Finite-valued invariant metrics and a classification of natural groups

Abstract

For every group $G$, of arbitrary cardinality, we construct a right-invariant metric with at most $32$ values whose isometries are exactly the permutations preserving every right-invariant metric on $G$. The proof combines subgroup-entry ranks and sign-variation colorings with a short-word rigidity theorem of Leemann and de la Salle. Their nonabelian orientation-rigidity theorem and direct regular-subgroup arguments yield the complete classification of natural groups in the right-translation sense: an abelian group $A$ is natural if and only if $2A=A$ or $2A=\{0\}$, and a nonabelian group is natural if and only if it is not generalized dicyclic. In particular, the additive group of every field is natural. The bound improves to $17$ for abelian groups and $5$ for Boolean groups, and the Boolean bound is sharp: $C_2^3$ admits no such metric with fewer than five values. Complementary constructions give one countable-valued hull metric realizing precisely the affine sign isometries simultaneously on all subgroups containing fixed coordinate markers, and signed-basis metrics with at most $p+5$ values over $\mathbb{F}_p$ for odd $p$. No other bound is claimed optimal, and no uncolored graphical regular representation is asserted.