Outer Automorphism Group - Imported Knowlpedia Content

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

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

where $\operatorname{Aut}(G)$ is the automorphism group and $\operatorname{Inn}(G)$ is the subgroup of inner automorphisms.

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.

Remarks

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.