Left 3-Engel Elements in Locally Finite 2-Groups

TL;DR

This paper constructs an infinite family of locally finite 2-groups with left 3-Engel elements whose normal closures are not nilpotent, expanding understanding of Engel elements in group theory.

Contribution

It generalizes previous constructions by providing new examples of such groups using Lie algebras and classical combinatorial theorems.

Findings

Constructed infinite families of groups with specific Engel properties

Demonstrated non-nilpotency of normal closures of certain elements

Utilized Lie algebra techniques and Lucas's theorem in the construction

Abstract

We give an infinite family of examples that generalise the construction given in arXiv:1811.12074 of a locally finite 2-group $G$ containing a left 3-Engel element $x$ where ${\langle x \rangle}^G$, the normal closure of $x$ in $G$, is not nilpotent. The construction is based on a family of Lie algebras that are of interest in their own right and make use of a classical theorem of Lucas, regarding when $\binom{m}{n}$ is even.