On a Combinatorial Problem in Group Theory

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 X0 ⊆ X, with 2 ≤ |X0| ≤ n + 1 and a function f : {0, 1, 2, . . . , k} −→ X0, with f(0) 6= f(1) and non-zero integers t0, t1, . . . , tk such that [x0 0 , x t1 1 , . . . , x tk k ] = 1, where xi := f(i), i = 0, . . . , k, and xj ∈ H whenever x tj j ∈ H, for some subgroup H 6= D x tj j E 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).