On second differences in product form (Q1381537)

\textit{J. Aczél, J. K. Chung} and \textit{C. T. Ng} [T. M. Rassias (ed.), Topics in mathematical analysis. Singapore: World Scientific, 1--22 (1989; Zbl 0731.39010)] solved the system of functional equations \(f(xy)+f(xy^{-1})-2f(x)=g(x)h(y)\), \(f(xyz)=f(xzy)\) for functions \(f,g,h\) mapping a group into a quadratically closed field of characteristic different from 2 and 3. In partial analogy, the authors offer the general solutions \(f,g,h\), mapping the Cartesian square of a group into a quadratically closed field of characteristic different fom 2 (characteristic 3 not excluded), for the system of functional equations \[ f(xy,uv) + f(xy^{-1},uv^{-1}) - f(xy,uv^{-1}) -f(xy^{-1},uv) = g(x,u)h(y,v), \] \[ f(s,xyz) = f(s,xzy), f(xyz,t) = f(xzy,t). \] The exhaustive technicality of the proof could be alleviated by \textit{L. Székelyhidi}'s remark [Lemma 2 in Pitman Res. Notes Math. Ser. 376, 266--269 (1997; Zbl 1049.47041); see also his book ``Convolution type functional equations on topological abelian groups''. Singapore etc.: World Scientific (1991; Zbl 0748.39003), pp. 136--137] that the last two equations (called here (KC) = Kannappan condition), which help solve the equation preceding them for noncommutative groups, can be used to directly reduce the noncommutative case to the commutative one (also in the first paper quoted). Connections to known results are also established.