D'Alembert's functional equations on metabelian groups
A group is metabelian if its commutator subgroup is contained in its centre. This paper offers the general solution, mapping a metabelian group \(G\) into the multiplicative group of the nonzero complex numbers, of the (generalized) d'Alembert equation \(g(xy)+g(xh(y))=2g(x)g(y)\), \(x,y\in G,\) where \(h\) is an involution of \(G.\) Also, the equation \(g(xy)+g(yx)+g(xh(y))+g(h(x)y)=4g(x)g(y)\), \(x,y\in G\), is reduced to the above d'Alembert equation.
scientific article; zbMATH DE number 1496062