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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 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.1 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 homomorphismGiven two groups G ) and H ·), a group homomorphism from G ) to H ·) is a function h : G H such that for all u and v in G it holds that : h ''u v h ''u · h ''v From this property, one can deduce that h maps the identity element e of G to the identity elem, ring homomorphismIn abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. See Also Ring theory. If R and S are rings and f : R S is a function, we require f ''a + b f ''a + f ''b for all a and b, and linear operator).
2 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 AutC(X), or simply Aut(X) if the category is clear from context.
3 Examples
- 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 into its own automophism group. (See the discussion on inner automorphisms below).
- 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 nontrivial field automorphisms (this follows from the fact that such automorphisms are order-preserving). 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 (see the paper by Yale cited below). 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.
- An automorphism of a differentiable manifold M is a diffeomorphism from M to itself. The automorphism group is sometimes denoted Diff(M).
