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Automorphism group

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, 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

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

• Closure: composition of two endomorphisms is another endomorphism.
• Associativity: morphism composition is associative by definition.
• Identity: the identity is the identity morphism from an object to itself which exists by definition.
• 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.

• 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 is the center of G. Thus, if G is centerless it can be embedded it's own automophism group. (See the discussion on inner automorphisms below).

• 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. The automorphism group of V is just the general linear group, GL(V).

• A field automorphism is a bijective ring homomorphism from a field to itself. In the case of the rational numbers, Q, or the real numbers, R, there are no nontrival field automorphisms. In the case of the complex numbers, C, there is a unique nontrivial automorphism: complex conjugation. 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 set of integers, Z, considered as a group 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.

• In graph theory an automorphism of a graph is a permutation of the nodes that maps the graph to itself.

• 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 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 groups, rings, and Lie algebras—it is possible to separate automorphisms into two classes:

• inner automorphisms
• outer automorphisms

The former corresponding to automorphisms coming from "conjugation" by elements of the object itself, and the latter being everything else.

In group theory, for example, let a be an element of a group G. Conjugation by a is the group homomorphism φ_a : G → G given by φ_a(g) = aga⁻¹. One can easily check that conjugation by a is actually a group automorphism. An inner automorphism is then an automorphism corresponding to conjugation by some element a. The set of all inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G). The quotient group Aut(G) / Inn(G) is usually denoted by Out(G).

The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different.

See also

• endomorphism
• endomorphism ring
• antiautomorphism