Editing Automorphism

{{Short description|Isomorphism of an object to itself}}
[[File:Klein-automorphism.svg|thumb|right|400px|An [[w:Automorphism|automorphism]] of the [[w:Klein_four-group|Klein four-group]] shown as a mapping between two [[w:Cayley_graph|Cayley graphs]], a permutation in [[w:Cycle_notation|cycle notation]], and a mapping between two [[w:Cayley_table|Cayley tables]].]]

In [[mathematics]], an '''automorphism''' is an [[isomorphism]] from a [[mathematical object]] to itself. It is, in some sense, a [[symmetry]] of the object, and a way of [[map (mathematics)|mapping]] the object to itself while preserving all of its structure. The [[Set (mathematics)|set]] of all automorphisms of an object forms a [[group (mathematics)|group]], called the [[automorphism group]]. It is, loosely speaking, the [[symmetry group]] of the object.

==Definition==
In an [[algebraic structure]] such as a [[group (mathematics)|group]], a [[ring (mathematics)|ring]], or [[vector space]], an ''automorphism'' is simply a [[bijective]] [[homomorphism]] of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, [[group homomorphism]], [[ring homomorphism]], and [[linear operator]].) 

More generally, for an object in some [[category (mathematics)|category]], an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism <math>f: X\to X</math> is an automorphism if there is a morphism <math>g: X\to X</math> such that <math>g\circ f= f\circ g = \operatorname {id}_X,</math> where <math>\operatorname {id}_X</math> is the [[identity morphism]]  of {{mvar|X}}. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the [[identity function]], and is often called the ''trivial automorphism''.

==Automorphism group==
{{main|Automorphism group}}
The automorphisms of an object {{mvar|X}} form a [[group (mathematics)|group]] under [[Function composition|composition]] of [[morphism]]s, which is called the ''[[automorphism group]]'' of {{mvar|X}}. This results straightforwardly from the definition of a category.

The automorphism group of an object {{math|''X''}} in a category {{math|''C''}} is often denoted {{math|Aut<sub>''C''</sub>(''X'')}}{{math|}}, or simply Aut(''X'') if the category is clear from context.

==Examples==
* In [[set theory]], an arbitrary [[permutation]] of the elements of a set ''X'' is an automorphism. The automorphism group of ''X'' is also called the symmetric group on ''X''.
* In [[elementary arithmetic]], the set of [[integer]]s, {{tmath|\Z}}, 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 [[abelian group]], but not of a ring or field.
* A group automorphism is a [[group isomorphism]] from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group ''G'' there is a natural group homomorphism ''G'' → Aut(''G'') whose [[image (mathematics)|image]] is the group Inn(''G'') of [[inner automorphism]]s and whose [[kernel (algebra)|kernel]] is the [[center (group theory)|center]] of ''G''. Thus, if ''G'' has [[Trivial group|trivial]] center it can be embedded into its own automorphism group.<ref name=Pahl>{{cite book |chapter-url=https://books.google.com/books?id=kvoaoWOfqd8C&pg=PA376 |page=376 |chapter=§7.5.5 Automorphisms |title=Mathematical foundations of computational engineering |edition=Felix Pahl translation |author=PJ Pahl, R Damrath |isbn=3-540-67995-2 |year=2001 |publisher=Springer}}</ref>
* In [[linear algebra]], an endomorphism of a [[vector space]] ''V'' is a [[linear transformation|linear operator]] ''V'' → ''V''. An automorphism is an invertible linear operator on ''V''. When the vector space is finite-dimensional, the automorphism group of ''V'' is the same as the [[general linear group]], GL(''V'').  (The algebraic structure of [[Endomorphism algebra|all endomorphisms of ''V'']] is itself an algebra over the same base field as ''V'', whose [[Group of units|invertible elements]] precisely consist of GL(''V'').)
* A field automorphism is a [[bijection|bijective]] [[ring homomorphism]] from a [[field (mathematics)|field]] to itself. 
**The field <math>\Q </math> of the [[rational number]]s has no other automorphism than the identity, since an automorphism must fix the [[additive identity]] {{math|0}} and the [[multiplicative identity]]  {{math|1}}; the sum of a finite number of {{math|1}} must be fixed, as well as the additive inverses of these sums (that is, the automorphism fixes all [[integers]]); finally, since every rational number is the quotient of two integers, all rational numbers must be fixed by any automorphism.
**The field <math>\R </math> of the [[real number]]s 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 <math>x<y</math> is equivalent to <math>\exists z\mid y-x=z^2,</math> and the latter property is preserved by every automorphism; finally every real number must be fixed since it is the [[least upper bound]] of a sequence of rational numbers.
** The field <math>\Complex</math> of the [[complex number]]s has a unique nontrivial automorphism that fixes the real numbers. It is the [[complex conjugation]], which maps <math>i</math> to <math>-i.</math> The [[axiom of choice]] implies the existence of [[uncountable|uncountably many]] automorphisms that do not fix the real numbers.<ref>{{cite journal | last = Yale | first = Paul B. | journal = Mathematics Magazine | title = Automorphisms of the Complex Numbers | volume = 39 | issue = 3 |date=May 1966 | pages = 135–141 | url = http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/PaulBYale.pdf | doi = 10.2307/2689301 | jstor = 2689301}}</ref><ref>{{citation |last=Lounesto |first=Pertti |year=2001 |publisher= Cambridge University Press |title=Clifford Algebras and Spinors | edition = 2nd |pages= 22–23|isbn=0-521-00551-5 }}</ref> 
** The study of automorphisms of [[algebraic field extension]]s is the starting point and the main object of [[Galois theory]].
* The automorphism group of the [[quaternion]]s ({{tmath|\mathbb H}}) as a ring are the inner automorphisms, by the [[Skolem–Noether theorem]]: maps of the form {{nowrap|''a'' ↦ ''bab''<sup>−1</sup>}}.{{refn|{{citation|year=2003|title=Handbook of algebra|volume=3|publisher=[[Elsevier]]|page=453}}}} This group is [[Quaternions_and_spatial_rotation|isomorphic]] to [[SO(3)]], the group of rotations in 3-dimensional space.
* The automorphism group of the [[octonions]] ({{tmath|\mathbb O}}) is the [[Exceptional Lie algebra|exceptional]] [[Lie group]] [[G2_(mathematics)|G<sub>2</sub>]].
* In [[graph theory]] an [[graph automorphism|automorphism of a graph]] 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.
* In [[geometry]], an automorphism may be called a [[motion (geometry)|motion]] of the space. Specialized terminology is also used:
** In [[metric geometry]] an automorphism is a self-[[isometry]]. The automorphism group is also called the [[isometry group]].
** In the category of [[Riemann surface]]s, an automorphism is a [[biholomorphy|biholomorphic]] map (also called a [[conformal map]]), from a surface to itself. For example, the automorphisms of the [[Riemann sphere]] are [[Möbius transformation]]s.
** An automorphism of a differentiable [[manifold]] ''M'' is a [[diffeomorphism]] from ''M'' to itself. The automorphism group is sometimes denoted Diff(''M'').
** In [[topology]], morphisms between topological spaces are called [[Continuous function (topology)|continuous maps]], and an automorphism of a topological space is a [[homeomorphism]] of the space to itself, or self-homeomorphism (see [[homeomorphism group]]). In this example it is ''not sufficient'' for a morphism to be bijective to be an isomorphism.

==History==
One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician [[William Rowan Hamilton]] in 1856, in his [[icosian calculus]], where he discovered an order two automorphism,<ref>{{Cite journal
|title=Memorandum respecting a new System of Roots of Unity
|author=Sir William Rowan Hamilton
|author-link=William Rowan Hamilton
|url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf |archive-date=2022-10-09 |url-status=live
|journal=[[Philosophical Magazine]]
|volume=12
|year=1856
|pages=446
}}</ref> writing:
<blockquote>so that <math>\mu</math> is a new fifth root of unity, connected with the former fifth root <math>\lambda</math> by relations of perfect reciprocity.</blockquote>

==Inner and outer automorphisms==
{{main article|Inner automorphism|Outer automorphism group}}
In some categories—notably [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[Lie algebra]]s—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.

In the case of groups, the [[inner automorphism]]s are the conjugations by the elements of the group itself. For each element ''a'' of a group ''G'', conjugation by ''a'' is the operation {{nowrap|''φ''<sub>''a''</sub> : ''G'' → ''G''}} given by {{nowrap|1=''φ''<sub>''a''</sub>(''g'') = ''aga''<sup>−1</sup>}} (or ''a''<sup>−1</sup>''ga''; usage varies). One can easily check that conjugation by ''a'' is a group automorphism. The inner automorphisms form a [[normal subgroup]] of Aut(''G''), denoted by Inn(''G''); this is called [[Goursat's lemma]].

The other automorphisms are called [[outer automorphism]]s. The [[quotient group]] {{nowrap|Aut(''G'') / Inn(''G'')}} is usually denoted by Out(''G''); the non-trivial elements are the [[cosets]] that contain the outer automorphisms.

The same definition holds in any [[unital algebra|unital]] [[ring (mathematics)|ring]] or [[algebra over a field|algebra]] where ''a'' is any [[Unit (ring theory)|invertible element]]. For [[Lie algebra]]s the definition is slightly different.

==See also==
* [[Antiautomorphism]]
* [[Mathematics of Sudoku#Automorphic Sudokus|Automorphism]] (in Sudoku puzzles)
* [[Characteristic subgroup]]
* [[Endomorphism ring]]
* [[Frobenius automorphism]]
* [[Morphism]]
* [[Order automorphism]] (in [[order theory]]).
* [[Isomorphism#Relation-preserving isomorphism|Relation-preserving automorphism]]
* [[Fractional Fourier transform]]

==References==
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==External links==
* [http://www.encyclopediaofmath.org/index.php/Automorphism ''Automorphism'' at Encyclopaedia of Mathematics]
* {{MathWorld | urlname=Automorphism | title = Automorphism}}

[[Category:Morphisms]]
[[Category:Abstract algebra]]
[[Category:Symmetry]]