  Let $X$ be a countable discrete Abelian group containing no elements of order
2, $\alpha$ be an automorphism of $X$, $\xi_1$ and $\xi_2$ be independent
random variables with values in the group $X$ and distributions $\mu_1$ and
$\mu_2$. The main result of the article is the following statement. The
symmetry of the conditional distribution of the linear form $L_2 = \xi_1 +
\alpha\xi_2$ given $L_1 = \xi_1 + \xi_2$ implies that $\mu_j$ are shifts of the
Haar distribution of a finite subgroup of $X$ if and only if the automorphism
$\alpha$ satisfies the condition ${\rm Ker}(I+\alpha)=\{0\}$. This theorem is
an analogue for discrete Abelian groups the well-known Heyde theorem where
Gaussian distribution on the real line is characterized by the symmetry of the
conditional distribution of one linear form of independent random variables
given another. We also prove some generalisations of this theorem.
