Rodion I. Gvozdev, Yakov N. Nuzhin, Tatyana B. Shaipova, “On generation of the groups $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ and $\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$ by three involutions, two of which commute”, Bulletin of Irkutsk State University. Series Mathematics, 2022, Volume 40,Pages 49

This article is cited in 2 papers

Algebraic and logical methods in computer science and artificial intelligence

On generation of the groups $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ and $\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$ by three involutions, two of which commute

Rodion I. Gvozdev^a, Yakov N. Nuzhin^a, Tatyana B. Shaipova^b

^a Siberian Federal University, Krasnoyarsk, Russian Federation
^b Krasnoyarsk Scientific Center of the Siberian Branch Russian Academy of Sciences, Krasnoyarsk, Russian Federation

Abstract: M. C. Tamburini and P. Zucca proved that the special linear group of dimension greater than 13 over the ring of Gaussian integers is generated by three involutions, two of which commute (J. of Algebra, 1997). A similar result for projective special linear groups of dimension greater than 6 was established by D. V. Levchuk and Ya. N. Nuzhin (J. Sib. Fed. Univ. Math. Phys., 2008, Bulletin of Novosibirsk State Univ., 2009). We consider the remaining small dimensions. It is proved that the projective special linear group of dimension other than 5 and 6 over the ring of Gaussian integers if and only if is generated by three involutions, two of which commute when its dimension is greater than 6. For dimension 5 and 6, it was possible to find only generators triples of involutions without the condition that two of which commute.

Keywords: special and projective special linear groups, the ring of Gaussian integers, generating triples of involutions.

UDC: 512.5

MSC: 20G15

Received: 27.12.2021
Revised: 24.03.2022
Accepted: 07.04.2022

DOI: 10.26516/1997-7670.2022.40.49