On a functional equation involving group actions |
Aequationes mathematicae 77, 25-32, 2009.
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 Mathematicae, Vol. 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 × G × X→ K satisfying
for all g_{1}, g_{2}, g_{3}∈ G and x∈ X.
f(g_{1}, g_{2}, x)+f(g_{1}g_{2}, g_{3}, x)=f(g_{2}, g_{3}, g_{1}^{-1}x)+f(g_{1}, g_{2}g_{3}, x)^{.}_{.}
This problem was solved in a particular case by B. Ebanks. (Cf. 19. Remark in "Report of Meeting", Aequationes Mathematicae, Vol. 67 (2004) p. 313.) We present the general solution of this problem.
On a functional equation involving group actions |