On profinite groups with Engel-like conditions

TL;DR

This paper investigates profinite groups where elements or specific subgroup elements have powers that are Engel, proving such groups are locally virtually nilpotent under these conditions.

Contribution

It establishes that profinite groups with elements having Engel powers are locally virtually nilpotent, extending understanding of Engel conditions in profinite group structures.

Findings

Profinite groups with elements having Engel powers are locally virtually nilpotent.

Finitely generated profinite groups with certain Engel-like conditions have locally virtually nilpotent derived subgroups.

The results generalize Engel conditions to broader classes of profinite groups.

Abstract

Let $G$ be a profinite group in which for every element $x\in G$ there exists a natural number $q=q(x)$ such that $x^q$ is Engel. We show that $G$ is locally virtually nilpotent. Further, let $p$ be a prime and $G$ a finitely generated profinite group in which for every $\gamma_k$-value $x\in G$ there exists a natural $p$-power $q=q(x)$ such that $x^q$ is Engel. We show that $\gamma_k(G)$ is locally virtually nilpotent.