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_pG]/Fit_ZpG](M){\#M\,\leq\,\#{\bf Z}_{p}G]/{{\rm Fit}}_{{\bf Z}_{p}G]}(M)} holds. Here, Fit_ZpG](M){\rm Fit}_{{\bf Z}_{p}G]}(M) is the Z _p G]-Fitting ideal of M.