The d'Alembert functional equation on metabelian groups (Q1296954)

The author offers the general solution of the functional equation \(f(xy)+f(xy^{-1})=2f(x)f(y),\) where \(f\) maps a metabelian group into a quadratically closed field. In the proof he eliminates the often used condition \(f(xyz)=f(xzy)\) by restricting \(f\) to a subgroup. Regarding this idea see also \textit{L. Székelyhidi}, Convolution type functional equations on topological abelian groups (1991; Zbl 0748.39003), pp. 136-137.