A Pexider-Jensen functional equation on groups (Q2567999)

Let \((G,\cdot)\) be a group, \((H,+)\) an abelian group, and \(f,g,h:G\to H.\) The Pexider-Jensen functional equation \[ f(x.y)+g(x.y^{-1})=h(x) \] is studied. The author obtains the solution of this equation on free groups and outlines a process to find the solution on other groups. Some results on Jensen's equation are extended. The results obtained are illustrated with examples which include certain linear groups, the Heisenberg group and semidirect products.