On Generation of the Groups $SL_n(\mathbb{Z}+i\mathbb{Z})$ and $PSL_n(\mathbb{Z}+i\mathbb{Z})$ by Three Involutions, Two of Which Commute

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