On Generation of the Group $PGL_n(\mathbb{Z}+i\mathbb{Z})$ by Three Involutions, Two of which Commute

• Published in: Известия Иркутского государственного университета: Серия "Математика"
• Main Authors: Ya.N. Nuzhin, T. B. Shaipova
• Format: Article
• Language: English
• Published: Irkutsk State University 2024-12-01
• Subjects: projective general linear group; the ring of gaussian integers; generating triples of involutions
• Online Access: https://mathizv.isu.ru/en/article/file?id=1515
• ISSN: 1997-7670; 2541-8785

The results of the paper relate to the following general problem. Find natural finite generating  sets of elements of a given linear group over a finitely generated commutative ring. Of particular interest are coefficient rings that are generated by a single element, for example, the ring of integers or the ring of Gaussian integers. We prove that a projective general linear group of dimension $n$ over the ring of Gaussian integers is generated by three involutions two of which commute if and only if $n$ is greater than $4$ and $4$ does not divide $n$. Earlier, M.\,A.\,Vsemirnov, R.\,I.\,Gvozdev, D.\,V.\,Lev\-chuk and the authors of this paper solved a similar problem for the special and projective special linear groups.