Outer automorphism group

Definition

Let $G$ be a group. The outer automorphism group of $G$ is the quotient group $\operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G)$.

Remarks

The quotient exists because $\operatorname{Inn}(G)$ is a normal subgroup of $\operatorname{Aut}(G)$. An automorphism that is not inner is called an outer automorphism, but the elements of $\operatorname{Out}(G)$ are cosets, not automorphisms. For an abelian group, $\operatorname{Inn}(G)$ is trivial, so $\operatorname{Out}(G) \cong \operatorname{Aut}(G)$. The symmetric group $S_6$ is the only symmetric group $S_n$ with nontrivial outer automorphism group.

Explanations

L1 human

Some symmetries of a group come from conjugation by its own elements, and some do not. The outer automorphism group keeps only the second kind, up to the first kind. For many groups it is small or trivial.