On the generation of certain matrix groups by three involutions two of which commute

T. B. Shaipova

Institute of Mathematics and Computer Science of SibFU, 79 Svobodny Ave., Krasnoyarsk 660041, Russia

Abstract: A group generated by three involutions two of which commute, is called $(2\times 2,2)$-generated. It is known that the special linear group $SL_n(\mathbb{Z}+i\mathbb{Z})$ over the ring of the Gaussian integers $\mathbb{Z}+i\mathbb{Z}$ (respectively, its quotient group by the center $PSL_n(\mathbb{Z}+i\mathbb{Z})$) is $(2\times 2,2)$-generated if and only if $n\geq 5$ and $n\neq 6$ (respectively, when $n\geq 5$). It is clear that the general linear group $GL_n(\mathbb{Z}+i\mathbb{Z})$ is not $(2\times 2,2)$-generated, since it contains matrices with determinant different from $\pm 1$, and the determinant of any of its involutions is equal to $\pm 1$. It is also known that the group $PGL_n(\mathbb{Z}+i\mathbb{Z})$ is generated by three involutions if and only if two of them commute when $n\geq 5 $ and $4$ does not divide $n$. In this paper we consider the problem on $(2\times 2,2)$-generation for the matrix group $GL_n^{\pm 1}(\mathbb{Z}+i\mathbb{Z})$ with determinant $\pm 1$ over the ring of the Gaussian integers and for its quotient group by the center $PGL_n^{\pm 1}(\mathbb{Z}+i\mathbb{Z})$.

Key words: general and projective linear groups, the ring of Gaussian integers, generating triples of involutions.

UDC: 512.54

MSC: 20Н25

Received: 29.04.2025

DOI: 10.46698/a1967-7824-2561-m