For a finite Abelian group G define the two-dimensional Markov shift X(G) = (x is-an-element-of G(Z2): x(i,j) + x(i+1,j) + x(i,j+1) = 0 for all (i,j) is-an-element-of Z2}. Let mu(G) be the Haar measure on the subgroup X(G) subset-of G(Z2). The group Z2 acts on the measure space (X(G), mu(G)) by shifts. We prove that if G1 and G2 are p-groups and E(G1) not-equal E(G2), where E(G) is the least common multiple of the orders of the elements of G, then the shift actions on (X(G1),mu(G1)) and (X(G2),mu(G2)) are not measure-theoretically isomorphic. For any finite Abelian groups G1 and G2 the shift actions on X(G1) and X(G2) are topologically conjugate if and only if G1 and G2 are isomorphic
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