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.

An automorphism of the Klein four-group shown as a mapping between two Cayley graphs, a permutation in cycle notation, and a mapping between two Cayley tables.

Definition edit

In an algebraic structure such as a group, a ring, or vector space, an automorphism is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.)

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   is an automorphism if there is a morphism   such that   where   is the identity morphism of X. 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 edit

The automorphisms of an object X form a group under composition of morphisms, which is called the automorphism group of X. This results straightforwardly from the definition of a category.

The automorphism group of an object X in a category C is often denoted AutC(X), or simply Aut(X) if the category is clear from context.

Examples edit

  • 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 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 VV. 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.
    • The field   of the rational numbers has no other automorphism than the identity, since an automorphism must fix the additive identity 0 and the multiplicative identity 1; the sum of a finite number of 1 must be fixed, as well as the additive inverses of these sums (that is, the automorphism fixes all integers); finally, since every rational number is the quotient of two integers, all rational numbers must be fixed by any automorphism.
    • The field   of the real numbers has no other automorphism than the identity. Indeed, the rational numbers must be fixed by every automorphism, per above; an automorphism must preserve inequalities since   is equivalent to   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   of the complex numbers, has a unique nontrivial automorphism, the that sends   into   the complex conjugation, but there are infinitely (uncountably) many "wild" automorphisms, if the axiom of choice is assumed.[2][3]
    • The study of automorphisms of algebraic field extensions is the starting point and the main object of Galois theory.
  • The automorphism group of the quaternions (H) as a ring are the inner automorphisms, by the Skolem–Noether theorem: maps of the form abab−1.[4] 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 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:

History edit

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,[5] 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 edit

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 φa : GG given by φa(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.

See also edit

References edit

  1. ^ 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.
  2. ^ Yale, Paul B. (May 1966). "Automorphisms of the Complex Numbers" (PDF). Mathematics Magazine. 39 (3): 135–141. doi:10.2307/2689301. JSTOR 2689301.
  3. ^ Lounesto, Pertti (2001), Clifford Algebras and Spinors (2nd ed.), Cambridge University Press, pp. 22–23, ISBN 0-521-00551-5
  4. ^ Handbook of algebra, vol. 3, Elsevier, 2003, p. 453
  5. ^ Sir William Rowan Hamilton (1856). "Memorandum respecting a new System of Roots of Unity" (PDF). Philosophical Magazine. 12: 446. Archived (PDF) from the original on 2022-10-09.

External links edit