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| 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 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.
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| ==Definition==
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| The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called [[category theory]]. Category theory deals with abstract objects and [[morphism]]s between those objects.
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| In category theory, an automorphism is an [[endomorphism]] (i.e. a [[morphism]] from an object to itself) which is also an [[isomorphism]] (in the categorical sense of the word).
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| This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.
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| In the context of [[abstract algebra]], for example, a mathematical object is an [[algebraic structure]] such as a [[group (mathematics)|group]], [[ring (mathematics)|ring]], or [[vector space]]. An isomorphism is simply a [[bijective]] [[homomorphism]]. (The definition of a homomorphism depends on the type of algebraic structure; see, for example: [[group homomorphism]], [[ring homomorphism]], and [[linear operator]]).
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| The [[identity morphism]] ([[identity mapping]]) is called the '''trivial automorphism''' in some contexts. Respectively, other (non-identity) automorphisms are called '''nontrivial automorphisms'''.
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| ==Automorphism group==
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| If the automorphisms of an object ''X'' form a set (instead of a proper [[class (set theory)|class]]), then they form a [[group (mathematics)|group]] under [[Function composition|composition]] of [[morphism]]s. This group is called the '''automorphism group''' of ''X''. That this is indeed a group is simple to see:
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| * [[Closure (binary operation)|Closure]]: composition of two endomorphisms is another endomorphism.
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| * [[Associativity]]: composition of morphisms is ''always'' associative.
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| * [[Identity element|Identity]]: the identity is the identity morphism from an object to itself which exists by definition.
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| * [[Inverse element|Inverses]]: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.
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| The automorphism group of an object ''X'' in a category ''C'' is denoted Aut<sub>''C''</sub>(''X''), or simply Aut(''X'') if the category is clear from context.
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| ==Examples==
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| * 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''.
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| * In [[elementary arithmetic]], the set of [[integer]]s, '''Z''', considered as a group under addition, has a unique nontrivial automorphism: negation. Considered as a [[ring (mathematics)|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.
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| * 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>
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| {{cite book |url=http://books.google.com/?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}}
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| </ref>
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| * 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'').
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| * A field automorphism is a [[bijection|bijective]] [[ring homomorphism]] from a [[field (mathematics)|field]] to itself. In the cases of the [[rational number]]s ('''Q''') and the [[real number]]s ('''R''') there are no nontrivial field automorphisms. Some subfields of '''R''' have nontrivial field automorphisms, which however do not extend to all of '''R''' (because they cannot preserve the property of a number having a square root in '''R'''). In the case of the [[complex number]]s, '''C''', there is a unique nontrivial automorphism that sends '''R''' into '''R''': [[complex conjugate|complex conjugation]], but there are infinitely ([[uncountable|uncountably]]) many "wild" automorphisms (assuming the [[axiom of choice]]).<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> Field automorphisms are important to the theory of [[field extension]]s, in particular [[Galois extension]]s. In the case of a Galois extension ''L''/''K'' the [[subgroup]] of all automorphisms of ''L'' fixing ''K'' pointwise is called the [[Galois group]] of the extension.
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| * 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.
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| * For relations, see [[Isomorphism#A relation-preserving isomorphism|relation-preserving automorphism]].
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| ** In [[order theory]], see [[order automorphism]].
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| * In [[geometry]], an automorphism may be called a [[motion (geometry)|motion]] of the space. Specialized terminology is also used:
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| ** In [[metric geometry]] an automorphism is a self-[[isometry]]. The automorphism group is also called the [[isometry group]].
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| ** In the category of [[Riemann surface]]s, an automorphism is a bijective [[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.
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| ** An automorphism of a differentiable [[manifold]] ''M'' is a [[diffeomorphism]] from ''M'' to itself. The automorphism group is sometimes denoted Diff(''M'').
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| ** 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.
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| ==History==
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| 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
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| |title=Memorandum respecting a new System of Roots of Unity
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| |author=Sir William Rowan Hamilton
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| |author-link=William Rowan Hamilton
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| |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf
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| |journal=[[Philosophical Magazine]]
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| |volume=12
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| |year=1856
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| |pages=446
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| }}</ref> writing:
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| {{quotation|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.}}
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| ==Inner and outer automorphisms==
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| 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.
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| 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 φ<sub>''a''</sub> : ''G'' → ''G'' given by <math> \varphi_a (g) = a g a^{-1} </math> (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]].
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| The other automorphisms are called [[outer automorphism]]s. The [[quotient group]] Aut(''G'') / Inn(''G'') is usually denoted by Out(''G''); the non-trivial elements are the cosets that contain the outer automorphisms.
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| 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.
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| ==See also==
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| * [[endomorphism ring]]
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| * [[antiautomorphism]]
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| * [[Frobenius automorphism]]
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| * [[morphism]]
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| * [[characteristic subgroup]]
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| ==References==
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| <!-- See [[Wikipedia:Footnotes]] for instructions. -->
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| <references />
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Automorphism ''Automorphism'' at Encyclopaedia of Mathematics]
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| * {{MathWorld | urlname=Automorphism | title = Automorphism}}
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| [[Category:Morphisms]]
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| [[Category:Abstract algebra]]
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| [[Category:Symmetry]]
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Emilia Shryock is my name but you can call me anything you like. Hiring is my occupation. To do aerobics is a factor that I'm completely addicted to. California is exactly where her house is but she needs to transfer because of her family.
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