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Automorphism

  • Page ID
    17886
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    An isomorphism from a group (G,*) to itself is called an automorphism of this group. It is a bijection f : GG such that

    f (g) * f (h) = f (g * h)

    An automorphism preserves the structural properties of a group, e.g.:

    • The identity element of G is mapped to itself.
    • Subgroups are mapped to subgroups, normal subgroups to normal subgroups.
    • Conjugacy classes are mapped to conjugacy classes (the same or another).
    • The image f(g) of an element g has the same order as g.

    The composition of two automorphisms is again an automorphism, and with composition as binary operation the set of all automorphisms of a group G, denoted by Aut(G), forms itself a group, the automorphism group of G.


    Automorphism is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Online Dictionary of Crystallography.