 Centralizer of a regular permutation subgroup

Identify a regular orbit with the group using a chosen base point. A permutation commuting with all left translations satisfies $c(x)=xc(1)$. Thus the centralizer consists of right translations. Defining $R_g(x)=x{g}^{-1}$ makes $R_g{R}_h=R_{gh}$ and gives an isomorphism to $G$, rather than an unmentioned reversal of multiplication.