On a Combinatorial Problem in Group Theory

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 X0 ⊆ X, with 2 ≤ |X0 | ≤ n + 1 and a function f: {0, 1, 2,..., k} − → X0, with f(0) � = f(1) and non-zero integers t0, t1,..., tk such that [x t0 0, xt1 1,..., xt k] = 1, where xi: = f(i), i = 0,..., k, and xj ∈ H whenever k x t j j ∈ H, for some subgroup H �=�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)