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auto- +‎ morphism


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, Volker Diekert, Bruno Durand (editors), STACS 2005: 22nd Annual Symposium on Theoretical Aspects of Computer Science, Springer, LNCS 3404, page 1,
      In mathematics, automorphisms of algebraic structures play an important role. Automorphisms capture the symmetries inherent in the structures and many important results have been proved by analyzing the automorphism group of the structure.
    • 2014, Alexei Belov, Leonid Bokut, Louis Rowen, Jie-Tai Yu, The Jacobian Conjecture, Together with Specht and Burnside-Type Problems, 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 notesEdit

  • (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.


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



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