D'Alembert's functional equations on metabelian groups

Summary. We extend d'Alembert's classical functional equation by replacing the domain of definition Bbb R {Bbb R} of the solutions by a metabelian group G and simultaneously replacing the group involution by an arbitrary involution of G. We find all complex valued solutions. In particular we show that the continuous solutions have the same form as in the abelian case if G is connected.