  The general form of the solutions of the Kac--Bernstein functional equation
$$ f(x+y)g(x-y)=f(x)f(y)g(x)g(-y), \ x, y\in X, $$ on an arbitrary Abelian
group $X$ in the class of positive functions is obtained. We also study the
solutions of this equation in the class of complex-valued functions that do not
vanish and satisfy the Hermitian condition.
