On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if ξj are independent random variables, αj, βj are nonzero constants such that βi α ±βjα−1j ≠ 0 for all i≠ j and the conditional distribution of L2=β1 ξ1 + ··· + βn ξn given L1=α1 ξ1 + ··· +αn ξn is symmetric, then all random variables ξj are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients αj,βj are automorphisms of X.