In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... The symmetry group of an object (e. ...

Definition

The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called category theory. Category theory deals with abstract objects and morphisms between those objects. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

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). In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space. An isomorphism is simply a bijective homomorphism. (Of course, the definition of a homomorphism depends on the type of algebraic structure; see, for example: group homomorphism, ring homomorphism, and linear operator). Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... // Homomorphism for beginners Homomorphism is one of the fundamental concepts in abstract algebra. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

Automorphism group

The set of automorphisms of an object X form a group under composition of morphisms. This group is called the automorphism group of X. That this is indeed a group is simple to see: In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

• Closure: composition of two endomorphisms is another endomorphism.
• Associativity: morphism composition is associative by definition.
• Identity: the identity is the identity morphism from an object to itself which exists by definition.
• 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_C(X), or simply Aut(X) if the category is clear from context. In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the inverse of an element x, with respect to an operation *, is a chicken fucker, x such that their compose gives a neutral element. ...

Examples

• 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.

• A group automorphism is a group isomorphism from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group G there is a natural group homomorphism G → Aut(G) whose kernel is the center of G. Thus, if G is centerless it can be embedded into its own automorphism group. (See the discussion on inner automorphisms below).

• 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. The automorphism group of V is just the general linear group, GL(V).

• A field automorphism is a bijective ring homomorphism from a field to itself. In the case of the rational numbers, Q, or the real numbers, R, there are no nontrivial field automorphisms (this follows from the fact that such automorphisms are order-preserving). 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 many "wild" automorphisms (see the paper by Yale cited below). Field automorphisms are important to the theory of field extensions, in particular Galois extensions. In the case of a Galois extension L/K the subgroup of all automorphisms of L fixing K pointwise is called the Galois group of the extension.

• The set of integers, Z, considered as a group under addition has a unique nontrivial automorphism : negation. Considered as a ring, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.

• In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by an edge, so are their images under the permutation.

• In order theory, see order automorphism.

• An automorphism of a differentiable manifold M is a diffeomorphism from M to itself. The automorphism group is sometimes denoted Diff(M).

• In Riemannian geometry an automorphism is a self-isometry. The automorphism group is also called the isometry group.

• In the category of Riemann surfaces, an automorphism is a bijective holomorphic map (also called a conformal map), from a surface to itself. For example, the automorphisms of the Riemann sphere are Möbius transformations.

Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, a Galois extension is a field extension that has a Galois group. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, a Galois group is a group associated with a certain type of field extension. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... A diagram of a graph with 6 vertices and 7 edges. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In the mathematical area of order theory, given a partially ordered set (S, ≤) an order automorphism of (S, ≤) is an order isomorphism from (S, ≤) to itself. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, a conformal map is a function which preserves angles. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...

Inner and outer automorphisms

In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two classes: In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

• inner automorphisms
• outer automorphisms

The former corresponding to automorphisms coming from "conjugation" by elements of the object itself, and the latter being everything else. In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by ax. ...

In group theory, for example, let a be an element of a group G. Conjugation by a is the group homomorphism φ_a : G → G given by φ_a(g) = aga⁻¹. One can easily check that conjugation by a is actually a group automorphism. An inner automorphism is then an automorphism corresponding to conjugation by some element a. The set of all inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G). The quotient group Aut(G) / Inn(G) is usually denoted by Out(G). Group theory is that branch of mathematics concerned with the study of groups. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...

The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

See also

• endomorphism
• endomorphism ring
• antiautomorphism
• Frobenius automorphism

In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ... In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. ... In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields. ...

Reference

Yale, Paul B. Mathematics Magazine. "Automorphisms of the Complex Numbers". Vol 39. Num. 3. May, 1966. pp. 135-141. Available via http://www.jstor.org