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).
- 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 fg : G -> G given by a conjugacy by a fixed element g of the group G, that is, for all h in G, the map fg is of the form fg(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).