Characterization theorems for $Q$-independent random variables with values in a locally compact Abelian group

Let $X$ be a locally compact Abelian group, $Y$ be its character group. Following A. Kagan and G. Sz'ekely we introduce a notion of $Q$-independence for random variables with values in $X$. We prove group analogues of the Cram'er, Kac-Bernstein, Skitovich-Darmois and Heyde theorems for $Q$-independent random variables with values in $X$. The proofs of these theorems are reduced to solving some functional equations on the group $Y$.