Outer Automorphism Group

For a group $G$, the outer automorphism group is the quotient

where $\operatorname{Aut}(G)$ is the automorphism group and $\operatorname{Inn}(G)$ is the subgroup of inner automorphisms . This is a quotient group , and it measures the “new” automorphisms not coming from conjugation.

Saying $\operatorname{Out}(G)$ is trivial means every automorphism of $G$ is inner.

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 the trivial group.
• For many groups, $\operatorname{Out}(G)$ is small even when $\operatorname{Aut}(G)$ is large, reflecting that “most” automorphisms are induced by conjugation.