A group generated by three involutions, two of which commute, will be called $(2\times 2,2)$-generated. The class of such groups is closed with respect to homomorphic images, if, by definition, we consider the identity group as such and do not exclude the coincidence of two or all three involutions. For finite simple groups, the question of generation by three involutions, two of which commute, was formulated by V.D. Mazurov in the Kourovka notebook in 1980. The answer to this question is known, and it is positive, excluding three alternating groups, some groups of Lie type of rank no more than three, and four sporadic groups. This article considers the $(2\times 2,2)$-generation of the general linear group over a finite field and its projective image $PGL_n(q)$. It is proven that $GL_n(q)$ (respectively $PGL_n(q)$) is $(2\times 2,2)$-generated if and only if a) $q=2$ and $n=2$ or $n\geq5$, or b) $q=3$ and $n\geq5$ (respectively, when either a) $n=2$ and any $q$, or b) $n\geq 4$ and $(n,q-1) =2$, or c) $n\geq 5$ and $(n,q-1)=1$).
Keywords: general and projective linear groups, finite field, generating triples of involutions
Received April 11, 2024
Revised April 25, 2024
Accepted April 28, 2025
Published online May 12, 2025
Funding Agency: This work was supported by Russian Science Foundation, project 25-21-20059, https://rscf.ru/project/25-21-20059/.
Irina Aleksandrovna Markovskaya, doctoral student, Institute of Mathematics and Computer Science of the Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: mark.i.a@mail.ru
Yakov Nifantievich Nuzhin, Dr. Phys.-Math. Sci., Prof. Institute of Mathematics and Computer Science of the Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: nuzhin2008@rambler.ru
REFERENCES
1. The Kourovka notebook. Unsolved problems in group theory / eds. V.D. Mazurov, E.I. Khukhro. 20th ed. Novosibirsk: Inst. Math. SO RAN Publ., 2022. 269 p. URL: https://kourovka-notebook.org/ .
2. Nuzhin Ya.N. Generating sets of involutions of finite simple groups. Algebra and Logic, 2019, vol. 58, no. 3, pp. 288–293 https://doi.org/10.1007/s10469-019-09547-x
3. Sjerve D., Cherkassoff M. On groups generated by three involutions, two of which commute. CRM Proceedings and Lecture Notes, Providence, Amer. Math. Soc., 1994, pp. 169–185.
https://doi.org/10.1090/crmp/006/09
4. Conder M., Oliveros D. The intersection condition for regular polytopes. J. Combin. Theory Ser. A, 2013, vol. 120, no. 6, pp. 1291–1304.
5. Leemans D. String C-group representations of almost simple groups: A survey. Contemporary Math., 2021, vol. 764, pp. 157–178.
6. Suprunenko D.A. Matrix groups. Providence: AMS, 1976, 252 p. ISBN: 0821813412 . Original Russian text published in Suprunenko D.A. Gruppy matrits, Moscow: Nauka Publ., 1972, 352 p.
7. 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
8. Scott L.L. Matricies and cohomology. Annals of Math., 1977, vol. 105, no. 3, pp. 473–492. https://doi.org/10.2307/1970920
9. Levchuk V.M. Generating sets of root elements of Chevalley groups over a field. Algebra and Logic, 1983, vol. 22, pp. 362–371. https://doi.org/10.1007/BF01982113
10. Levchuk V.M. Remark on a theorem of L. Dickson. Algebra and Logic, 1983, vol. 22, pp. 306–316. https://doi.org/10.1007/BF01979677
11. Markovskaya I.A., Nuzhin Ya.N. On generation of the groups $GL_n(Z)$ and $PGL_n(Z)$ by three involutions, two of which commute. J. Siberian Federal Univ. Mathematics & Physics, 2023, vol. 16, no. 4, pp. 413–419.
Cite this article as: I.A. Markovskaya, Ya.N. Nuzhin. On generation of the groups $GL_n(q)$ and $PGL_n(q)$ by three involutions, two of which commute. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 247–259.
