# Kernel of Inner Automorphism Group is Center

## Theorem

Let the mapping $\kappa: G \to \Inn G$ from a group $G$ to its inner automorphism group $\Inn G$ be defined as:

$\map \kappa a = \kappa_a$

where $\kappa_a$ is the inner automorphism of $G$ given by $a$.

Then $\kappa$ is a group epimorphism, and its kernel is the center of $G$:

$\map \ker \kappa = \map Z G$

## Proof

Let $\kappa: G \to \Aut G$ be a mapping defined by $\map \kappa x = \kappa_x$.

It is clear that $\Img \kappa = \Inn G$.

It is also clear that $\kappa$ is a homomorphism:

 $\ds \map \kappa x \map \kappa y$ $=$ $\ds \kappa_x \circ \kappa_y$ $\ds$ $=$ $\ds \kappa_{x y}$ Inner Automorphism is Automorphism $\ds$ $=$ $\ds \map \kappa {x y}$

Note that $\forall \kappa_x \in \Inn G: \exists x \in G: \map \kappa x = \kappa_x$.

Thus $\kappa: G \to \Inn G$ is a surjection and therefore an group epimorphism.

Now we investigate the kernel of $\kappa$:

 $\ds \map \ker \kappa$ $=$ $\ds \set {x \in G: \kappa_x = I_G}$ Definition of Kernel of Group Homomorphism $\ds$ $=$ $\ds \set {x \in G: \forall g \in G: \map {\kappa_x} g = \map {I_G} g}$ Equality of Mappings $\ds$ $=$ $\ds \set {x \in G: \forall g \in G: x g x^{-1} = g}$ Definition of $\kappa_x$ $\ds$ $=$ $\ds \set {x \in G: \forall g \in G: x g = g x}$ $\ds$ $=$ $\ds \map Z G$ Definition of Center of Group

So the kernel of $\kappa$ is the center of $G$.

$\blacksquare$