automorphism

automorphism

English

Etymology

auto- +‎ morphism

Noun

automorphism (plural automorphisms)

1. (algebra) An isomorphism of a mathematical object or system of objects onto itself.
   • 1971, Norman Biggs, Finite Groups of Automorphisms: Course Given at the University of Southampton‎, Cambridge University Press, page 25:

     Since every linear automorphism of V fixes 0 our interest in the transitivity properties of GL(V) is confined to its action on V* = V - {0}. GL(V) is transitive on V* since any two elements of V* may be chosen as the initial members of two ordered bases; it is not in general 2-transitive because there is no linear automorphism taking an independent pair to a dependent pair.

   • 2005, Maninda Agrawal, Nitin Saxena, “Automorpisms of Finite Rings and Applications to Complexity of Problems”, in Volker Diekert, Bruno Durand, editors, STACS 2005: 22nd Annual Symposium on Theoretical Aspects of Computer Science‎, Springer, LNCS 3404, page 1:

   • 2014, Alexei Belov, Leonid Bokut, Louis Rowen, Jie-Tai Yu, “The Jacobian Conjecture, Together with Specht and Burnside-Type Problems”, in Ivan Cheltsov, Ciro Ciliberto, Hubert Flenner, James McKernan, Yuri G. Prokhorov, Mikhail Zaidenberg, editors, Automorphisms in Birational and Affine Geometry‎, Springer, page 274:

     A tame automorphism is a product of elementary automorphisms, and a non-tame automorphism is called wild. The “tame automorphism problem” asks whether any automorphism is tame.

2. The ascription to others of one's own characteristics.

Usage notes

• (algebra):
  • An automorphism is characterised by the structure it preserves, which is usually specified as an object type. Thus one may speak of a group automorphism or ring automorphism.
  • The identity mapping is sometimes called the trivial automorphism; any other automorphism may then be called a nontrivial automorphism.

Synonyms

• (isomorphism of a mathematical object or system of objects onto itself): self-map
• (ascription to others of one's own characteristics): projection

Hypernyms

• (algebra): isomorphism, endomorphism

Hyponyms

• (algebra): inner automorphism, outer automorphism, Möbius transformation

Derived terms

• automorphism group

Translations

isomorphism of a mathematical object or system of objects onto itself

Catalan: automorfisme m Danish: automorfi c Finnish: automorfismi German: Automorphismus m Icelandic: sjálfmótun f Irish: uathmhorfacht f Italian: automorfismo (it) m Portuguese: automorfismo m Russian: автоморфи́зм (ru) m (avtomorfízm) Serbo-Croatian: automorfizam (sh) m Spanish: automorfismo m Swedish: automorfi (sv), automorfism (sv)

ascription to others of one's own characteristics

See also

• self-map

Further reading

• Automorphism in the Encyclopædia Britannica (11th edition, 1911)