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 $\langle g,x\rangle$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 $Y(g)$.