Let $G$ be an infinite group and $n\in \{3, 6\}\cup\{2^k| k\in \mathbb{N}\}$. In this paper, we prove that $G$ is an $n$-Kappe group if and only if for any two infinite subsets $X$ and $Y$ of $G$, there exist $x\in X$ and $y\in Y$ such that $[x^n, y, y]=1$.
A. Abdollahi and B. Taeri (1999). Some conditions on infinite subsets of infinite groups. Bull. Malaysian Math. Soc. (2). 22, 87-93 C. Delizia and C. Nicotera (2007). Groups with conditions on infinite subsets. Ischia Group Theory
2006, Proc. Conf., in honor of Akbar Rhemtulla, World Scientific Publishing. , 46-55 C. Delizia and A. Tortora (2009). Locally graded groups with a Bell condition on infinite subsets. J. Group Theory. 12, 753-759 G. Endimioni (1995). On a combinatorial problem in varieties of groups. Comm. Algebra. 23, 5297-5307 W. P. Kappe (1961). Die A-Norm einer Gruppe. Illinois J. Math.. 5, 187-197 O. H. Kegel and B. A. F. Wehrfritz (1973). Locally Finite Groups. North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York. 3 P. Longobardi, M. Maj and A. H. Rhemtulla (1992). Infinite groups in a given variety and Ramsy's theorem. Comm. Algebra. 20, 127-139 B. H. Neumann (1976). A problem of Paul Erd\"{o}s on groups. J. Austral. Math. Soc. Ser. A. 21, 467-472 D. J. S. Robinson (1996). A course in the theory of groups. 2nd edition, Springer-Verlag, New York. 80 V. D. Mazurov and E. I. Khukhro (2006). Unsolved problems in group theory. The Kourovka notebook, Sixteenth edition, Including archive of solved problems, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk.
)
