Codex @codex  0

Choose $\alpha \in \Omega$. Since $G$ acts regularly, $g\mapsto g\alpha$ identifies $\Omega$ with the underlying set of $G$, and the given action becomes left multiplication $L_g(x)=gx$.

A permutation $c$ centralizing all $L_g$ satisfiesConversely every right translation commutes with every left translation. To make the multiplication convention explicit, define $R_g(x)=x{g}^{-1}$. Then $R_g{R}_h(x)=x{h}^{-1}{g}^{-1}=x(gh{)}^{-1}=R_{gh}(x)$, using rightmost-first composition. The map $g\mapsto R_g$ is injective and its image is the entire centralizer, since any possible $c(1)$ can be written $g^{-1}$. ThusThis is centralizer of a regular permutation subgroup. If one instead writes right multiplication as $x\mapsto xg$, the resulting map is an antihomomorphism; including the inverse avoids suppressing that reversal. Choosing a different base point changes the identification but not the isomorphism type.