A family of Markov shifts (almost) classified by periodic points

Let G be a finite group, and let XG = {x = (x(s,t)) Î GZ2 : x(s,t) = x(s,t-1)·x(s+1,t-1)for all (s,t) Î Z2}. The compact zero-dimensional set XG carries a natural shift Z2-action sG and the pair SG = (XG,sG) is a two-dimensional topological Markov shift. Using recent work by Crandall, Dilcher and Pomerance on the Fermat quotient, we show the following: if G is abelian, and the order of G is not divisible by 1024, nor by the square of any Wieferich prime larger than 4×1012, and H is any abelian group for which SG has the same periodic point data as SH, then G is isomorphic to H. This result may be viewed as an example of the ``rigidity'' properties of higher-dimensional Markov shifts with zero entropy.