Jensen's functional equation on the symmetric group ${\mathbf{S}}_{\mathbf{n}}$

Two natural extensions of Jensen’s functional equation on the real line are the equations $f(xy)+f(x{y}^{-1})=2f(x)$ and $f(xy)+f(y^{-1}x)=2f(x)$, where $f$ is a map from a multiplicative group $G$ into an abelian additive group $H$. In a series of papers \cite{Ng1}, \cite{Ng2}, \cite{Ng3}, C. T. Ng has solved these functional equations for the case where $G$ is a free group and the linear group $G{L}_n(R)$, R=\z,\r, a quadratically closed field or a finite field. He has also mentioned, without detailed proof, in the above papers and in \cite{Ng4} that when $G$ is the symmetric group $S_n$ the group of all solutions of these functional equations coincides with the group of all homomorphisms from $(S_n,\cdot )$ to $(H,+)$. The aim of this paper is to give an elementary and direct proof of this fact.