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.
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64 \alpha$