Automorphism

In mathematics, an automorphism is a structure-preserving bijection of a mathematical object onto itself, that is, an isomorphism between the object and itself. Very informally, it 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).

The set of automorphisms of an object X together with the operation of [function composition]? forms a group called the [automorphism group]?, Aut(X).

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 correspondance 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).

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