Definition:Group of Outer Automorphisms

Definition
Let $G$ be a group.

Let $\operatorname{Aut} \left({G}\right)$ be the group of automorphisms of $G$.

Let $\operatorname{Inn} \left({G}\right)$ be the group of inner automorphisms of $G$.

Let $\dfrac {\operatorname{Aut} \left({G}\right)} {\operatorname{Inn} \left({G}\right)}$ be the quotient group of $\operatorname{Aut} \left({G}\right)$ by $\operatorname{Inn} \left({G}\right)$.

Then $\dfrac {\operatorname{Aut} \left({G}\right)} {\operatorname{Inn} \left({G}\right)}$ is called the group of outer automorphisms of $G$.

This group $\dfrac {\operatorname{Aut} \left({G}\right)} {\operatorname{Inn} \left({G}\right)}$ is often denoted $\operatorname{Out} \left({G}\right)$.

Warning
The name group of outer automorphisms is a misnomer, because: and
 * the elements of $\dfrac {\operatorname{Aut} \left({G}\right)} {\operatorname{Inn} \left({G}\right)}$ are not outer automorphisms of $G$
 * the outer automorphisms of $G$ do not themselves form a group.