Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group

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 $10$