1.9: Automorphism
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 .