Kernel of the second order Cauchy difference on groups (Q372383)

Let (\(G, \cdot\)) be a group, (\(H, +\)) be an abelian group, \(f : G \to H\), and \[ C^{(2)} f(x,y,z) = f(xyz)-f(xy)-f(yz)-f(xz) + f(x) + f(y) + f(z) \] be the second-order Cauchy difference of the function \(f\). In this paper, the authors determine the general solution of the functional equation \(C^{(2)} f(x,y,z) = 0\) for all \(x, y, z \in G\), when \(G\) is a free group. They also present the solutions of this equation on other selected groups such as symmetric groups \(S_n\), finite cyclic groups \(C_n\), and dihedral groups \(D_n\). This reviewer finds this paper very interesting.