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

TL;DR

This paper provides a straightforward proof that solutions to certain Jensen-like functional equations on the symmetric group are exactly the group homomorphisms, extending known results to this specific case.

Contribution

It offers an elementary, direct proof that solutions to the functional equations on the symmetric group are precisely the homomorphisms, filling a gap in the literature.

Findings

Solutions coincide with group homomorphisms on $S_n$

Provides an elementary proof for the symmetric group case

Extends previous results to $S_n$

Abstract

Two natural extensions of Jensen's functional equation on the real line are the 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 has solved these functional equations for the case where $G$ is a free group and the linear group $GL_n(R)$, $R=\z,\r$, 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.