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Two natural extensions of Jensen's functional equation on the real line arethe equations $f(xy)+f(xy^{-1}) = 2f(x)$ 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 hassolved these functional equations for the case where $G$ is a free group andthe linear group $GL_n(R)$, $R=\z,\r$, a quadratically closed field or a finitefield. He has also mentioned, without detailed proof, in the above papers andin \cite{Ng4} that when $G$ is the symmetric group $S_n$ the group of allsolutions of these functional equations coincides with the group of allhomomorphisms from $(S_n,\cdot)$ to $(H,+)$. The aim of this paper is to givean elementary and direct proof of this fact.

Jensen's functional equation on the symmetric group $\bold{S_n}$