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 \(x_ix_j=x_jx_i\). 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.
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Ahmadkhah, N., Marzang, S. & Zarrin, M. A Generalization of Neumann’s Question. Results Math 73, 80 (2018). https://doi.org/10.1007/s00025-018-0844-3
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DOI: https://doi.org/10.1007/s00025-018-0844-3