MATHEMATICA, Tome 44 (67), N o 2, 2002, pp. 121–128 WILSON’S FUNCTIONAL EQUATION ON METABELIAN GROUPS Ilie Corovei Abstract. Consider the Wilson functional equation f,g : G → K, f (xy)+ f (xy -1 )=2f (x)g(y) where G is a group and K a quadratically closed field. Acz´ el, Chung and Ng in 1989 have solved Wilson’s equation, assuming that the function g satisfies Kannappan’s condition g(xyz)= g(xzy) and f (xy)= f (yx) for all x, y, z ∈ G and K is a quadratically closed field of char K = 2. Investigations of Wilson’s equation on non-abelian groups show that there exist solutions different of those obtained by Acz´ el, Chung and Ng. In the present paper we obtain the general solution of Wilson’s equation when G is a metabelian group all of whose commutators have finite order, and K a field with char K = 0 generalizing our result from [6] where this was obtained for P3-groups. MSC 2000. Primary 39B52; Secondary 20B99. Key words. Wilson’s equation, metabelian group, P3-group. REFERENCES [1] Acz´ el, J. and Dhombres, J., Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989. [2] Acz´ el, J., Chung, J.K. and Ng, C.T., Symmetric second differences in product form on groups. Topics in mathematical analysis, Ser. Pure Math. 11 (1989), 1-12. [3] Corovei, I., The functional equation f (xy)+f (yx)+f (xy -1 )+f (y -1 x)=4f (x)f (y) for nilpotent groups, Buletinul S ¸tiint ¸ific al Institutului Politehnic Cluj-Napoca 20 (1977), 25-28. [4] Corovei, I., The functional equation f (xy)+f (xy -1 )=2f (x)g(y) for nilpotent groups, Mathematica (Cluj) 22(45) (1980), 33-41. [5] Corovei, I., The d’Alembert functional equation on metabelian groups, Aequationes Math. 57 (1999), 201-205. [6] Corovei, I., Wilson’s functional equation on P3-groups, Aequationes Mathematicae, 61 (2001), 212-220. [7] Friis, P., Trigonometric Functional Equations on Groups, Progress report. Department of Mathematics. Aarhus University. December 1998. [8] Ng, C.T., Jensen’s functional equation on groups, Aequationes Math. 39 (1990), 85-89. [9] Penney, R.C. and Rukhin, A.L., D’Alembert’s functional equation on groups, Proc. Amer. Math. Soc. 77 (1979), 73-80. [10] Stetkaer, H., Wilson’s functional equation on groups, Aequationes Math. 49 (1995), 252-275. [11] Stetkaer, H., D’Alembert’s functional equation on metabelian groups, Aequationes Math., 59 (2000), 306-320. Received: June 8, 2000