The Heyde Theorem on a Group $$\mathbb {R}^n \times D$$, Where D is a Discrete Abelian Group

Heyde proved that the Gaussian distribution on real line is characterized by symmetry of conditional one linear form given another. The present article devoted to a group analogue theorem. We describe distributions independent random variables $$\xi _1$$ and _2$$ with values in $$X=\mathbf {R}^n\times D$$ , where D discrete Abelian group, which are $$L_2 = \xi _1 + \delta $$L_1 $$\delta $$ topological automorphism X such $$\mathrm{Ker}(I+\delta )=\{0\}$$ .