= Right-invariant group metric
{title2=$d(xa,ya)=d(x,y)$}
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.
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