A generalization of Neumann's Question

Abstract

Let $G$ be a group, $m\ge 2$ and $n\ge 1$. We say that $G$ is an $\mathcal{T}(m,n)$-group if for every $m$ subsets $X_1,X_2,\dots ,X_m$ of $G$ of cardinality $n$, there exists $i\ne j$ and $x_i\in X_i,x_j\in X_j$ such that $x_i{x}_j=x_j{x}_i$. In this paper, we give some examples of finite and infinite non-abelian $\mathcal{T}(m,n)$-groups and we discuss finiteness and commutativity of such groups. We also show solvability length of a solvable $\mathcal{T}(m,n)$-group is bounded in terms of $m$ and $n$.

10 pages