Center of Group is Kernel of Conjugacy Action

Theorem

Let $G$ be a group.

Let $Z$ be the kernel of the conjugacy action.

Then $Z$ is the center of $G$.

Proof

\(\ds \)

\(\)

\(\ds x \text { is in the kernel of the conjugacy action}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall y \in G: \, \)

\(\ds \)

\(\)

\(\ds x y x^{-1} = y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \forall y \in G: \, \)

\(\ds \)

\(\)

\(\ds x y = y x\)

Division Laws for Groups

\(\ds \leadstoandfrom \ \ \)

\(\ds \)

\(\)

\(\ds x \text { is in the center of } G\)

$\blacksquare$