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).