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
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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
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214:
182:
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1022:
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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,
1369:
1302:
715:
and the latter property is preserved by every automorphism; finally every real number must be fixed since it is the
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1319:
305:. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the
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273:
17:
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1406:
152:, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism
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831:
1484:
810:
1075:—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.
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must be fixed, as well as the additive inverses of these sums (that is, the automorphism fixes all
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has no other automorphism than the identity, since an automorphism must fix the
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has a unique nontrivial automorphism that fixes the real numbers. It is the
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are the conjugations by the elements of the group itself. For each element
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513:. When the vector space is finite-dimensional, the automorphism group of
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352:. This results straightforwardly from the definition of a category.
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is a new fifth root of unity, connected with the former fifth root
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940:
928:), from a surface to itself. For example, the automorphisms of the
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891:
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339:
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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
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for a morphism to be bijective to be an isomorphism.
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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
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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
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527:all endomorphisms of
419:elementary arithmetic
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33:
1327:Mathematics Magazine
1028:
1017:{\displaystyle \mu }
1008:
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756:
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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:
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705:
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345:automorphism group
322:Automorphism group
316:Automorphism group
287:
260:
206:
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135:group homomorphism
92:automorphism group
56:
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
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1225:Antiautomorphism
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1043:
1041:
1040:
1035:
1023:
1021:
1020:
1015:
998:icosian calculus
996:in 1856, in his
918:Riemann surfaces
844:. This group is
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804:uncountably many
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381:, or simply Aut(
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1339:10.2307/2689301
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1170:
1157:Goursat's lemma
1145:normal subgroup
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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
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746:complex numbers
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500:linear operator
398:, an arbitrary
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143:linear operator
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1456:"Automorphism"
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1438:
1437:External links
1435:
1433:
1432:
1394:
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1370:
1352:
1333:(3): 135–141.
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1168:quotient group
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1013:
989:
986:
985:
984:
983:
982:
979:not sufficient
959:
948:diffeomorphism
937:
930:Riemann sphere
914:
911:isometry group
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926:conformal map
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815:Galois theory
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752:, which maps
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53:
52:Cayley tables
49:
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44:Cayley graphs
41:
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19:
18:Automorphisms
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1444:Automorphism
1443:
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1413:
1397:
1384:
1379:
1361:
1355:
1330:
1326:
1313:
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1284:
1260:order theory
1230:Automorphism
1213:Lie algebras
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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
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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:)
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