From Online Dictionary of Crystallography
Automorphisme (Fr). Automorphismus (Ge). Automorfismo (It). 自己同形 (Ja). Automorfismo (Sp).
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.
An inner automorphism of a group G is a function
fa : G → G
fa(g) = aga−1
for all g in G, where a is a given fixed element of G.
The operation aga−1 is called conjugation by a (see also conjugacy class).
The inner automorphisms form a normal subgroup of Aut(G), called the inner automorphism group and denoted by Inn(G).
The outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). The outer automorphism group is usually denoted by Out(G).
For Abelian groups the mapping g → g-1 is an outer automorphism, whereas for non-Abelian groups this mapping is not even a homomorphism.