¹Siberian Federal University, Krasnoyarsk, Russian Federation

The results of the paper relate to the following general problem. Find natural finite generating sets of elements of a given linear group over a finitely generated commutative ring. Of particular interest are coefficient rings that are generated by a single element, for example, the ring of integers or the ring of Gaussian integers. We prove that a projective general linear group of dimension n over the ring of Gaussian integers is generated by three involutions two of which commute if and only if n is greater than 4 and 4 does not divide n. Earlier, M. A. Vsemirnov, R. I. Gvozdev, D. V. Levchuk and the authors of this paper solved a similar problem for the special and projective special linear groups.

Yakov N. Nuzhin, Dr. Sci. (Phys.-Math.), Prof., Siberian Federal University, Krasnoyarsk,660041, Russian Federation, nuzhin2008@rambler.ru

Tatyana B. Shaipova, Sen. Lec., Siberian Federal University, Krasnoyarsk,660041, Russian Federation, 663431@mail.ru

Nuzhin Ya. N., Shaipova T. B. On Generation of the Group PGL_n(Z+iZ) by Three Involutions, Two of which Commute. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 50, pp. 143–151. (in Russian)

https://doi.org/10.26516/1997-7670.2024.50.143

1. Vsemirnov М.А., Gvozdev R.I., Nuzhin Ya.N., Shaipova Т.B. On generation of the groups PSL_n(Z + iZ) и PSL_n(Z + iZ) by three involutions,two of which commute. Mathematical Notes, 2024, vol. 114, no. 3, pp. 289–300. https://doi.org/10.1134/S0001434624030015
2. Gvozdev R.I., Nuzhin Ya.N., Shaipova Т.B. On Generation Groups PSL_n(Z + iZ) and PSL_n(Z + iZ) by three involutions,two of which commute. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 40, pp. 49–62. https://doi.org/10.26516/1997-7670.2022.40.49
3. Kargapolov М.I., Merzlyakov Yu.I. Fundamentals of group theory. Moscow, Nauka Publ., 1982.
4. Kostrikin A.I. Introduction to algebra. Moscow, Nauka Publ., 1977.
5. Levchuk D. V. On generation of the group PSL₇(Z + iZ) by three involutions, two of which commute. Bulletin of Novosibirsk State Univ., 2009, vol. 9, no. 1, pp. 35–38.
6. Nuzhin Ya. N. On generating sets of involutions of simple finite groups. Algebra and Logic, 2019, vol. 58, no. 3, pp. 426–434. https://doi.org/10.33048/alglog.2019.58.310
7. Steinberg R. Lectures on Chevalley groups. Moscow, Mir Publ., 1975.
8. Timofeenko I.A. Generating multiplets of linear groups over the ring of integers. Cand. sci. diss. Abstr. Krasnoyarsk, 2017, 72 p. (in Russian)
9. Levchuk D.V., Nuzhin Ya.N. On generation of the group PSL_n(Z + iZ) by three involutions, two of which commute.J. Sib. Fed. Univ. Math. Phys., 2008, no. 2, pp. 133–139.