On Jensen's functional equation on groups (Q2386094)

The classical Jensen's functional equation is known as [see \textit{J. Aczél}, Lectures on functional equatons and their applications (Academic Press, London) (1966; Zbl 0139.09301)] \[ f\Biggl({x+y\over 2}\Biggr)= {f(x)+ f(y)\over 2} \] which with \(x= u+v\), \(y= u-v\) becomes \(f(u+ v)+ f(u- v)= 2f(u)\), transparent for generalization for a group taking instead of additive group \((\mathbb{R},+)\) an arbitrary group \(G\). That is what the author is doing: \[ f(xy)+ f(xy^{-1})= 2f(x),\quad x,y\in G\tag{1} \] for complex valued functions defined on a group \(G\). The most exhaustive study of Jensen's functional equation on groups has been accomplished by \textit{C. T. Ng} [Jensen's functional equation on groups, Aequationes Math. 39, No. 1, 85--99 (1990; Zbl 0688.39007); Jensen's functional equation on groups, II: Aequationes Math. 58, No. 3, 311--320 (1999; Zbl 0938.39024)]. The purpose of the present paper is to develop a coherent theory for Jensen's functional equation on groups that includes most of the results obtained by \textit{C. T. Ng} (cited papers), \textit{I. Corovei} [On semi-homomorphisms and Jensen's equation, Mathematica 37, No. 1--2, 59--64 (1995; Zbl 0881.39018)] and \textit{P. de Place Friis} [d'Alembert's and Wilson's equations on Lie groups, Aequationes Math. 67, No. 1--2, 12--25 (2004; Zbl 1060.39026)]. As the autor says in the introduction he will: (a) find explicit formulas for solutions of Jensen's functional equation in a reasonable general setting; (b) parametrize \(S(G,C)\) modulo \(\Hom(G,C)\); (c) give sufficient condition on \(G\) to ensure that \(S(G,C)= \Hom(G,C)\), for example to ensure that \(S(G,C)= \{0\}\), where \(S(G,C)\) is the complex vector space of odd solutions of (1) on \(G\), and \(\Hom(G,C)\) the set of homomorphisms of \(G\) into \((C,+)\). The present paper differs from the previous ones by its discovery of the central roles played by the subgroup \([G,[G,G]]\) and the commutator group \([G,G]\). If I am allowed to say the present paper is a remarkable contribution to the extended theory of functional equations enlarging to a highest degree the area of study.