A nilpotency-like condition for infinite groups

If k is a positive integer, a group G is said to have the $FE_{k}$-property if for each element g of G there exists a normal subgroup of finite index X(g) such that the subgroup is nilpotent of class at most k for every element x of X(g) . Thus, $FE_{1}$ -groups are precisely those groups with finite conjugacy classes (FC-groups) and the aim of this paper is to extend properties of FC-groups to the case of groups with the $FE_{k}$-property for k>1. The class of $FE_{k}$-groups contains the relevant subclass $FE_{k}^{ast }$ , consisting of all groups G for which to every element g there corresponds a normal subgroup of finite index Y(g) such that is nilpotent of class at most k, whenever U is a nilpotent subgroup of class at most k of Y(g).