Automorphism

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.

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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.

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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.

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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.

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