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 mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.
Definition
In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself..The identity morphism is called the trivial automorphism in some contexts. Respectively, other automorphisms are called nontrivial automorphisms.
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 which is also an isomorphism.
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.
Automorphism group
If the automorphisms of an object form a set, then they form a group under composition of morphisms. This group is called the automorphism group of.;Closure: Composition of two automorphisms is another automorphism.
;Associativity: It is part of the definition of a category that composition of morphisms is associative.
;Identity: The identity is the identity morphism from an object to itself, which is an automorphism.
;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 AutC, or simply Aut 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 integers, 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 whose image is the group Inn of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.
- In linear algebra, an endomorphism of a vector space V is a 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.
- A field automorphism is a bijective ring homomorphism from a field to itself. In the cases of the rational numbers and the real numbers there are no nontrivial field automorphisms. Some subfields of R have nontrivial field automorphisms, which however do not extend to all of R. In the case of the complex numbers, C, there is a unique nontrivial automorphism that sends R into R: complex conjugation, but there are infinitely many "wild" automorphisms. Field automorphisms are important to the theory of field extensions, in particular Galois extensions. 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.
- The automorphism group of the quaternions as a ring are the inner automorphisms, by the Skolem–Noether theorem: maps of the form. This group is isomorphic to SO, the group of rotations in 3-dimensional space.
- The automorphism group of the octonions is the exceptional Lie group G2.
- 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 geometry, an automorphism may be called a 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 surfaces, an automorphism is a biholomorphic map, from a surface to itself. For example, the automorphisms of the Riemann sphere are Möbius transformations.
- * An automorphism of a differentiable manifold M is a diffeomorphism from M to itself. The automorphism group is sometimes denoted Diff.
- * In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism. In this example it is not sufficient for a morphism to be bijective to be an isomorphism.
History
so that is a new fifth root of unity, connected with the former fifth root by relations of perfect reciprocity.
Inner and outer automorphisms
In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. For each element a of a group G, conjugation by a is the operation given by . One can easily check that conjugation by a is a group automorphism. The inner automorphisms form a normal subgroup of Aut, denoted by Inn; this is called Goursat's lemma.
The other automorphisms are called outer automorphisms. The quotient group is usually denoted by Out; the non-trivial elements are the cosets that contain the outer automorphisms.
The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different.