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).

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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

In particular, for groups, an *inner automorphism* is an automorphism *f _{g}* :

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