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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[1], 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[2], 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[3], 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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