A generalization of Neumann's Question

Original · EN

Let G be a group, m≥2 and n≥1. We say that G is an T(m,n)-group if for every m subsets X₁, X₂,, Xₘ of G of cardinality n, there exists i≠ j and xᵢ ∈ Xᵢ, xⱼ ∈ Xⱼ such that xᵢxⱼ=xⱼxᵢ. In this paper, we give some examples of finite and infinite non-abelian T(m,n)-groups and we discuss finiteness and commutativity of such groups. We also show solvability length of a solvable T(m,n)-group is bounded in terms of m and n.