AUTOMORPHISM

In mathematics, an automorphism of a mathematical object is a mapping that:

• maps the object onto itself, i.e. the domain and codomain are the same
• is a homomorphism, i.e. structure preserving
• is bijective
• its inverse map is also a homomorphism

Very informally, an automorphism is a symmetry of the object, a way of showing its internal regularity (whichever side of a regular polygon you choose as it basis, it looks the same).

For example, in graph theory an automorphism of a graph is a permutation of the nodes that maps the graph to itself. In group theory, an automorphism of a group G is a bijective homomorphism of G onto itself (that is, a one-to-one map G -> G that preserves the group operation; informally, a way of shuffling the elements of the group which doesn't affect the structure).

The set of automorphisms of an object X together with the operation of function composition forms a group called the automorphism group of X, Aut(X). That this is indeed a group is simple to see:

• Closure: composition of two bijections is a bijection, composition of homomorphisms is a homomorphism
• Identity: the identity automorphism is simply "do nothing": the identity mapping of the object onto itself
• Inverse: since an automorphism is by definition a bijection, it has an inverse. This satisfies the same properties, and is therefore an automorphism itself
• Associativity: function composition is trivially associative

When it is possible to build transformation of an object by selecting one of its elements and applying operations to the object, one can separate

• inner automorphisms
• outer automorphisms

The latter being transformations for which there is no correspondence with an element of the object.

In particular, for groups, an inner automorphism is an automorphism f_g : G -> G given by a conjugacy by a fixed element g of the group G, that is, for all h in G, the map f_g is of the form f_g(h) = g^-1 hg. The 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).

See also Isomorphism, Endomorphism, Morphism