The Engel elements in generalized FC-groups

Abstract

We generalize to $FC^{*}$, the class of generalized $FC$-groups introduced in [ Serdica Math. J. 28 (2002) 241–254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an $FC^{*}$-group $G$ coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of $G$ are subgroups and the former coincides with the hypercentre. We also give an example of an $FC^{*}$-group for which the set of right Engel elements contains properly the set of bounded right Engel elements.