Automorphism

Automorphisme (Fr). Automorfismo (It). 自己同形 (Ja).

Definition

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 (u) * f (v) = f (u * v)

An automorphism always maps the identity to itself. The image under an automorphism of a conjugacy class is always a conjugacy class (the same or another). The image of an element has the same order as that element.

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

Inner automorphism

An inner automorphism of a group G is a function

f : G → G

defined by

f(x) = axa⁻¹

where a is a given fixed element of G, for all x in G.

The operation axa⁻¹ is called conjugation (see also conjugacy class).

Outer 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 Out(G).

For all Abelian groups there is at least the automorphism that replaces the group elements by their inverses. All other automorphisms are outer automorphisms. Non-Abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.