Jensen's functional equation on the symmetric group Sₙ

الأصل · EN

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⁻¹x) = 2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers Ng1, Ng2, Ng3, C. T. Ng has solved these functional equations for the case where G is a free group and the linear group GLₙ(R), R=,, a quadratically closed field or a finite field. He has also mentioned, without detailed proof, in the above papers and in Ng4 that when G is the symmetric group Sₙ the group of all solutions of these functional equations coincides with the group of all homomorphisms from (Sₙ,·) to (H,+). The aim of this paper is to give an elementary and direct proof of this fact.