:
1285:
1281:
1272:
1269:
1267:
1264:
1261:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1237:
1234:
1231:
1228:
1226:
1223:
1222:
1216:
1214:
1210:
1206:
1202:
1198:
1195:
1190:
1188:
1184:
1178:
1174:
1169:
1165:
1160:
1158:
1154:
1150:
1146:
1142:
1138:
1135:
1130:
1126:
1121:
1117:
1111:
1107:
1102:
1098:
1093:
1089:
1085:
1081:
1076:
1074:
1070:
1066:
1060:
1056:
1045:
1031:
1011:
1001:
999:
995:
980:
976:
972:
971:homeomorphism
968:
964:
960:
957:
953:
949:
945:
942:
938:
935:
931:
927:
926:conformal map
923:
922:biholomorphic
919:
915:
912:
908:
904:
900:
899:
897:
893:
889:
886:
882:
878:
875:
869:
866:
862:
858:
854:
851:
847:
842:
838:
833:
829:
825:
821:
816:
815:Galois theory
812:
808:
805:
801:
785:
782:
779:
759:
752:, which maps
751:
747:
721:
718:
702:
697:
693:
689:
686:
683:
680:
677:
674:
651:
648:
645:
637:
611:
608:
597:
590:
586:
560:
559:
557:
553:
550:
546:
543:
539:
535:
531:
530:
524:
520:
516:
512:
508:
504:
501:
497:
494:
490:
486:
483:
479:
475:
471:
467:
463:
459:
455:
451:
447:
443:
439:
435:
432:
431:abelian group
428:
424:
421:, the set of
420:
416:
413:
409:
405:
401:
397:
393:
392:
386:
384:
378:
373:
365:
359:
353:
347:
346:
341:
337:
333:
323:
313:
312:
308:
300:
282:
278:
257:
252:
248:
244:
241:
238:
235:
232:
229:
226:
223:
203:
197:
194:
191:
171:
165:
162:
159:
151:
146:
144:
140:
136:
132:
129:
125:
121:
117:
113:
109:
99:
97:
93:
90:, called the
89:
85:
81:
77:
73:
69:
65:
61:
53:
52:Cayley tables
49:
45:
44:Cayley graphs
41:
37:
32:
19:
1459:
1444:Automorphism
1443:
1419:
1413:
1397:
1384:
1379:
1361:
1355:
1330:
1326:
1313:
1294:
1284:
1260:order theory
1230:Automorphism
1213:Lie algebras
1204:
1191:
1182:
1176:
1172:
1161:
1152:
1148:
1140:
1136:
1133:
1128:
1124:
1119:
1115:
1109:
1105:
1100:
1096:
1091:
1087:
1083:
1077:
1073:Lie algebras
1062:
1003:
991:
978:
955:
951:
943:
881:graph theory
860:
840:
836:
827:
636:real numbers
541:
533:
528:
522:
514:
510:
506:
502:
495:
493:vector space
477:
473:
457:
449:
445:
441:
426:
411:
407:
403:
382:
376:
371:
363:
357:
354:
343:
325:
310:
147:
131:homomorphism
126:is simply a
124:automorphism
123:
120:vector space
105:
64:automorphism
63:
57:
36:automorphism
35:
1086:of a group
865:exceptional
824:quaternions
558:to itself.
476:. Thus, if
400:permutation
336:composition
68:isomorphism
60:mathematics
1474:Categories
1277:References
846:isomorphic
722:The field
612:The field
561:The field
464:and whose
396:set theory
216:such that
110:such as a
102:Definition
1480:Morphisms
1461:MathWorld
1113:given by
1032:λ
1012:μ
868:Lie group
863:) is the
857:octonions
780:−
684:−
678:∣
672:∃
549:bijective
340:morphisms
239:∘
227:∘
201:→
169:→
128:bijective
1490:Symmetry
1424:Archived
1405:(1856).
1389:Elsevier
1251:Morphism
1219:See also
1175:) / Inn(
1104: :
1004:so that
963:topology
941:manifold
907:isometry
892:geometry
607:integers
594:and the
536:, whose
452:) whose
423:integers
389:Examples
150:category
76:symmetry
1422:: 446.
1347:2689301
1207:is any
1201:algebra
1147:of Aut(
988:History
744:of the
634:of the
583:of the
554:from a
482:trivial
468:is the
330:form a
297:is the
80:mapping
70:from a
38:of the
1368:
1345:
1301:
1211:. For
1203:where
1194:unital
1187:cosets
1166:. The
1071:, and
1065:groups
896:motion
470:center
466:kernel
448:→ Aut(
334:under
270:where
141:, and
106:In an
66:is an
1427:(PDF)
1410:(PDF)
1343:JSTOR
1323:(PDF)
1069:rings
950:from
946:is a
850:SO(3)
556:field
521:, GL(
498:is a
460:) of
454:image
332:group
122:, an
118:, or
112:group
88:group
62:, an
1366:ISBN
1299:ISBN
1258:(in
1197:ring
1171:Aut(
1132:(or
1127:) =
1057:and
932:are
798:The
649:<
480:has
145:.)
116:ring
114:, a
1335:doi
1199:or
1129:aga
961:In
901:In
890:In
883:an
879:In
848:to
841:bab
772:to
544:).)
487:In
472:of
417:In
394:In
369:Aut
348:of
338:of
301:of
84:set
58:In
34:An
1476::
1458:.
1420:12
1418:.
1412:.
1341:.
1331:39
1329:.
1325:.
1293:.
1262:).
1159:.
1137:ga
1108:→
1067:,
958:).
839:↦
505:→
425:,
279:id
249:id
137:,
1464:.
1349:.
1337::
1307:.
1205:a
1183:G
1179:)
1177:G
1173:G
1153:G
1149:G
1141:a
1134:a
1125:g
1123:(
1120:a
1116:φ
1110:G
1106:G
1101:a
1097:φ
1092:a
1088:G
1084:a
956:M
952:M
944:M
936:.
913:.
876:.
873:2
871:G
861:O
859:(
837:a
828:H
826:(
817:.
786:.
783:i
760:i
731:C
703:,
698:2
694:z
690:=
687:x
681:y
675:z
652:y
646:x
621:R
603:1
599:1
592:0
570:Q
542:V
534:V
529:V
523:V
515:V
511:V
507:V
503:V
496:V
478:G
474:G
458:G
450:G
446:G
442:G
427:Z
414:.
412:X
408:X
404:X
383:X
379:)
377:X
375:(
372:C
364:C
358:X
350:X
328:X
303:X
283:X
258:,
253:X
245:=
242:g
236:f
233:=
230:f
224:g
204:X
198:X
195::
192:g
172:X
166:X
163::
160:f
54:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.