On the Generation of Certain Matrix Groups by Three Involutions, Two of which Commute

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/a1967-7824-2561-m On the Generation of Certain Matrix Groups by Three Involutions, Two of which Commute Shaipova, T. B. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 3. Abstract: A group generated by three involutions two of which commute, is called \((2\times 2,2)\)-generated. It is known that the special linear group \(SL_n(\mathbb{Z}+i\mathbb{Z})\) over the ring of the Gaussian integers \(\mathbb{Z}+i\mathbb{Z}\) (respectively, its quotient group by the center \(PSL_n(\mathbb{Z}+i\mathbb{Z})\)) is \((2\times 2,2)\)-generated if and only if \(n\geq 5\) and \(n\neq 6\) (respectively, when \(n\geq 5\)). It is clear that the general linear group \(GL_n(\mathbb{Z}+i\mathbb{Z})\) is not \((2\times 2,2)\)-generated, since it contains matrices with determinant different from \(\pm 1\), and the determinant of any of its involutions is equal to \(\pm 1\). It is also known that the group \(PGL_n(\mathbb{Z}+i\mathbb{Z})\) is generated by three involutions if and only if two of them commute when \(n\geq 5 \) and \(4\) does not divide \(n\). In this paper we consider the problem on \((2\times 2,2)\)-generation for the matrix group \(GL_n^{\pm 1}(\mathbb{Z}+i\mathbb{Z})\) with determinant \(\pm 1\) over the ring of the Gaussian integers and for its quotient group by the center \(PGL_n^{\pm 1}(\mathbb{Z}+i\mathbb{Z})\). Keywords: general and projective linear groups, the ring of Gaussian integers, generating triples of involutions Language: Russian Download the full text Download JATS XML For citation: Shaipova, T. B. On the Generation of Certain Matrix Groups by Three Involutions, Two of which Commute, Vladikavkaz Math. J., 2025, vol. 27, no. 3, pp. 127-135 (in Russian). DOI 10.46698/a1967-7824-2561-m + References 1. Levchuk, D. V. and Nuzhin, Ya. N. On Generation of the Group \(PSL_n (\mathbb{Z}+ i\mathbb{Z})\) by Three Involutions, Two of Which Commute, Journal of Siberian Federal University. Mathematics & Physics, 2008, vol. 1, no. 2, pp. 133-139. 2. Levchuk, D. V. On the Generation of the Group \(SL_7(\mathbb{Z} + i\mathbb{Z})\) by Three Involutions, Two of Which Commute, Vestnik Novosibirskogo Gosudarstvennogo Universiteta. Seriya Matematika, Mekhanika, Informatika, 2009, vol. 9, no. 1 pp. 35-38 (in Russian). 3. Gvozdev, R. I., Nuzhin, Ya. N. and Shaipova, T. B. 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, The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 40, pp. 49-62 (in Russian). DOI: 10.26516/1997-7670.2022.40.49. 4. Vsemirnov, M. A., Gvozdev, R. I., Nuzhin, Ya. N. and Shaipova T. B. On the 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. II, Mathematical Notes, 2018, vol. 115, pp. 289-300. DOI: 10.1134/S0001434624030015. 5. Nuzhin, Ya. N. and Shaipova, T. B. On Generation of the Groups \(PGL_n(\mathbb{Z}+i\mathbb{Z})\) by Three Involutions, Two of which Commute, The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 50, pp. 143-151 (in Russian). DOI: 10.26516/1997-7670.2024.50.143. 6. Nuzhin, Ya. N. Generating Sets of Involutions of Finite Simple Groups, Algebra and Logic, 2019, vol. 58, no. 3, pp. 288-293. DOI: 10.1007/s10469-019-09547-x. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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