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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 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).