Linear groups with almost right Engel elements

TL;DR

This paper investigates linear groups where each element's iterated commutators eventually lie in a finite set, leading to the conclusion that such groups are finite-by-hypercentral, thus extending understanding of their structure.

Contribution

It introduces a new condition on linear groups involving finite sets of commutators and proves these groups are finite-by-hypercentral, advancing the classification of linear groups.

Findings

Groups are finite-by-hypercentral under the given condition

Commutator behavior constrains group structure

Extends known results on linear groups

Abstract

Let G be a linear group such that for every g in G there is a finite set R(g) with the property that for every x in G all sufficiently long commutators [g,x,x,...,x] belong to R(g). It is proved that G is finite-by-hypercentral.