# Definition:Outer Automorphism

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## Definition

Let $G$ be a group.

Let $\phi: G \to G$ be an automorphism which is not an inner automorphism.

Then $\phi$ is an **outer automorphism**.

## Also defined as

Some sources use the term **outer automorphism** to mean an element of $\dfrac {\Aut G} {\Inn G}$: the quotient group of the automorphism group by the inner automorphism group.

Because of such ambiguity in understanding what is meant, it is important to specify which definition is meant whenever the term is used.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the former definition, that is an element of $\Aut G \setminus \Inn G$, is meant.

## Sources

- 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.10$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{AA}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64$