Right-invariant group metric ($d(xa,ya)=d(x,y)$)

Right translations are isometries for a right-invariant group metric. Given a compatible left-invariant group metric $d$, the formula $d_R(x,y)=d(x^{-1},y^{-1})$ defines a compatible right-invariant group metric, since inversion is a homeomorphism and $(xa{)}^{-1}=a^{-1}{x}^{-1}$. Compatible invariant metrics need not be Lipschitz equivalent; on the additive group $\mathbb{R}$, both $|x-y|$ and $\sqrt{|x-y|}$ are compatible invariant distances.