Automorphism - Examples

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.
• 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).
• 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). 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 non-edges. In particular, if two nodes are joined by an edge, so are their images under the permutation.
• For relations, see relation-preserving automorphism.
  • In order theory, see order automorphism.
• 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.