Definition:Group of Outer Automorphisms

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Definition

Let $G$ be a group.

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

Let $\Inn G$ be the inner automorphism group 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 $\Out G$.


Warning

The name group of outer automorphisms is a misnomer, because:

$(1): \quad$ the elements of $\dfrac {\Aut G} {\Inn G}$ are not outer automorphisms of $G$

and

$(2): \quad$ the outer automorphisms of $G$ do not themselves form a group.


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