Let K be a field with at least 19 elements. It is proved that any subgroup of GL(4, K) generated by a pair of 2-tori contains unipotent elements of rank 1 or 2. Taking into account previous papers of N. A. Vavilov and the author, this result is valid for any general linear group. It is one of the first steps in studying subgroups generated by a pair of microweight tori in Chevalley groups.
Similar content being viewed by others
References
Z. I. Borevich, “A description of the subgroups of the complete linear group that contain the group of diagonal matrices,” J. Math. Sci., 17, 1718–1641 (1981).
N. A. Vavilov, “Subgroups of Chevalley groups containing a maximal torus,” Trudy Leningr. Mat. Ob., 1, 64–109 (1990).
N. A. Vavilov, “Geometry of 1-tori in GLn,” St. Petersburg Math. J., 19, No. 3, 407–429 (2008).
N. A. Vavilov and V. V. Nesterov, “Geometry of microweight tori,” Vladikavkaz Mat. Zh., 10, No. 1, 10–23 (2008).
N. A. Vavilov and V. V. Nesterov, “Subgroups generated by a pair of 2-tori in GL(5,K),” in print (2021).
R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, Cambridge (1985).
H. Azad, “Root groups,” J. Algebra, 76, No. 1, 211–213 (1982).
A. M. Cohen, H. Cuypers, and H. Sterk, “Linear groups generated by reflection tori,” Canad. J. Math., 51, No. 6, 1149–1174 (1999).
E. Cline, B. Parshall, and L. Scott, “Minimal elements of N(H,P) and conjugacy of Levi complements in finite Chevalley groups,” J. Algebra, 34, No. 3, 521–523 (1975).
A. J. Hahn and O. T. O’Meara, The Classical Groups and K-theory, Springer, Berlin et al. (1989).
V. V. Nesterov and N. A. Vavilov, “Pairs of microweight tori in GLn,” Chebyshev Sb., 21, No. 3, 256–266 (2020).
G. Seitz, “Small rank permutation representations of finite Chevalley groups,” J. Algebra, 28, No. 3, 508–517 (1974).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 492, 2020, pp. 134–148.
Translated by the author.
Rights and permissions
About this article
Cite this article
Nesterov, V. Extraction of Small Rank Unipotent Elements in GL(4, K). J Math Sci 264, 86–95 (2022). https://doi.org/10.1007/s10958-022-05982-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-05982-x