Center (group Theory) - Conjugation

Conjugation

Consider the map f: G → Aut(G) from G to the automorphism group of G defined by f(g) = ϕ_g, where ϕ_g is the automorphism of G defined by

.

The function f is a group homomorphism, and its kernel is precisely the center of G, and its image is called the inner automorphism group of G, denoted Inn(G). By the first isomorphism theorem we get

The cokernel of this map is the group of outer automorphisms, and these form the exact sequence