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According to the well-known Heyde theorem the Gaussian distribution on thereal line is characterized by the symmetry of the conditional distribution ofone linear form of independent random variables given another. We studyanalogues of this theorem for some locally compact Abelian groups. We considerlinear forms of two independent random variables with values in a locallycompact Abelian group X. We assume that the characteristic functions of theseindependent random variables do not vanish. Unlike most previous works, we donot impose any restrictions on coefficients of the linear forms. They arearbitrary topological automorphisms of X.

Generalization of the Heyde theorem to finite Abelian groups and groups of the form RxG, where G is a finite Abelian group