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.
