Right 4-Engel elements of a group

TL;DR

This paper proves that in certain locally nilpotent groups without elements of orders 2, 3, or 5, the set of right 4-Engel elements forms a subgroup and their normal closures are nilpotent of class at most 7.

Contribution

It establishes the subgroup property of right 4-Engel elements in specific groups and bounds the nilpotency class of their normal closures.

Findings

Right 4-Engel elements form a subgroup in the specified groups.

Normal closure of a right 4-Engel element is nilpotent of class at most 7.

The result applies to locally nilpotent groups without elements of orders 2, 3, or 5.

Abstract

We prove that the set of right 4-Engel elements of a group $G$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $<x>^G$ is nilpotent of class at most 7 for each right 4-Engel elements $x$ of $G$.