, an automorphism
of a mathematical object is a mapping that
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 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).
See also Isomorphism, Endomorphism, Morphism