A generalization of Neumann's Question

TL;DR

This paper introduces a new class of groups called (m,n)-groups, explores their properties, examples, and establishes bounds on their solvability length, extending the understanding of group commutativity conditions.

Contribution

It generalizes Neumann's question by defining (m,n)-groups, providing examples, and analyzing their finiteness, commutativity, and solvability length bounds.

Findings

Examples of finite and infinite non-abelian (m,n)-groups provided.

Finiteness and commutativity conditions discussed.

Bound on solvability length in terms of m and n established.

Abstract

Let $G$ be a group, $m\geq2$ and $n\geq1$. 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 $i\neq j$ and $x_i \in X_i, x_j \in X_j$ such that $x_ix_j=x_jx_i$. 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 $n$.