What is an outer automorphism? I know that an automorphism of a group is just an isomorphism of that group with itself. I know that an _inner_ automorphism is an isomorphism/automorphism of the form $\phi_g$ where $\phi_g(x) = gxg^{-1}$. I have heard people talking about an outer automorphism. What are these?

An **outer automorphism** is just an automorphism that is not an inner automorphism. But be careful: We know that the group $\operatorname{Inn}(G)$ of inner automorphisms is a normal subgroup of the group $\operatorname{Aut}(G)$ of all automorphisms of $G$, and the quotient $\operatorname{Out}(G)=\operatorname{Aut}(G)/\operatorname{Inn}(G)$ is called the **outer automorphism group**. So confusingly, the elements of the outer automorphism group are _not_ outer automorhims, but rather equivalence classes of outer automorphisms ...