On generating sets of matrix groups over finite fields

Irina A. Markovskaya

Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract: It is proved that the group of all $(n\times n)$-matrices with determinant $\pm 1$ over a finite field of $q$ elements for odd $q$ is generated by three involutions, two of which commute, if and only if $n\geqslant 5$. It is also established that for $n\geqslant 5$ this group is the automorphism group of a regular $3$-polytope of type $[4p,n]$ or $[4p,2n]$, where $q$ is a power of a prime number $p$.

Keywords: general linear group, finite fields, generating triples of involutions, string C-groups, regular polytopes.

UDC: 512.5

Received: 10.06.2025
Received in revised form: 20.07.2025
Accepted: 04.09.2025

Language: English