Parallel spinors on flat manifolds

Let p(M) be the dimension of the vector space of parallel spinors on a closed spin manifold M. We prove that every finite group G   is the holonomy group of a closed flat spin manifold M(G)M(G) such that p(M(G))>0. If the holonomy group Hol(M)Hol(M) of M   is cyclic, then we give an explicit formula for p(M) another than that given in [R.J. Miatello, R.A. Podesta, The spectrum of twisted Dirac operators on compact flat manifolds, Trans. Am. Math. Soc., in press]. We answer the question when p(M)>0 if Hol(M)Hol(M) is a cyclic group of prime order or dim⁡M≤4dim⁡M≤4.