Fitting ideals and the Gorenstein property

Let p be a prime number and G be a finite commutative group such that p ² does not divide the order of G. In this note we prove that for every finite module M over the group ring Z _p [G], the inequality #M  ￡  #Z_p[G]/Fit_Zp[G](M){#M,leq,#{bf Z}_{p}[G]/{{rm Fit}}_{{bf Z}_{p}[G]}(M)} holds. Here, Fit_Zp[G](M){rm Fit}_{{bf Z}_{p}[G]}(M) is the Z _p [G]-Fitting ideal of M.