On Non-Abelian Tensor Analogues of 3-Engel and 4-Engel Groups

A group G is said to be an n(circle times)-Engel, if [y, (n-1)x]circle times x=1 for all x, yG, and we say a group G is tensor nilpotent of class at most n, if . In this article, we show that if G is a 3(circle times)-Engel group, thenx, x(y)is tensor nilpotent of class at most 2, for all x, yG. We also prove that if G is a 4(circle times)-Engel group and G circle times G is torsion-free, thenx, x(y)is tensor nilpotent of class at most 4, for all x, y is an element of G.