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

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 ).

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 >.

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.

* 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 ).

Given a ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.

An automorphism of a Lie algebra is called an inner automorphism if it is of the form Ad < sub > g </ sub >, where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is.

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.

symmetric bilinear form with orthonormal basis v < sub > i </ sub >, the map sending a lattice to its dual lattice gives an automorphism with square the identity, giving the permutation σ that sends each label to its negative modulo n. The image of the above homomorphism is generated by σ and τ and is isomorphic to the dihedral group D < sub > n </ sub > of order 2n ; when n = 3, it gives the whole of S < sub > 3 </ sub >.

The compact form of G < sub > 2 </ sub > can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO ( 7 ) that preserves any chosen particular vector in its 8-dimensional real spinor representation.

The compact form is simply connected and its outer automorphism group is the trivial group.

The fundamental group of the complex form, compact real form, or any algebraic version of E < sub > 6 </ sub > is the cyclic group Z / 3Z, and its outer automorphism group is the cyclic group Z / 2Z.

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.

The compact real form of E < sub > 6 </ sub > as well as the noncompact forms EI = E < sub > 6 ( 6 )</ sub > and EIV = E < sub > 6 (- 26 )</ sub > are said to be inner or of type < sup > 1 </ sup > E < sub > 6 </ sub > meaning that their class lies in H < sup > 1 </ sup >( k, E < sub > 6, ad </ sub >) or that complex conjugation induces the trivial automorphism on the Dynkin diagram, whereas the other two real forms are said to be outer or of type < sup > 2 </ sup > E < sub > 6 </ sub >.

Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces ( the complex plane and the hyperbolic plane ).

Given then a normal extension L of K, with automorphism group Aut ( L / K ) = G, and containing α, any element g ( α ) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates.

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 ).

The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group < var > G </ var >, denoted by Aut (< var > G </ var >), forms itself a group, the automorphism group of < var > G </ var >.

There is one more automorphism with this property: multiplying all elements of Z < sub > 7 </ sub > by 5, modulo 7.

In other words, an automorphism of E / F is an isomorphism α from E to E such that α ( x ) = x for each x in F. The set of all automorphisms of E / F forms a group with the operation of function composition.

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 composition of two inner automorphisms is again an inner automorphism ( as mentioned above: ( x < sup > a </ sup >)< sup > b </ sup >= x < sup > ab </ sup >, and with this operation, the collection of all inner automorphisms of G is itself a group, the inner automorphism group of G denoted Inn ( G ).

By associating the element a in G with the inner automorphism ƒ ( x ) = x < sup > a </ sup > in Inn ( G ) as above, one obtains an isomorphism between the quotient group G / Z ( G ) ( where Z ( G ) is the center of G ) and the inner automorphism group:

The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.

It is bijective and compatible with the arithmetical operations, and hence is a field automorphism.

Over the finite field with q = p < sup > r </ sup > elements, F < sub > q </ sub >, there is a unique quadratic extension field, F < sub > q² </ sub >, with order 2 automorphism ( the rth power of the Frobenius automorphism ).

The compact manifolds with sol geometry are either the mapping torus of an Anosov map of the 2-torus ( an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, such as ), or quotients of these by groups of order at most 8.

A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M. In the sequel we will write G multiplicatively and M additively.

For n-dimensional V, the automorphism group of V is identified with a subset of complex square-matrices of order n. The automorphism group of V is given the structure of a smooth manifold using this identification.