Geodesic
On the Generation of the Groups $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ and $\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$
Matematičeskie zametki, Tome 115 (2024) no. 3, pp. 317-329 Cet article a éte moissonné depuis la source Math-Net.Ru

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We complete the solution of the problem on the existence of generating triplets of involutions two of which commute for the special linear group $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ and the projective special linear group $\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$ over the ring of Gaussian integers. The answer has only been unknown for $\mathrm{SL}_5$, $\mathrm{PSL}_6$, and $\mathrm{SL}_{10}$. We explicitly indicate the generating triples of involutions in these three cases, and we make a significant use of computer calculations in the proof. Taking into account the known results for the problem under consideration, as a consequence, we obtain the following two statements. The group $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ ($\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$, respectively) is generated by three involutions two of which commute if and only if $n\geqslant 5$ and $n\neq 6$ (if $n\geqslant 5$, respectively).
Keywords: special and projective special linear groups, ring of Gaussian integers, generating triplet of involutions. 
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M. A. Vsemirnov; R. I. Gvozdev; Ya. N. Nuzhin; T. B. Shaipova. On the Generation of the Groups $\mathrm{SL}_n(\mathbb{Z}+i\mathbb{Z})$ and $\mathrm{PSL}_n(\mathbb{Z}+i\mathbb{Z})$. Matematičeskie zametki, Tome 115 (2024) no. 3, pp. 317-329. http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a0/

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