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.

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.

** 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.

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

It is the automorphism group of the Riemann sphere ( when considered as a Riemann surface ) and is sometimes denoted

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 >.

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 ).

The Möbius group is usually denoted as it is the automorphism group of the Riemann sphere.

The outer automorphism group is usually denoted Out ( G ).

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.

* For any subgroup H of Gal ( E / F ), the corresponding field, usually denoted E < sup > H </ sup >, is the set of those elements of E which are fixed by every automorphism in H.

** An automorphism of a differentiable manifold M is a diffeomorphism from M to itself.

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.

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

The automorphism group can be augmented ( by a symmetry which is not realized by a symmetry of the tiling ) to yield the Mathieu group M < sub > 24 </ sub >.

Conversely, given a left kG module M, then M is a k vector space, and multiplication with an element g of G yields a k-linear automorphism of M ( since g is invertible in kG ), which describes a group homomorphism G → GL ( M ).

In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group.

If M is an oriented manifold, Aut ( M ) would be the orientation-preserving automorphisms of M and so the mapping class group of M ( as an oriented manifold ) would be index two in the mapping class group of M ( as an unoriented manifold ) provided M admits an orientation-reversing automorphism.

M < SUB > 11 </ SUB > ( 2 classes, fused by an outer automorphism )

M < SUB > 12 </ SUB >: 2 ( Two classes, fused by an outer automorphism )

Then sections of E are smooth functions such that is the identity automorphism of M. The jet of a section s over a neighborhood of a point p is just the jet of this smooth function from M to E at p.

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 ).

for every automorphism φ of G ( where φ ( H ) denotes the image of H under φ ).

is an automorphism of G ( known as an inner automorphism ).

The image under an automorphism of a conjugacy class is always a conjugacy class ( the same or another ).

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 ).

is known as the outer automorphism group Out ( G ).

Every non-inner automorphism yields a non-trivial element of Out ( G ), but different non-inner automorphisms may yield the same element of Out ( G ).

This is a consequence of the first isomorphism theorem, because Z ( G ) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism ( conjugation changes nothing ).

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

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 ).

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 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 ).