Outer Automorphism Group

For a group $G$ , the subgroup $\operatorname{Inn}(G)$ of inner automorphisms is normal in the automorphism group $\operatorname{Aut}(G)$ . The outer automorphism group is the quotient group

\operatorname{Out}(G) := \operatorname{Aut}(G)\big/\operatorname{Inn}(G),

which measures automorphisms not arising from conjugation.

Examples

If $G$ is abelian, then $\operatorname{Inn}(G)$ is trivial, so $\operatorname{Out}(G)=\operatorname{Aut}(G)$ .
If $\operatorname{Aut}(G)=\operatorname{Inn}(G)$ , then $\operatorname{Out}(G)$ is trivial.

Remarks

The group $\operatorname{Out}(G)$ is trivial exactly when every automorphism of $G$ is inner.