Jensen's functional equation on groups (Q1263751)

Let f be a map from a group G into an abelian group H satisfying the equation \(f(xy)+f(xy^{-1})=2f(x)\) for all \(x,y\in G\) with the normalisation condition \(f(e)=0\) where e denotes the identity of G and 0 that of H. Let S(G,H) denote the set of all solutions of the functional equation under reference. The author provides a description of S(G,H) for special groups G, including the free group on 2 generators, \(GL_ 2(Z)\) and the multiplicative group of some fields.