Automorphism Group

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 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 , Ring Homomorphism , and Linear Operator ).

AUTOMORPHISM GROUP

The automorphisms of an object ''X'' form a Group under composition of Morphism s. 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 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 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 , but there are infinitely many "wild" automorphisms (see the paper by Yale cited below). 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.

The set of , 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 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 Surface s, 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 Transformation s.

INNER AND OUTER AUTOMORPHISMS

In some categories—notably Group s, Ring s, and Lie Algebra s—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 Automorphism s. 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 or Algebra where ''a'' is any Invertible Element . For Lie Algebra s 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

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	CATEGORIES ABOUT AUTOMORPHISM

	abstract algebra

	algebra

	category theory

	symmetry

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Automorphism Group