Commutator lengths in general linear group over a skew-field

TL;DR

This paper investigates the bounds on the commutator length of elements in the elementary subgroup of the general linear group over a skew-field, relating it to the commutator length in the skew-field's multiplicative group.

Contribution

It provides new upper and lower estimates for the commutator length in the general linear group over a skew-field, linking it to the properties of the skew-field's multiplicative group.

Findings

Established bounds for commutator lengths in the elementary subgroup

Connected commutator length in the group to that in the skew-field's multiplicative group

Enhanced understanding of algebraic structure of linear groups over skew-fields

Abstract

We give an upper and lower estimate for the maximal commutator length of a noncentral element of the elementary subgroup of the general linear group over a skew-field based on the maximal commutator length of an element of the multiplicative group of that skew-field.