In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an automorphism is an
isomorphism from a
mathematical object to itself. It is, in some sense, a
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of the object, and a way of
mapping the object to itself while preserving all of its structure. The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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 the context of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a mathematical object is an
algebraic structure such as a
group,
ring, or
vector space. 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 with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example,
group homomorphism,
ring homomorphism, and
linear operator.)
The
identity morphism (
identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms.
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.
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, meaning there exists a right and left inverse endomorphism).
This is a very abstract definition since, in category theory, morphisms are not necessarily
functions and objects are not 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.
Automorphism group
If the automorphisms of an object form a set (instead of a proper
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
), then they form a
group under
composition of
morphisms. This group is called the
automorphism group of .
;
Closure: Composition of two automorphisms is another automorphism.
;
Associativity: It is part of the definition of a
category that composition of morphisms is associative.
;
Identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
: The identity is the identity morphism from an object to itself, which is an automorphism.
;
Inverses: By definition every isomorphism has an inverse that 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.
Examples
* In
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concern ...
, an arbitrary
permutation 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, the set of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, 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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
, 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 is the group Inn(''G'') of
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
s and whose
kernel is the
center of ''G''. Thus, if ''G'' has
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
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 matrice ...
, 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 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. In the cases 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 (e.g. ). The set of all ra ...
s (Q) and 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (R) there are no nontrivial field automorphisms. Some subfields of R have nontrivial field automorphisms, which however do not extend to all of R (because they cannot preserve the property of a number having a square root in R). In the case 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 fo ...
s, 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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
). 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 automorphism group of the
quaternions (H) as a ring are the inner automorphisms, by the
Skolem–Noether theorem: maps of the form . This group is
isomorphic to
SO(3), the group of rotations in 3-dimensional space.
* The automorphism group of the
octonions (O) is the
exceptional Lie group
G2.
* In
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
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, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an automorphism may be called a
motion of the space. Specialized terminology is also used:
** In
metric geometry an automorphism is a self-
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
. 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), from a surface to itself. For example, the automorphisms of the
Riemann sphere are
Möbius transformations.
** An automorphism of a differentiable
manifold ''M'' is a
diffeomorphism from ''M'' to itself. The automorphism group is sometimes denoted Diff(''M'').
** In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, morphisms between topological spaces are called
continuous maps, and an automorphism of a topological space is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
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. Homeomorphism groups are very important ...
). 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
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
in 1856, in his
icosian calculus, where he discovered an order two automorphism,
writing:
so that is a new fifth root of unity, connected with the former fifth root by relations of perfect reciprocity.
Inner and outer automorphisms
In some categories—notably
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
,
rings, and
Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.
In the case of groups, the
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
s 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 of Aut(''G''), denoted by Inn(''G''); this is called
Goursat's lemma.
The other automorphisms are called
outer automorphisms. The
quotient group is usually denoted by Out(''G''); the non-trivial elements are the
cosets
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
that contain the outer automorphisms.
The same definition holds in any
unital ring or
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
where ''a'' is any
invertible element. For
Lie algebras 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