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Products of Commutators of Involutions in Skew Linear Groups
Nguyen Thi Thai Ha
,
Phan Hoang Nam
,
Tran Nam Son
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.