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.

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.

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.

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.

Two vertices x and y of H are called symmetric if there exists an automorphism such that.

Two edges and are said to be symmetric if there exists an automorphism such that.

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 ).

* 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.

In general, S is an antihomomorphism, so S < sup > 2 </ sup > is a homomorphism, which is therefore an automorphism if S was invertible ( as may be required ).

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.

One can define a projective space axiomatically in terms of an incidence structure ( a set of points P, lines L, and an incidence relation I specifying which points lie on which lines ) satisfying certain axioms – an automorphism of a projective space thus defined then being an automorphism f of the set of points and an automorphism g of the set of lines, preserving the incidence relation ,< ref group =" note ">" Preserving the incidence relation " means that if point p is on line l then is in ; formally, if then .</ ref > which is exactly a collineation of a space to itself.

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.

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 ).

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 )

If the first equation is relaxed to just have, then f is a partial automorphism ( with inverse g ).

if σ is an automorphism of the simple Lie algebra associated to an automorphism of its Dynkin diagram, the twisted loop algebra consists of-valued functions f on the real line which satisfy

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:

The dynamics of this homeomorphism is the simplest when f is a pseudo-Anosov map: in this case, there are two fixed points on the Thurston boundary, one attracting and one repelling, and the homeomorphism behaves similarly to a hyperbolic automorphism of the Poincaré half-plane.

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.

* 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 order of the automorphism group is related, being the number of polygons times the number of edges in the polygon in both cases.

The Schur multiplier and the outer automorphism group of the Thompson group are both trivial.

An invertible endomorphism of X is called an automorphism.

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

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.