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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2022. Vol 40

On Generation of the Groups 𝑆𝐿𝑛(Z + 𝑖Z) and 𝑃 𝑆𝐿𝑛(Z + 𝑖Z) by Three Involutions, Two of Which Commute

Author(s)
Rodion I. Gvozdev1, Yakov N. Nuzhin1, Tatyana B. Shaipova2

1Siberian Federal University, Krasnoyarsk, Russian Federation

2Krasnoyarsk Scientific Center of the Siberian Branch Russian Academy of Sciences, Krasnoyarsk, Russian Federation

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.
About the Authors

Rodion I. Gvozdev, Student, Institute of Mathematics and Computer Science, Siberian Federaln University, Krasnoyarsk, 660041, Russian Federation, gvozdev.rodion@bk.ru

Yakov N. Nuzhin, Dr. Sci. (Phys.–Math.), Prof., Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, nuzhin2008@rambler.ru

Tatyana B. Shaipova, Postgraduate, Krasnoyarsk Scientific Center SB RAS, Krasnoyarsk, 660036, Russian Federation, 663431@mail.ru

For citation
Gvozdev R. I., Nuzhin Ya. N., Shaipov T.B. On Generation of the Groups 𝑆𝐿𝑛(Z + 𝑖Z) and 𝑃 𝑆𝐿𝑛(Z + 𝑖Z) by Three Involutions, Two of Which Commute. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 40, pp. 49–62. (in Russian) https://doi.org/10.26516/1997-7670.2022.40.49
Keywords
special and projective special linear groups, the ring of Gaussian integers, generating triples of involutions
UDC
512.5
MSC
20G15
DOI
https://doi.org/10.26516/1997-7670.2022.40.49
References

1. Kostrikin A.I. Introduction to algebra. Moscow, Nauka Publ., 1977.

2. Levchuk D.V. On generation of the group 𝑃 𝑆𝐿7(Z + 𝑖Z) by three involutions, two of which commute. Bulletin of Novosibirsk State Univ., 2009, vol. 9, no. 1, pp. 35–38.

3. Nuzhin Ya.N. Generating triples of involutions of Chevalley groups over a finite field of characteristic 2. Algebra and Logic, 1990, vol. 29, no. 2, pp. 192–206. https://doi.org/10.1007/2FBF02001358

4. Nuzhin Ya.N. Generating triples of involutions for Lie type groups over a finite field of odd characteristic. II. Algebra and Logic, 1997, vol. 36, no. 4, pp. 422–440.

5. Nuzhin Ya.N. On generation of the group 𝑃 𝑆𝐿𝑛(Z) by three involutions, two of which commute. Vladikavkaz. Math. J., 2008. vol. 10, no. 1, pp. 68–74.

6. Nuzhin Ya.N. Tensor representations and generating sets of involutions of some matrix groups, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 133–141. https://doi.org/10.21538/0134-4889-2020-26-3-133-141

7. Steinberg R. Lectures on Chevalley groups. Moscow, Mir Publ., 1975.

8. Suprunenko D.A. Matrix groups. Moscow, Nauka Publ., 1972.

9. Levchuk D.V., Nuzhin Ya.N. On generation of the group 𝑃 𝑆𝐿𝑛(Z + 𝑖Z) by three involutions, two of which commute. J. Sib. Fed. Univ. Math. Phys., 2008, vol. 1, no 2, pp. 133–139.

10. Tamburini M.C., Zucca P. Generation of Certain Matrix Groups by Three Involutions, Two of Which Commute. J. of Algebra, 1997, vol. 195, no. 2, pp. 650–661.


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