Automorphism

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.

==Definition==

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.

In category theory, an '''automorphism''' is an [[endomorphism]] (i.e. a [[morphism]] from an object to itself) which is also an [[Category theory#Some properties of morphisms|isomorphism]] (in the categorical sense of the word).

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.

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]]. (Of course, the definition of a homomorphism depends on the type of algebraic structure; see, for example: [[group homomorphism]], [[ring homomorphism]], and [[linear operator]]).

==Automorphism group==
The automorphisms of an object ''X'' 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:
* [[Closure (binary operation)|Closure]]: composition of two endomorphisms is another endomorphism.
* [[Associativity]]: composition of functions is ''always'' associative.
* [[Identity element|Identity]]: the identity is the identity morphism from an object to itself which exists by definition.
* [[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.

The automorphism group of an object ''X'' in a category ''C'' is denoted Aut''C''(''X''), or simply Aut(''X'') if the category is clear from context.

==Examples==
*In [[set theory]], an automorphism of a set ''X'' is an arbitrary [[permutation]] of the elements of ''X''. The automorphism group of ''X'' is also called the [[symmetric group]] on ''X''.

*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.

*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 [[kernel (algebra)|kernel]] is the [[center of a group|center]] of ''G''. Thus, if ''G'' is centerless it can be embedded into its own automorphism group. (See the discussion on inner automorphisms below).

*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'').

*A field automorphism is a [[bijection|bijective]] [[ring homomorphism]] from a [[field (mathematics)|field]] to itself. In the case of the [[rational number]]s, '''Q''' there are no nontrivial field automorphisms; in the case of the [[real number]]s '''R''' there are no nontrivial [[Monotonic function|order-preserving]] field automorphisms (that is, automorphisms of the [[ordered field]]). 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 | month = May | year = 1966 | pages = 135–141 | url = http://www.jstor.org/view/0025570x/di021045/02p0089z/0}}</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.

*In [[graph theory]] an 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.

*For relations, see {{ml|Isomorphism|A_relation-preserving_isomorphism|relation-preserving automorphism}}.
**In [[order theory]], see [[order automorphism]].

*An automorphism of a differentiable [[manifold]] ''M'' is a [[diffeomorphism]] from ''M'' to itself. The automorphism group is sometimes denoted Diff(''M'').

*In [[Riemannian 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 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.

==Inner and outer automorphisms==

In some categories—notably [[group (mathematics)|group]]s, [[ring (mathematics)|ring]]s, 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 φ''a'' : ''G'' → ''G'' given by φ''a''(''g'') = ''aga''−1 (or ''a''−1''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]] 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]] [[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==
*[[endomorphism ring]]
*[[antiautomorphism]]
*[[Frobenius automorphism]]

==References==

<references />

==External links==
*{{MathWorld | urlname=Automorphism | title = Automorphism}}

[[Category:Abstract algebra]]
[[Category:Algebra]]
[[Category:Category theory]]
[[Category:Symmetry]]

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