The set of all automorphisms of an object forms a group, called the automorphism group.

This group is called the automorphism group of X.

The automorphism group of X is also called the symmetric group on X.

* In elementary arithmetic, the set of integers, Z, considered as a group under addition, has a unique nontrivial automorphism: negation.

Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.

* A group automorphism is a group isomorphism from a group to itself.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

The automorphism group is also called the isometry group.

The automorphism group is sometimes denoted Diff ( M ).

** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).

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:

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

The exact definition of an automorphism depends on the type of " mathematical object " in question and what, precisely, constitutes an " isomorphism " of that object.

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 ).

* Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.

More formally, a " representation " means a homomorphism from the group to the automorphism group of an object.

* An automorphism is an endomorphism which is also an isomorphism, i. e., an isomorphism from an object to itself.

Given an object X, a functor G ( taking for simplicity the first functor to be the identity ) and an isomorphism proof of unnaturality is most easily shown by giving an automorphism that does not commute with this isomorphism ( so ).

* In set theory, an automorphism of a set X is an arbitrary permutation of the elements of X.

An invertible endomorphism of X is called an automorphism.

The set of all automorphisms is a subset of End ( X ) with a group structure, called the automorphism group of X and denoted Aut ( X ).

More strongly, if one wishes to prove that X and G ( X ) are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism η, there is some A with which it does not commute ; in some cases a single automorphism A works for all candidate isomorphisms η, while in other cases one must show how to construct a different A < sub > η </ sub > for each isomorphism.

The set of automorphisms of a hypergraph H (= ( X, E )) is a group under composition, called the automorphism group of the hypergraph and written Aut ( H ).

His idea was that if F is the Frobenius automorphism over the finite field, then the number of points of the variety X over the field of order q < sup > m </ sup > is the number of fixed points of F < sup > m </ sup > ( acting on all points of the variety X defined over the algebraic closure ).

A deck transformation or automorphism of a cover p: C → X is a homeomorphism f: C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut ( p ).

Any simplicial automorphism φ of X defines a permutation π of Z / n Z such that label ( φ ( M ))

* The map X P < sup >− 1 </ sup > XP is an automorphism of the associative algebra of all n-by-n matrices, as the one-object case of the above category of all matrices.

In purely algebraic terms, for a given field K, this is the automorphism group over K of the field K ( X, Y ) of rational functions in two variables.

The points of X that are defined over F < sub >< span > p < sup > n </ sup ></ span ></ sub > are those fixed by F < sup > n </ sup >, where F is the Frobenius automorphism in characteristic p.

In fact, it has no automorphism other than the identity, because it is contained in the real numbers and X < sup > 3 </ sup > − 2 has just one real root.

Given a pair of spaces ( X, A ) the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of ( X, A ) is defined as an automorphism of X that preserves A, i. e. f: X → X is invertible and f ( A )

** In the category of Riemann surfaces, an automorphism is a bijective biholomorphic map ( also called a conformal map ), from a surface to itself.

In category theory, given any family P < sub > n </ sub > of invertible n-by-n matrices defining a similarity transformation for all rectangular matrices sending the m-by-n matrix A into P < sub > m </ sub >< sup >− 1 </ sup > AP < sub > n </ sub >, the family defines a functor that is an automorphism of the category of all matrices, having as objects the natural numbers and morphisms from n to m the m-by-n matrices composed via matrix multiplication.

A translation functor on a category D is an automorphism ( or for some authors, an auto-equivalence ) T from D to D. One usually uses the notation

In the case of the complex numbers, C, there is a unique nontrivial automorphism that sends R into R: complex conjugation, but there are infinitely ( uncountably ) many " wild " automorphisms ( assuming the axiom of choice ).

* Gal ( C / R ) has two elements, the identity automorphism and the complex conjugation automorphism.

From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group ( reversing the Dynkin diagram A < sub > n </ sub >, which corresponds to transpose inverse ) and the field automorphism of the extension C / R ( namely complex conjugation ).

An automorphism in Split ( C ) is of the form, with inverse satisfying:

A ( partial ) involution in Split ( C ) is a self-inverse ( partial ) automorphism.

This has two connected components where Z < sub > 2 </ sub > acts as an automorphism, which is the composition of an involutive outer automorphism of SU ( 3 )< sub > C </ sub > with the interchange of the left and right copies of SU ( 2 ) with the reversal of U ( 1 )< sub > B − L </ sub >.