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.

This group is called the automorphism group of X.

* 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.

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

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

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 ).

* automorphism if f is both an endomorphism and an isomorphism.

If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

An invertible endomorphism of X is called an automorphism.

Furthermore, if, and if the Frobenius endomorphism of F is an automorphism, g may be written as, and in particular, ; a contradiction of the irreducibility of f. Therefore, if F possesses an inseparable irreducible ( non-zero ) polynomial, then the Frobenius endomorphism of F cannot be an automorphism ( where F is assumed to have prime characteristic p ).

) More generally, if F is any field of ( non-zero ) prime characteristic for which the Frobenius endomorphism is not an automorphism, F possesses an inseparable algebraic extension.

By the argument outlined in the above paragraphs, it follows that F is perfect if and only if F has characteristic zero, or F has ( non-zero ) prime characteristic p and the Frobenius endomorphism of F is an automorphism.

An important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the ( constant sequence of the ) endomorphism.

* Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x → x < sup > p </ sup > is an automorphism of k

More generally, a ring of characteristic p ( p a prime ) is called perfect if the Frobenius endomorphism is an automorphism.

An injective endomorphism can be extended to an automorphism of a magma extension — the colimit of the constant sequence of the endomorphism.

In particular, every endomorphism of a central simple k-algebra is an inner automorphism.

For the Chevalley groups, the automorphism is the Frobenius endomorphism of F, while for the Steinberg groups the automorphism is the Frobenius endomorphism times an automorphism of the Dynkin diagram.

* Gal ( Q (√ 2 )/ Q ) has two elements, the identity automorphism and the automorphism which exchanges √ 2 and −√ 2.

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.

A result of Wolfgang Gaschütz says that if G is a finite non-abelian p-group, then G has an automorphism of p-power order which is not inner.

The image of this function was called an automorphism of G. ( Similarly, a homomorphic function ( or homomorph ) was a function between groups which preserved the product while a homomorphism was the image of a homomorph.

A semilinear transformation is a transformation which is linear " up to a twist ", meaning " up to a field automorphism under scalar multiplication ".

In mathematics, the orthogonal group of a symmetric bilinear form or quadratic form on a vector space is the group of invertible linear operators on the space which preserve the form: it is a subgroup of the automorphism group of the vector space.

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 ).

In particular, parallel transport around a closed curve starting at a point x defines an automorphism of the tangent space at x which is not necessarily trivial.

This has fundamental group Z / 3Z, has maximal compact subgroup the compact form ( see below ) of E < sub > 6 </ sub >, and has an outer automorphism group non-cyclic of order 4 generated by complex conjugation and by the outer automorphism which already exists as a complex automorphism.

* The compact form ( which is usually the one meant if no other information is given ), which has fundamental group Z / 3Z and outer automorphism group Z / 2Z.

* The split form, EI ( or E < sub > 6 ( 6 )</ sub >), which has maximal compact subgroup Sp ( 4 )/(± 1 ), fundamental group of order 2 and outer automorphism group of order 2.

* The quasi-split form EII ( or E < sub > 6 ( 2 )</ sub >), which has maximal compact subgroup SU ( 2 ) × SU ( 6 )/( center ), fundamental group cyclic of order 6 and outer automorphism group of order 2.

* EIII ( or E < sub > 6 (- 14 )</ sub >), which has maximal compact subgroup SO ( 2 ) × Spin ( 10 )/( center ), fundamental group Z and trivial outer automorphism group.

* EIV ( or E < sub > 6 (- 26 )</ sub >), which has maximal compact subgroup F < sub > 4 </ sub >, trivial fundamental group cyclic and outer automorphism group of order 2.

Over an algebraically closed field, this and its triple cover are the only forms ; however, over other fields, there are often many other forms, or “ twists ” of E < sub > 6 </ sub >, which are classified in the general framework of Galois cohomology ( over a perfect field k ) by the set H < sup > 1 </ sup >( k, Aut ( E < sub > 6 </ sub >)) which, because the Dynkin diagram of E < sub > 6 </ sub > ( see below ) has automorphism group Z / 2Z, maps to H < sup > 1 </ sup >( k, Z / 2Z ) = Hom ( Gal ( k ), Z / 2Z ) with kernel H < sup > 1 </ sup >( k, E < sub > 6, ad </ sub >).

The automorphism group is also called the isometry group.

** 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.

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