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.

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 (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. This group is called the automorphism group of X. That this is indeed a group is simple to see:

Closure: Composition of two automorphisms is another automorphism.
Associativity: Composition of morphisms is always associative.
Identity: The identity is the identity morphism from an object to itself, which is an automorphism.
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 puzzles, automorphism exists when elements of the puzzle have a type of symmetry among the elements and their positions, such as an automorphic Sudoku.
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 finite-dimensional, the automorphism group of V is the same as the general linear group, GL(V).
An example of an automorphism is a similarity transform, which leaves the geometrical form of a figure unchanged.^[2]
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).^[3]^[4] 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 a ↦ bab⁻¹.^[5]
In graph theory 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, an automorphism may be called a 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 bijective 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 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,^[6] 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 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⁻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.

The other automorphisms are called outer automorphisms. 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.

The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different.

References

↑ PJ Pahl, R Damrath (2001). "§7.5.5 Automorphisms". Mathematical foundations of computational engineering (Felix Pahl translation ed.). Springer. p. 376. ISBN 3-540-67995-2.
↑ Klaus Maintzer: Local activity principle: The cause of complexity and symmetry breaking, Chapter 12 (pages 146–159). In: Andrew Adamatzky; Guanrong Chen (2 January 2013). Chaos, CNN, Memristors and Beyond: A Festschrift for Leon ChuaWith DVD-ROM, composed by Eleonora Bilotta. World Scientific. pp. 149–150. ISBN 978-981-4434-81-2.
↑ Yale, Paul B. (May 1966). "Automorphisms of the Complex Numbers" (PDF). Mathematics Magazine. 39 (3): 135–141. JSTOR 2689301. doi:10.2307/2689301.
↑ Lounesto, Pertti (2001), Clifford Algebras and Spinors (2nd ed.), Cambridge University Press, pp. 22–23, ISBN 0-521-00551-5
↑ Handbook of algebra, 3, Elsevier, 2003, p. 453
↑ Sir William Rowan Hamilton (1856). "Memorandum respecting a new System of Roots of Unity" (PDF). Philosophical Magazine. 12: 446.

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