Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group
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Theorem
Let $G$ be a group.
Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as:
- $\forall x \in G: \map \kappa x := \kappa_x$
where $\kappa_x$ is the inner automorphism on $x$:
- $\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
Then $\Img \kappa$ is the inner automorphism group of $G$.
Proof
Let $\Inn G$ denote the inner automorphism group of $G$.
For all $x \in G$, $\map \kappa x = \kappa_x \in \Inn G$.
Hence $\Img \kappa \subseteq \Inn G$.
Let $\phi \in \Inn G$. Then:
- $\exists y \in G: \forall g \in G: \map \phi g = y g y^{-1}$
Then $\map \kappa y = \phi$.
Hence $\Inn G \subseteq \Img \kappa$.
Therefore $\Img \kappa = \Inn G$.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $10$