A characterization of the Gaussian distribution on Abelian groups

It is well known that the independence of two linear forms with nonzero coefficients of independent random variables implies that the random variables are Gaussian (the Skitovich-Darmois theorem). The analogous result holds true for two linear forms of independent random vectors with nonsingular matrices as coefficients (the Ghurye-Olkin theorem). In this paper we give the complete description of locally compact Abelian groups X for which the independence of two linear forms of independent random variables with values in X having distributions with nonvanishing characteristic functions (coefficients of the forms are topological automorphisms of X) implies that the random variables are Gaussian.