Let A be the ring of integers of a number field K. Let \(G \subseteq GL_3(A)\) be a finite group. Let G act linearly on \(R = A[X,Y, Z]\) (fixing A) and let \(S = R^G\) be the ring of invariants. Assume the Veronese subring \(S^{<m>}\) of S is standard graded. We prove that if for all primes p dividing |G|, the Sylow p-subgroup of G has exponent p then for all \(l \gg 0\) the Veronese subring \(S^{<ml>}\) of S is Cohen-Macaulay. We prove a similar result if for all primes p dividing |G|, the prime p is unramified in K.
