The Engel elements in generalized FC-groups

TL;DR

This paper extends Baer's results on Engel elements to FC*-groups, showing that certain Engel element sets coincide with classical group substructures like the Fitting subgroup and hypercentre.

Contribution

It generalizes known properties of Engel elements from FC-groups to the broader class of FC*-groups, establishing new equalities and subgroup structures.

Findings

Left and bounded left Engel elements equal the Fitting subgroup.

Right and bounded right Engel elements form subgroups, with the right Engel set matching the hypercentre.

An example demonstrates the proper inclusion of bounded right Engel elements in right Engel elements.

Abstract

We generalize to FC*, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, 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.