Engel groups with an identity

TL;DR

This paper proves that residually finite Engel groups satisfying an identity are locally nilpotent and explores properties of right Engel elements and varieties defined by Engel conditions.

Contribution

It establishes that such groups are locally nilpotent and characterizes the structure of right Engel elements within these groups.

Findings

Residually finite Engel groups with an identity are locally nilpotent.

The set of right Engel elements lies in the Hirsch-Plotkin radical.

Certain classes of groups defined by Engel conditions form varieties.

Abstract

We give an affrmative answer to the question whether a residually finite Engel group satisfying an identity is locally nilpotent. More generally, for a residually finite group G with an identity, we prove that the set of right Engel elements of G is contained in the Hirsch-Plotkin radical of G. Given an arbitrary word w, we also show that the class of all groups G in which the w-values are right n-Engel and w(G) is locally nilpotent is a variety.