Counterexamples to a Proposed Stam Inequality on Finite Groups

Gibilisco and Isola have recently proposed a definition of Fisher information for random variables taking values in a finite group that is analogous to the definition for real valued random variables with a density. Based on this Fisher information concept, they claim to prove a Stam inequality for finite-group valued random variables that is analogous to the one in the case of real values. In this note we show these results, unfortunately, do not hold for nonabelian groups in general, by constructing explicit counterexamples.