Inner Automorphisms form Normal Subgroup of Automorphism Group
Theorem
Let $G$ be a group.
Then the set $\Inn G$ of all inner automorphisms of $G$ is a normal subgroup of the automorphism group $\Aut G$ of $G$:
- $\Inn G \lhd \Aut G$
Proof
Let $G$ be a group whose identity is $e$.
From Inner Automorphisms form Subgroup of Automorphism Group, $\Inn G$ forms a subgroup of $\Aut G$.
It remains to be shown that $\Inn G$ is normal in $\Aut G$.
Let $\kappa_x: G \to G$ be the inner automorphism defined as:
- $\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
Let $\phi \in \Aut G$.
If we can show that:
- $\forall \phi \in \Aut G: \forall \kappa_x \in \Inn G: \phi \circ \kappa_x \circ \phi^{-1} \in \Inn G$
then by the Normal Subgroup Test:
- $\Inn G \lhd \Aut G$
Fix $\kappa_x \in \Inn G$.
We claim $\phi \circ \kappa_x \circ \phi^{-1} = \kappa_{\map \phi x}$.
Since $\phi \in \Aut G$ then $\phi$ is, in particular, a homomorphism.
Therefore:
\(\ds \forall g \in G: \, \) | \(\ds \map {\paren {\phi \circ \kappa_x \circ \phi^{-1} } } g\) | \(=\) | \(\ds \map {\paren {\phi \circ \kappa_x} } {\map {\phi^{-1} } g}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {x \phi^{-1} \paren g x^{-1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi x \map \phi {\map {\phi^{-1} } g} \map \phi {x^{-1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi x g \map \phi x^{-1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\kappa_{\map \phi x} } g\) |
Therefore:
- $\phi \circ \kappa_g \circ \phi^{-1} = \kappa_{\map \phi g} \in \Inn G$
Since $\kappa_x \in \Inn G$ and $\phi \in \Aut G$ were arbitrary:
- $\Inn G \lhd \Aut G$
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $3$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.8 \ \text{(b)}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64 \alpha$