Locally finite $p$-groups with a left 3-Engel element whose normal closure is not nilpotent

TL;DR

This paper constructs an example of a locally finite p-group with a left 3-Engel element whose normal closure is not nilpotent, challenging assumptions about the structure of such groups.

Contribution

It provides the first known example of a locally finite p-group with a left 3-Engel element having a non-nilpotent normal closure.

Findings

Existence of such a p-group disproves previous conjectures.

The normal closure of the left 3-Engel element is not necessarily nilpotent.

The example applies to all odd primes p.

Abstract

For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where $\langle x \rangle^G$ is not nilpotent.