Products of Commutators of Involutions in Skew Linear Groups

Abstract

In connection with [Theorem 4.6, Linear Algebra Appl. 646, 119–131, (2022)], we show that each matrix in the commutator subgroup of the general linear group over a centrally-finite division ring D, in which each element in the commutator subgroup of D is a product of at most s commutators, can be written as a product of at most \(3+3\left\lceil \frac{s}{\lfloor n/2 \rfloor } \right\rceil \) commutators of involutions if \(\mathrm {char\,}D\ne 2\), where \({\displaystyle \lceil x \rceil }\), \({\displaystyle \lfloor x \rfloor }\) denote the ceiling and floor functions of x, respectively. Moreover, we also present the special case when \(D= \mathbb {H}\), the division ring of quaternions, and an application in real group algebras.