# automorphism
## English
### Etymology

From auto- +‎ morphism.

### Pronunciation

* Audio (US):(file)

### Noun

automorphism (plural automorphisms)
* (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: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”, 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.
* The ascription to others of one's own characteristics or of one's own perceived characteristics.
** 1895 Hiram M. Stanley: Studies in the Evolutionary Psychology of Feeling. MacMillan
    Sensation for us is a complex of sensations plus perceptions and other cognitive and emotional elements which lie beyond early mind, but which by an inevitable automorphism we interpret into early forms. This automorphism with the child is complete, and is never perfectly effaced even in the most accomplished psychologist. . . . But when we come to interpret the psychoses of the lower animals in connection with sexuality we may still more easily slip into a doubtful automorphism. Thus to say with Darwin, "When we behold a male bird elaborately displaying ... before the female, ... it is impossible to doubt that she admires the beauty of her male partner" (Descent of Man), or more strongly still with Grant Allen, "Every crow must think its own mate beautiful" (Mind, v. 448), we too easily take for granted that these birds would feel like ourselves in corresponding circumstances.

#### 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

* antiautomorphism
* automorphism group
* gyroautomorphism

#### Translations
### See also

* anthropomorphism
* self-map

### Further reading

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