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 an object X in a category C is denoted Aut < sub > C </ sub >( X ), or simply Aut ( X ) if the category is clear from context.

* 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 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 ).

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 ).

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

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 ).

The identity morphism ( identity mapping ) is called the trivial automorphism in some contexts.

* 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.

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

Hence G is also a transformation group ( and an automorphism group ) because function composition preserves the partitioning of A.

Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.

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

When p is prime, GL ( n, p ) is the outer automorphism group of the group Z, and also the automorphism group, because Z is Abelian, so the inner automorphism group is trivial.

It is the automorphism group of the Fano plane and of the group Z, and is also known as PSL ( 2, 7 ).

In the case when and are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.

It is known ( for compact, orientable S ) that this is isomorphic with the automorphism group of the fundamental group of S. This is consistent with the genus 1 case, stated above, if one takes into account that then the fundamental group is Z < sup > 2 </ sup >, on which the modular group acts as automorphisms ( as a subgroup of index 2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant − 1 ).

The automorphism is also called global Cartan involution, and the diffeomorphism is called global Cartan decomposition.

As an automorphism of G, σ fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra of G, also denoted by σ, whose square is the identity.

He found that if a finite field of characteristic 2 also has an automorphism whose square was the Frobenius map, then an analogue of Steinberg's construction gave the Suzuki groups.

He also gave some complicated conditions satisfied by the automorphism σ.

These are clearly automorphisms of K. There is also the identity automorphism e which does not change anything, and the composition of f and g which changes the signs on both radicals:

Two-graphs have been studied because of their connection with equiangular lines and, for regular two-graphs, strongly regular graphs, and also finite groups because many regular two-graphs have interesting automorphism groups.

When viewed as a factor-algebra of g, this semisimple Lie algebra is also called the Levi factor of g. Moreover, Malcev ( 1942 ) showed that any two Levi subalgebras are conjugate by an ( inner ) automorphism of the form