The Skitovich-Darmois theorem for finite Abelian groups

Let X be a finite Abelian group, xi_i, i=1,2,…,n,n>1, be independent random variables with values in X and distributions mu_i. Let alpha_{ij},i,j=1,2,…,n, be automorphisms of X. We prove that the independence of n linear forms L_j=alpha_{1j}xi_1+alpha_{2j}xi_2+…+alpha_{nj}xi_n implies that all mu_i are shifts of the Haar distributions on some subgroups of the group X. This theorem is an analogue of the Skitovich-Darmois theorem for finite Abelian groups.