Groups generated by a finite Engel set

TL;DR

This paper investigates conditions under which groups generated by finite Engel sets, especially of size two, are nilpotent, contributing to understanding the structure of Engel-generated groups.

Contribution

It provides new criteria for when groups generated by finite Engel sets, particularly of size two, are guaranteed to be nilpotent.

Findings

Groups generated by finite Engel sets can be nilpotent under certain conditions.

Special focus on groups generated by two Engel elements.

Provides criteria for nilpotency in Engel-generated groups.

Abstract

A subset $S$ of a group $G$ is called an Engel set if, for all $x,y\in S$, there is a non-negative integer $n=n(x,y)$ such that $[x,\,_n y]=1$. In this paper we are interested in finding conditions for a group generated by a finite Engel set to be nilpotent. In particular, we focus our investigation on groups generated by an Engel set of size two.