On a functional equation on groups in two variables (Q6562892)

Let \(G\) be a (noncommutative) 2-divisible group. A two-variable function \(f\colon G\times G\to\mathbb{C}\) is called central if and only if \(f(x_1y_1,x_2)=f(y_1x_1,x_2)\) and \(f(x_1,x_2y_2)=f(x_1,y_2x_2)\) for all \(x_1,x_2,y_1,y_2\in G\).\N\NThe authors solve, in the class of central functions \(f,g\colon G\to \mathbb{C}\), the functional equation \N\[ f(x_1y_1,x_2y_2)+g(x_1y_1^{-1},x_2)+f(x_1,x_2y_2^{-1})=f(x_1y_1^{-1},x_2y_2^{-1})+g(x_1y_1,x_2)+f(x_1,x_2y_2) \N\]\Nfor all \(x_1,x_2,y_1,y_2\in G\). A pair \(f,g\) of central functions is a solution of the above equation if and only if \N\[\N f(x_1,x_2)=A(x_1,x_2)+\alpha(x_1)+\beta_1(x_2),\qquad x_1,x_2\in G,\N\]\N\[\Ng(x_1,x_2)=A(x_1,x_2)+\alpha(x_1)+\beta_2(x_2),\qquad x_1,x_2\in G,\N\]\Nwhere \(A\colon G\times G\to\mathbb{C}\) is a bihomomorphism and \(\alpha,\beta_1,\beta_2\colon G\to\mathbb{C}\) are central (one-variable) functions.\N\NUsing the above result, solutions of two other functional equations (which are modifications of the given one) are also derived.