# Kernel of Inner Automorphism Group is Center

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## 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 $G$.

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $10$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.11 \ \text{(b)}$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{AA}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $25$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Proposition $8.17$