On the Skitovich–Darmois theorem for discrete abelian groups

The following theorem is proved: Let $X$ be a discrete countable Abelian group, let $\xi_1,\xi_2$ be independent random variables with values in the group $X$ and with distributions $\mu_1,\mu_2$, and let $\alpha_j,\beta_j$, $j=1, 2$, be automorphisms of the group $X$. Then the independence of the linear statistics $L_1=\alpha_1\xi_1 + \alpha_2\xi_2$ and $L_2=\beta_1\xi_1 + \beta_2\xi_2$ implies that $\mu_1$ and $\mu_2$ are idempotent distributions.