Kernel of Inner Automorphisms is Center

Theorem

Let the mapping $\kappa: G \to \operatorname{Inn} \left({G}\right)$ from a group $G$ to its group of inner automorphisms $\operatorname{Inn} \left({G}\right)$ be defined as:

$\kappa \left({a}\right) = \kappa_a$

...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 $G$:

$\ker \left({\kappa}\right) = Z \left({G}\right)$

Proof

Let $\kappa: G \to \operatorname{Aut} \left({G}\right)$ be a mapping defined by $\kappa \left({x}\right) = \kappa_x$.

It is clear that $\operatorname{Im} \left({\kappa}\right) = \operatorname{Inn} \left({G}\right)$.

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

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \kappa \left({x}\right) \kappa \left({y}\right)\)	\(=\)	\(\displaystyle \kappa_x \circ \kappa_y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \kappa_{x y}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Inner Automorphism is Automorphism
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \kappa \left({x y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Note that $\forall \kappa_x \in \operatorname {Inn} \left({G}\right): \exists x \in G: \kappa \left({x}\right) = \kappa_x$.

Thus $\kappa: G \to \operatorname {Inn} \left({G}\right)$ is a surjection and therefore an group epimorphism.

• Now we investigate the kernel of $\kappa$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \ker \left({\kappa}\right)\)	\(=\)	\(\displaystyle \left\{ {x \in G: \kappa_x = I_G}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of kernel
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {x \in G: \forall g \in G: \kappa_x \left({g}\right) = I_G \left({g}\right)}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of equality of mappings
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {x \in G: \forall g \in G: x g x^{-1} = g}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of $\kappa_x$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {x \in G: \forall g \in G: x g = g x}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle Z \left({G}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of center

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

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $12.11 \ \text{(b)}$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$: Problem $\text{AA}$
• John F. Humphreys: A Course in Group Theory (1996): $\S 8$: Proposition $8.17$