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

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

- inner automorphisms
- outer automorphisms

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*is of the form

_{g}*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