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Published online by Cambridge University Press: 28 March 2018
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 
$\langle g,x\rangle$ is nilpotent of class at most 
$k$ for all 
$x\in X(g)$. 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 
$\langle g,U\rangle$ is nilpotent of class at most 
$k$, whenever 
$U$ is a nilpotent subgroup of class at most 
$k$ of 
$Y(g)$.
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