In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) The identity morphism (identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms. The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called category theory. Category theory deals with abstract objects and morphisms between those objects. In category theory, an automorphism is an endomorphism (i.e., a morphism from an object to itself) which is also an isomorphism (in the categorical sense of the word, meaning there exists a right and left inverse endomorphism). This is a very abstract definition since, in category theory, morphisms are not necessarily functions and objects are not necessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure. Automorphism group If the automorphisms of an object form a set (instead of a proper class), then they form a group under composition of morphisms. This group is called the automorphism group of . Closure Composition of two automorphisms is another automorphism. Associativity It is part of the definition of a category that composition of morphisms is associative. Identity The identity is the identity morphism from an object to itself, which is an automorphism. Inverses By definition every isomorphism has an inverse that is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism. The automorphism group of an object X in a category C is denoted AutC(X), or simply Aut(X) if the category is clear