  By the Heyde theorem, the Gaussian distribution on the real line is
characterized by the symmetry of the conditional distribution of one linear
form of of $n$ independent random variables given another. When $n=2$ we prove
analogues of this theorem in the case when independent random variables take
values in a locally compact Abelian group $X$ and coefficients of the linear
forms are topological automorphisms of $X$.

Probability;Group Theory