automorphism

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, an automorphism is an
isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

from a
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
to itself. It is, in some sense, a
symmetry Symmetry (from Greek συμμετρία ''symmetria'' "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 pre ...

of the object, and a way of
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, called the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
. It is, loosely speaking, the
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of the object.

# Definition

In the context of
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a mathematical object is an
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
such as a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, ring, or
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. An automorphism is simply a
bijective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 ''homom ...
of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example,
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,
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...
, and
linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
). The
identity morphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
( 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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

(in the categorical sense of the word). This is a very abstract definition since, in category theory, morphisms are not necessarily Function (mathematics), 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 (set theory), class), then they form a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
under Function composition, composition of morphisms. This group is called the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
of . ;Closure (binary operation), Closure: Composition of two automorphisms is another automorphism. ;Associativity: It is part of the definition of a Category (mathematics), category that composition of morphisms is associative. ;Identity element, Identity: The identity is the identity morphism from an object to itself, which is an automorphism. ;Inverse element, 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.

# Examples

* In set theory, 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 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. * 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 (mathematics), image is the group Inn(''G'') of inner automorphisms and whose kernel (algebra), kernel is the center (group theory), center of ''G''. Thus, if ''G'' has Trivial group, trivial center it can be embedded into its own automorphism group. * In linear algebra, an endomorphism of a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''V'' is a linear transformation, 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 Endomorphism algebra, all endomorphisms of ''V'' is itself an algebra over the same base field as ''V'', whose Group of units, invertible elements precisely consist of GL(''V'').) * A field automorphism is a bijection, bijective
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...
from a field (mathematics), field to itself. In the cases of the rational numbers (Q) and the real numbers (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 numbers, C, there is a unique nontrivial automorphism that sends R into R: complex conjugate, complex conjugation, but there are infinitely (uncountable, uncountably) many "wild" automorphisms (assuming the axiom of choice). 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 Quaternions_and_spatial_rotation, isomorphic to SO(3), the group of rotations in 3-dimensional space. * The automorphism group of the octonions (O) is the Exceptional Lie algebra, exceptional Lie group G2_(mathematics), G2. * In graph theory an graph automorphism, 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, an automorphism may be called a motion (geometry), motion 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 biholomorphy, 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, morphisms between topological spaces are called Continuous function (topology), continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism (see homeomorphism group). 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 group (mathematics), groups, ring (mathematics), 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 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 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 that contain the outer automorphisms. The same definition holds in any unital algebra, unital ring or algebra over a field, algebra where ''a'' is any Unit (ring theory), invertible element. For Lie algebras the definition is slightly different.