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.

In an algebraic structure such as a group, a ring, or vector space, an automorphism is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.)

More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism ${\displaystyle f:X\to X}$ is an automorphism if there is a morphism ${\displaystyle g:X\to X}$ such that ${\displaystyle g\circ f=f\circ g=\operatorname {id} _{X},}$ where ${\displaystyle \operatorname {id} _{X}}$ is the identity morphism of X. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the identity function, and is often called the trivial automorphism.

The automorphisms of an object X form a group under composition of morphisms, which is called the automorphism group of X. This results straightforwardly from the definition of a category.

The automorphism group of an object X in a category C is often denoted Aut_C(X), or simply Aut(X) if the category is clear from context.

One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus, where he discovered an order two automorphism, writing:

so that ${\displaystyle \mu }$ is a new fifth root of unity, connected with the former fifth root ${\displaystyle \lambda }$ by relations of perfect reciprocity.

In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.