In mathematics, an automorphism is an isomorphism from a mathematical object to itself. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics and related technical fields the term map or mapping is often a Synonym for function. The set of all automorphisms of an object forms a group, called the automorphism group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element It is, loosely speaking, the symmetry group of the object. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is
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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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Category theory deals with abstract objects and morphisms between those objects. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and
In category theory, an automorphism is an endomorphism (i. In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself e. a morphism from an object to itself) which is also an isomorphism (in the categorical sense of the word). In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets
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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added An isomorphism is simply a bijective homomorphism. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector (Of course, the definition of a homomorphism depends on the type of algebraic structure; see, for example: group homomorphism, ring homomorphism, and linear operator). In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that
The identity morphism (identity mapping) is called the trivial automorphism in some contexts. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that Respectively, other (non-identity) automorphisms are called nontrivial automorphisms.
Automorphism group
If the automorphisms of an object X form a set (instead of a proper class), then they form a group under composition of morphisms. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and This group is called the automorphism group of X. That this is indeed a group is simple to see:
- Closure: composition of two endomorphisms is another endomorphism. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set
- Associativity: composition of functions is always associative. In Mathematics, associativity is a property that a Binary operation can have
- Identity: the identity is the identity morphism from an object to itself which exists by definition. 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
- 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 Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to
The automorphism group of an object X in a category C is denoted AutC(X), or simply Aut(X) if the category is clear from context.
Examples
- In set theory, an automorphism of a set X is an arbitrary permutation of the elements of X. In several fields of Mathematics the term permutation is used with different but closely related meanings The automorphism group of X is also called the symmetric group on X. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying
- In elementary arithmetic, the set of integers, Z, considered as a group under addition, has a unique nontrivial automorphism: negation. Elementary arithmetic is the most basic kind of Mathematics: it concerns the operations of Addition, Subtraction, Multiplication, and division The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Considered as a ring, however, it has only the trivial automorphism. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the
- A group automorphism is a group isomorphism from a group to itself. 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 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 image is the group Inn(G) of inner automorphisms and whose kernel is the center of G. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G 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 In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the Thus, if G has trivial center it can be embedded into its own automorphism group. In Mathematics, a trivial group is a group consisting of a single element
- In linear algebra, an endomorphism of a vector space V is a linear operator V → V. Linear algebra is the branch of Mathematics concerned with In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that 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). In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation
- A field automorphism is a bijective ring homomorphism from a field to itself. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Ring theory or 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 In the cases of the rational numbers (Q) and the real numbers (R) there are no nontrivial field automorphisms. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways 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). Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. [1] Field automorphisms are important to the theory of field extensions, in particular Galois extensions. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Mathematics, a Galois extension is an algebraic field extension E / F satisfying certain conditions (described below one also says that the 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. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, a Galois group is a group associated with a certain type of Field extension.
- In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In graph-theoretical mathematics, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge-vertex connectivity In particular, if two nodes are joined by an edge, so are their images under the permutation.
- For relations, see relation-preserving automorphism. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective
- In order theory, see order automorphism. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In the mathematical field of Order theory an order isomorphism is a special kind of Monotone function that constitutes a suitable notion of Isomorphism
- 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. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function Topological equivalence redirects here see also Topological equivalence (dynamical systems. In this example it is not sufficient for a morphism to be bijective in order to be an isomorphism.
- An automorphism of a differentiable manifold M is a diffeomorphism from M to itself. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable The automorphism group is sometimes denoted Diff(M).
- In Riemannian geometry an automorphism is a self-isometry. Elliptic geometry is also sometimes called Riemannian geometry. For the Mechanical engineering and Architecture usage see Isometric projection. The automorphism group is also called the isometry group. In Mathematics, the isometry group of a Metric space is the set of all isometries from the metric space onto itself with the Function composition
- In the category of Riemann surfaces, an automorphism is a bijective biholomorphic map (also called a conformal map), from a surface to itself. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In the mathematical theory of functions of one or more complex variables, and also in Complex algebraic geometry, a biholomorphism or biholomorphic In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane For example, the automorphisms of the Riemann sphere are Möbius transformations. In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that Möbius transformations should not be confused with the Möbius transform or the Möbius function.
Inner and outer automorphisms
In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie
In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G For each element a of a group G, conjugation by a is the operation φa : G → G given by φa(g) = aga−1 (or a−1ga; usage varies). One can easily check that conjugation by a is a group automorphism. The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. Goursat's lemma is an Algebraic Theorem. Let G G' be groups and let H be a Subgroup of
The other automorphisms are called outer automorphisms. In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner The quotient group Aut(G) / Inn(G) is usually denoted by Out(G); the non-trivial elements are the cosets that contain the outer automorphisms. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G
The same definition holds in any unital ring or algebra where a is any invertible element. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i For Lie algebras the definition is slightly different. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie
See also
References
- ^ Yale, Paul B. In Abstract algebra, one associates to certain objects a ring, the object's 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 Commutative algebra and field theory, which are branches of Mathematics, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a (May 1966). "Automorphisms of the Complex Numbers" ( – Scholar search). Mathematics Magazine 39 (3): 135–141.
External links
- Eric W. Weisstein, Automorphism at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Dictionary
automorphism
-noun
- (mathematics) An isomorphism of a mathematical object or system of objects onto itself.
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