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) the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group (mathematics), 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 morphisms 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#Types of morphisms (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 ).

= Automorphism group =

The set of automorphisms of an object X form a group (mathematics) under composition of morphisms. This group is called the automorphism group of X . That this is indeed a group is simple to see:

Closure (binary operation): composition of two endomorphisms is another endomorphism.

Associativity: morphism composition is associative by definition.

Identity element: the identity is the identity morphism from an object to itself which exists by definition.

Inverse element: 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 .

*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) is the center of a group 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 V → V . An automorphism is an invertible linear operator on V . The automorphism group of V is just the general linear group, GL( V ).

*A field automorphism is a of the extension.

*The set of , but not of a ring or field.

*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.

*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 surfaces, an automorphism is a bijective holomorphic function map (also called a conformal map), from a surface to itself. For example, the automorphisms of the Riemann sphere are Möbius transformations.

= Inner and outer automorphisms =

In some categories—notably group (mathematics)s, ring (mathematics)s, and Lie algebras—it is possible to separate automorphisms into two classes.

In the case of groups:

The of Aut( G ), denoted by Inn( G ).

The other automorphisms are called outer automorphisms. The quotient group Aut( G ) / Inn( G ) is usually denoted by Out( G ); the non-trivial elements are the cosets containing the outer automorphisms.

The same definition holds in any Unital ring (mathematics) or algebra over a field where a is any Unit (ring theory). For Lie algebras the definition is slightly different.

= See also =

*Endomorphism *endomorphism ring *antiautomorphism *Frobenius automorphism

= Reference =

Yale, Paul B. Mathematics Magazine . Automorphisms of the Complex Numbers . Vol 39. Num. 3. May, 1966. pp. 135-141. Available via http://www.jstor.org