Centralizer of a regular permutation subgroup
= Centralizer of a regular permutation subgroup
{title2=$C_{\operatorname{Sym}(G)}(L(G))=R(G)\cong G$}
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 $R_g(x)=xg^{-1}$ makes $R_gR_h=R_{gh}$ and gives an isomorphism to $G$, rather than an unmentioned reversal of multiplication.