Automorphism

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.
Contents
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.
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).
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. (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 (nonidentity) 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. 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.
 Associativity: composition of morphisms is always associative.
 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.
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.
 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 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.^{[1]}
 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. When the vector space is finitedimensional, the automorphism group of V is the same as the general linear group, GL(V).
 A field automorphism is a bijective ring homomorphism from a 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 conjugation, but there are infinitely (uncountably) many "wild" automorphisms (assuming the axiom of choice).^{[2]} 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.
 In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and nonedges. In particular, if two nodes are joined by an edge, so are their images under the permutation.
 For relations, see relationpreserving automorphism.
 In order theory, see order automorphism.
 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 selfhomeomorphism (see homeomorphism group). In this example it is not sufficient for a morphism to be bijective to be an isomorphism.
 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 selfisometry. The automorphism group is also called the isometry group.
 In the category of Riemann surfaces, an automorphism is a bijective biholomorphic map (also called a conformal map), from a surface to itself. For example, the automorphisms of the Riemann sphere are Möbius transformations.
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,^{[3]} writing:
so that μ is a new fifth root of unity, connected with the former fifth root λ by relations of perfect reciprocity.
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 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 φ_{a} : G → G given by φ_{a}(g) = aga^{−1} (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 Aut(G) / Inn(G) is usually denoted by Out(G); the nontrivial elements are the cosets that contain the outer automorphisms.
The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different.
See also
 endomorphism ring
 antiautomorphism
 Frobenius automorphism
 morphism
 characteristic subgroup
References
 ^ PJ Pahl, R Damrath (2001). "§7.5.5 Automorphisms". Mathematical foundations of computational engineering (Felix Pahl translation ed.). Springer. p. 376. ISBN 3540679952. http://books.google.com/?id=kvoaoWOfqd8C&pg=PA376.
 ^ Yale, Paul B. (May 1966). "Automorphisms of the Complex Numbers". Mathematics Magazine 39 (3): 135–141. doi:10.2307/2689301. JSTOR 2689301. http://mathdl.maa.org/images/upload_library/22/Ford/PaulBYale.pdf.
 ^ Sir William Rowan Hamilton (1856). "Memorandum respecting a new System of Roots of Unity". Philosophical Magazine 12: 446. http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf.
External links
 Weisstein, Eric W., "Automorphism" from MathWorld.
Wikimedia Foundation. 2010.
Look at other dictionaries:
Automorphism — Au to*mor phism, n. Automorphic characterization. H. Spenser. [1913 Webster] … The Collaborative International Dictionary of English
automorphism — /aw teuh mawr fiz euhm/, n. Math. an isomorphism from a given set to itself. Cf. inner automorphism, outer automorphism. [1870 75; AUTO 1 + MORPH + ISM] * * * ▪ mathematics in mathematics, a correspondence that associates to every element… … Universalium
automorphism — /aw teuh mawr fiz euhm/, n. Math. an isomorphism from a given set to itself. Cf. inner automorphism, outer automorphism. [1870 75; AUTO 1 + MORPH + ISM] … Useful english dictionary
automorphism — noun Etymology: aut + isomorphism Date: 1903 an isomorphism of a set (as a group) with itself … New Collegiate Dictionary
automorphism — noun An isomorphism of a mathematical object or system of objects onto itself … Wiktionary
Outer automorphism group — In mathematics, the outer automorphism group of a group G is the quotient Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually … Wikipedia
Inner automorphism — In abstract algebra an inner automorphism is a function which, informally, involves a certain operation being applied, then another one (x) performed, and then the initial operation being reversed. Sometimes this has a net effect ( take off shoes … Wikipedia
Family automorphism — In mathematics, in the realm of group theory, an automorphism of a group is termed a family automorphism if it takes every element to an element generating a conjugate subgroup. In symbols, an automorphism sigma of a group G is a family… … Wikipedia
Extensible automorphism — In mathematics, an automorphism of a structure is said to be extensible if, for any embedding of that structure inside another structure, the automorphism can be lifted to the bigger structure.In group theory, an extensible automorphism of a… … Wikipedia
Class automorphism — In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the automorphism group. Some facts: Every inner… … Wikipedia