On a functional equation involving group actions

Abstract

During the forty-first ISFE in Noszvaj, Hungary, G. Guzik posed a problem on a functional equation involving group actions which arose in a generalization of Bargman theory occurring in Quantum Mechanics. (Cf. 18. Problem and Remark in “Report of Meeting", Aequationes Math., 67 (2004), 312–313.) Let (G, ·) be a group which is acting on a set X and let (K, +) be an abelian group. Describe all functions $$f : G \times G \times X \rightarrow K$$ satisfying $$f(g1, g2, x) + f(g1g2, g3, x) = f(g2, g3, g^{-1}_{1} x) + f(g1, g2g3, x)$$ for all $$g1, g2, g3 \in G$$ and $$x \in X$$ . This problem was solved in a particular case by B. Ebanks. (Cf. 19. Remark in “Report of Meeting", Aequationes Math. 67 (2004), p. 313.) We present the general solution of this problem.