For any integersn>m≥ 2, we say that a complete theoryTis (m, n)-homogeneous if, for each modelMofT, two n-tuples ⱥ,ƀ inMhave the same type if the corresponding m-tuples from ⱥ and ƀ have the same type. It was conjectured by H. Kikyo that, ifMis an infinite group, with possibly additional structure, then the theory ofMis not (m, n)-homogeneous. We prove a general result on structures with (m, n)-homogeneous theory which implies that, ifMis a counterexample to this conjecture, then there exists an integerhsuch that each abelian subgroup ofMhas at mosthelements. It follows that there exist an integerksuch thatMk= 1, and an integerlsuch that each finite subgroup ofMhas at mostlelements.