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 : G → G 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.