Definition:Group of Outer Automorphisms

Definition
Let $G$ be a group.

Let $\Aut G$ be the group of automorphisms of $G$.

Let $\Inn G$ be the group of inner automorphisms of $G$.

Let $\dfrac {\Aut G} {\Inn G}$ be the quotient group of $\Aut G$ by $\Inn G$.

Then $\dfrac {\Aut G} {\Inn G}$ is called the group of outer automorphisms of $G$.

This group $\dfrac {\Aut G} {\Inn G}$ is often denoted $\map {\operatorname {Out} }G$.

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