automorphism
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an automorphism is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
from a
mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
to itself. It is, in some sense, a
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
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.


Definition

In an
algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
such as a group, a ring, or
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, an ''automorphism'' is simply a bijective
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example,
group homomorphism In mathematics, 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) \cdot h(v) whe ...
, ring homomorphism, and
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
.) More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f: X\to X is an automorphism if there is a morphism g: X\to X such that g\circ f= f\circ g = \operatorname _X, where \operatorname _X is the identity morphism of . For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the identity function, and is often called the ''trivial automorphism''.


Automorphism group

The automorphisms of an object form a group under composition of morphisms, which is called the '' automorphism group'' of . This results straightforwardly from the definition of a category. The automorphism group of an object in a category is often denoted , or simply Aut(''X'') if the category is clear from context.


Examples

* In
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, an arbitrary
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
of the elements of a set ''X'' is an automorphism. The automorphism group of ''X'' is also called the symmetric group on ''X''. * In
elementary arithmetic Elementary arithmetic is a branch of mathematics involving addition, subtraction, multiplication, and Division (mathematics), division. Due to its low level of abstraction, broad range of application, and position as the foundation of all mathema ...
, the set of
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, , 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. * 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
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
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 Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, an endomorphism of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''V'' is a
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
''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 In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
, GL(''V''). (The algebraic structure of all endomorphisms of ''V'' is itself an algebra over the same base field as ''V'', whose invertible elements precisely consist of GL(''V'').) * A field automorphism is a bijective ring homomorphism from a field to itself. **The field \Q of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s has no other automorphism than the identity, since an automorphism must fix the additive identity and the
multiplicative identity In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
; the sum of a finite number of must be fixed, as well as the additive inverses of these sums (that is, the automorphism fixes all
integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
); finally, since every rational number is the quotient of two integers, all rational numbers must be fixed by any automorphism. **The field \R of the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s has no automorphisms other than the identity. Indeed, the rational numbers must be fixed by every automorphism, per above; an automorphism must preserve inequalities since x is equivalent to \exists z\mid y-x=z^2, and the latter property is preserved by every automorphism; finally every real number must be fixed since it is the least upper bound of a sequence of rational numbers. ** The field \Complex of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s has a unique nontrivial automorphism that fixes the real numbers. It is the complex conjugation, which maps i to -i. The
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
implies the existence of uncountably many automorphisms that do not fix the real numbers. ** The study of automorphisms of algebraic field extensions is the starting point and the main object of
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
. * The automorphism group of the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s () as a ring are the inner automorphisms, by the Skolem–Noether theorem: maps of the form . This group is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to SO(3), the group of rotations in 3-dimensional space. * The automorphism group of the octonions () is the exceptional
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
G2. * In
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
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
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, an automorphism may be called a
motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...
of the space. Specialized terminology is also used: ** In metric 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 biholomorphic map (also called a
conformal map In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...
), from a surface to itself. For example, the automorphisms of the Riemann sphere are Möbius transformations. ** An automorphism of a differentiable
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
''M'' is a diffeomorphism from ''M'' to itself. The automorphism group is sometimes denoted Diff(''M''). ** In
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
of the space to itself, or self-homeomorphism (see
homeomorphism group In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. They are important to the theory of top ...
). In this example it is ''not sufficient'' for a morphism to be bijective to be an isomorphism.


History

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:
so that \mu is a new fifth root of unity, connected with the former fifth root \lambda by relations of perfect reciprocity.


Inner and outer automorphisms

In some categories—notably groups, rings, and
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms. In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. For each element ''a'' of a group ''G'', conjugation by ''a'' is the operation given by (or ''a''−1''ga''; usage varies). One can easily check that conjugation by ''a'' is a group automorphism. The inner automorphisms form a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of Aut(''G''), denoted by Inn(''G''); this is called Goursat's lemma. The other automorphisms are called outer automorphisms. The
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
is usually denoted by Out(''G''); the non-trivial elements are the cosets that contain the outer automorphisms. The same definition holds in any unital ring or
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
where ''a'' is any invertible element. For
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s the definition is slightly different.


See also

* Antiautomorphism * Automorphism (in Sudoku puzzles) * Characteristic subgroup * Endomorphism ring * Frobenius automorphism * Morphism * Order automorphism (in
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
). * Relation-preserving automorphism * Fractional Fourier transform


References


External links


''Automorphism'' at Encyclopaedia of Mathematics
* {{MathWorld , urlname=Automorphism , title = Automorphism Morphisms Abstract algebra Symmetry