An alternative functional equation related to the quadratic equation.

Given a rational number α ≠ 2, we establish a criterion for the existence of the general solution of an alternative quadratic functional equation of the form f (x y) + f (x y-1) = 2f (x) + 2f (y) or f (x y) + f (x y-1) = αf (x) + 2f (y), where f is a mapping from an abelian group (...) to a uniquely divisible abelian group (...). We also find the general solution in the cases when G is a 6-divisible abelian group and G is a cyclic group. [ABSTRACT FROM AUTHOR]