Southeast Asian Bulletin of Mathematics c  SEAMS. 2003 Southeast Asian Bulletin of Mathematics (2003) 26: 1029–1039 On a Combinatorial Problem in Group Theory * Bijan Taeri Department of Mathematics, Isfahan University of Technology, Isfahan, Iran OR In- stitute for Studies in Theoretical Physics and Mathematics, Iran E-mail: b.taeri@cc.iut.ac.ir AMS Mathematical Subject Classification (2000): 20F99 Abstract. Let n be a positive integer or infinity (denote ∞). We denote by W * (n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X 0 ⊆ X, with 2 ≤|X 0 |≤ n + 1 and a function f : {0, 1, 2,...,k} -→ X0, with f (0) 6= f (1) and non-zero integers t0,t1,...,t k such that [x t 0 0 ,x t 1 1 ,...,x t k k ] = 1, where xi := f (i), i =0,...,k, and xj ∈ H whenever x t j j ∈ H, for some subgroup H 6= x t j j of G. If the integer k is fixed for every subset X we obtain the class W * k (n). Here we prove that (1) Let G ∈ W * (n), n a positive integer, be a finite group, p>n a prime divisor of the order of G, P a Sylow p-subgroup of G. Then there exists a normal subgroup K of G such that G = P × K. (2) A finitely generated soluble group has the property W * (∞) if and only if it is finite-by-nilpotent. (3) Let G ∈ W * k (∞) be a finitely generated soluble group, then G is finite-by- (nilpotent of k-bounded class). Keywords: combinatorial conditions, finitely generated soluble groups 1. Introduction and Results B. H. Neumann has proved [19] that a group is center-by-finite if and only if every infinite subset contains a commuting pair of distinct elements. This result was an affirmative answer to a question of P. Erd¨os. Other problems of this type have been the object of several articles, for example [1]-[12], [15]-[17], [19], [23]-[25]. Our notation and terminology are standard and can be found in [20]. In partic- ular for a group G and elements x, y, x 1 ,x 2 ,...,x k ∈ G we write [x 1 ,x 2 ]= x -1 1 x -1 2 x 1 x 2 = x -1 1 x x2 1 , [x 1 ,...,x k ] = [[x 1 ,...,x k-1 ],x k ] This research was in part supported by a grant from IPM.