In mathematics, automorphism is the isomorphism between a mathematical object and itself. In a sense, it is a symmetrical mirror image of the object itself, and it is a method to map the object to itself while maintaining its whole structure. All automorphisms of an object form a set called an automorphism group. Or, in general, a symmetric group of objects.

The exact definition of automorphism depends on the type of "ordered object" in the problem and what constitutes the "isomorphism" of the object. These nouns are most commonly used in the branch of abstract mathematics called category theory. Category theory deals with abstract objects and homomorphisms between them.

In category theory, automorphism is homomorphism (that is, homomorphism from an object to itself). It is also an isomorphism (in terms of category theory).

In abstract algebra, mathematical objects are algebraic structures, such as groups, rings and vector spaces. Isomorphism is a simple bijective homomorphism. The definition of homomorphism depends on the types of algebraic structures, such as group homomorphism, ring homomorphism and linear operators. )

Homomorphism (single mapping) is called ordinary automorphism in some places. Accordingly, other (non-simple) automorphisms are called very automorphisms.

Automorphism group