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
- The identity element of G is mapped to itself.
- Subgroups are mapped to subgroups, normal subgroups to
- 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
The inner automorphisms
The inner automorphism group is isomorphic to the quotient of G by its center Z(G). In particular, for Abelian groups the inner automorphism group consists just of the trivial automorphism.
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.