DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The outer automorphism group $Out (G) = Aut (G) / Inn (G)$

Definition

The outer automorphism group of a group $G$ is

$Out (G) = Aut (G) / Inn (G) .$

Here $Inn (G)$ is the inner automorphism group of Inner automorphisms and $Inn (G)$ . This quotient group is the one of The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ , and it is defined because $Inn (G)$ is a normal subgroup of $Aut (G)$ proves that $Inn (G)$ is normal in $Aut (G)$ .

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Group Theory (standard reference, not scraped)

The outer automorphism group Out⁡(G)=Aut⁡(G)/Inn⁡(G)Out(G)=Aut(G)/Inn(G)