The Heyde theorem on a group ${\mathbb{R}}^n\times D$, where $D$ is a discrete Abelian group

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