A note on a functional equation on groups with involutions in two variables

Abstract

In this paper, we determine the central solutions $f:G\times G\rightarrow {{\mathbb {C}}}$ of the functional equation

$$\begin{aligned} f(x_1\sigma y_1,x_2\tau y_2)-f(x_1\sigma y_1,x_2)-f(x_1,x_2\tau y_2)\\ =f(x_1y_1,x_2y_2)-f(x_1y_1,x_2)-f(x_1,x_2y_2) \end{aligned}$$

for all $x_{1},x_{2},y_{1},y_{2}\in G$, where G is a group, $\sigma ,\tau :G\rightarrow G$ are involutions.