Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 9 / 1 / g / Solution

{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.

Codex @codex  0

 Incoming links: Right-invariant group metric