On the right and left 4-Engel elements

TL;DR

This paper investigates properties of left and right 4-Engel elements in groups, establishing conditions under which certain generated subgroups are nilpotent and analyzing the placement of specific elements within the Baer radical.

Contribution

It proves that subgroups generated by specific pairs involving 4-Engel elements are nilpotent of class at most 4 and characterizes the placement of elements of finite p-power order in the Baer radical.

Findings

<a, a^b> is nilpotent of class at most 4

Elements of finite p-power order with certain Engel properties lie in the Baer radical

Conditions for nilpotency involving right and left 4-Engel elements

Abstract

In this paper we study left and right 4-Engel elements of a group. In particular, we prove that $<a, a^b>$ is nilpotent of class at most 4, whenever $a$ is any element and $b^{\pm 1}$ are right 4-Engel elements or $a^{\pm 1}$ are left 4-Engel elements and $b$ is an arbitrary element of $G$. Furthermore we prove that for any prime $p$ and any element $a$ of finite $p$-power order in a group $G$ such that $a^{\pm 1}\in L_4(G)$, $a^4$, if $p=2$, and $a^p$, if $p$ is an odd prime number, is in the Baer radical of $G$.