TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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$Inn (G)$ is a normal subgroup of $Aut (G)$

Statement

$Inn (G)$ is a normal subgroup of $Aut (G)$ .

Facts & Assumptions

Given: A group $G$ .

[L1]

Inner automorphisms are the maps $c_{g}$ (Inner automorphisms and $Inn (G)$ ).

[L2]

$Aut (G)$ is a group under composition (The automorphisms of a group form a group under composition).

[L3]

A conjugation-stable subgroup is normal (Equivalent characterisations of a normal subgroup by conjugates and left and right cosets).

[L4]

The inverse of a bijective homomorphism is a homomorphism (The inverse of a bijective group homomorphism is a group homomorphism).

Proof

technique · direct

1.1

For $φ \in Aut (G)$ and $c_{g} \in Inn (G)$ , direct evaluation gives $(φ c_{g} φ^{- 1}) (x) = φ (g) x φ (g)^{- 1} = c_{φ (g)} (x)$ .

L1L2L3L4givenalgebra

2.1

Thus conjugation by every element of $Aut (G)$ carries $Inn (G)$ into itself; applying the same statement to $φ^{- 1}$ gives equality.

step 1.1L1L2L3L4givenalgebra

3.1

The conjugation closure in step 2.1 proves normality.

step 2.1∎

Depends on

Used by

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

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Milne, Group Theory, Automorphisms of Groups (standard reference, not scraped)

Inn⁡(G)Inn(G) is a normal subgroup of Aut⁡(G)Aut(G)