On left Engel elements of a group relative to a subgroup of certain linear groups

Let X be a subset of a group G. An element g∈G is called a X-relative left Engel element of G, if for each x∈X, there is a positive integer n=n(g,x) such that [x,ng]=1. In this paper, we will study the set of X-relative left Engel elements when X is a subgroup of certain linear groups.