Almost Engel linear groups

TL;DR

This paper proves that linear groups with the almost Engel property are finite-by-hypercentral, and those with the almost nil property are finite-by-nilpotent, establishing structural classifications for these groups.

Contribution

It introduces the concepts of almost Engel and almost nil groups in the context of linear groups and proves their structural implications.

Findings

Almost Engel linear groups are finite-by-hypercentral.

Almost nil linear groups are finite-by-nilpotent.

Structural classifications for these classes of groups.

Abstract

We prove that if a linear group $G$ is almost Engel, then $G$ is finite-by-hypercentral. If $G$ is almost nil, then $G$ is finite-by-nilpotent.