Affine conjugation obstruction to a bi-invariant group metric
= Affine conjugation obstruction to a bi-invariant group metric
{title2=$D_aT_bD_a^{-1}=T_{ab}$}
In the <orientation-preserving affine group of the real line>, translations $T_{1/n}$ converge to the <identity element>, but <conjugation> by dilation $D_n$ sends each one to $T_1$. A compatible <bi-invariant group metric> would give $d(e,T_{1/n})=d(e,T_1)>0$ for every $n$, contradicting convergence. The group is nonetheless metrizable in its ordinary two real coordinates.