On a combinatorial problem in group theory. (Q1881616)

Let \(k,n\) be fixed positive integers and suppose that \(G\) is a group with the following properties: For every subset \(X\) of \(G\) of cardinality \(n+1\) there exist a subset \(X_0\) of \(X\) where \(|X_0|\geq 2\), a fuction \(f\colon\{0,\dots,k\}\to X_0\) with \(f(0)\neq f(1)\), and non-zero integers \(t_0,\dots,t_k\) such that \([f(0)^{t_0},f(1)^{t_1},\dots,f(k)^{t_k}]=1\) and \(f(j)\in H\) whenever \(f(j)^{t_j}\in H\) for some subgroup \(H\neq\langle f(j)^{t_j}\rangle\) of \(G\). The author obtains some results for finite groups having the above properties. He also shows that a finitely generated soluble group with these conditions and \(n\) replaced by \(\infty\) is finite by nilpotent.