Generation of the group $GL_6^{\pm 1}(\mathbb{Z}+i\mathbb{Z})$ by three involutions, two of which commute

Tatyana B. Shaipova

Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract: Previously, the author solved the problem of generation by three involutions, two of which commute, of the matrix group $GL_n^{\pm 1}(\mathbb{Z}+i\mathbb{Z})$ of dimension $n$ with determinant $\pm 1$ over the ring of Gaussian integers $\mathbb{Z}+i\mathbb{Z}$ and its quotient group by the center $PGL_n^{\pm 1}(\mathbb{Z}+i\mathbb{Z})$, with the exception of the group $GL_6^{\pm 1}(\mathbb{Z}+i\mathbb{Z})$. In this note, it is proved that the group $GL_6^{\pm 1}(\mathbb{Z}+i\mathbb{Z})$ is generated by three involutions, two of which commute.

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

UDC: 512.5

Received: 01.05.2025
Received in revised form: 01.06.2025
Accepted: 04.07.2025

Language: English