Abstract
Let K be a field containing at least seven elements. In the group G=GL(n, K) we describe the subgroups containing the group D of all diagonal matrices. This description is given in terms of the concept of a D-net subgroup, defined as a subgroup of G composed of matrices (a ij) with zero elementsa ij in some prescribed cells outside the main diagonal (the set of cells is subordinated to some condition of agreement). The main theorem is: Every subgroup of G containing D is contained between a uniquely determined D-net subgroup and its normalizer in G. The structure of all subgroups of G containing D is finite and does not depend on the field K (when card K ≥ 7).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
Z. I. Borevich, “Parabolic subgroups in linear groups over a semilocal ring,” Vestn. Leningr. Univ., No. 13, 16–24 (1976).
Yu. I. Merzlyakov, “Central series and series of commutants of matrix groups,” Algebra Logika,3, No. 4, 49–58 (1964).
N. S. Romanovskii, “Subgroups of the general and the special linear groups over a ring,” Mat. Zametki,9, 699–708 (1971).
M. Newman and I. Reiner, “Inclusion theorems for congruence subgroups,” Trans. Am. Math. Soc.,91, 369–379 (1959).
I. Reiner and J. D. Swift, “Congruence subgroups of matrix groups,” Pac. J. Math.,6, 529–540 (1956).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 12–29, 1976.
Rights and permissions
About this article
Cite this article
Borevich, Z.I. A description of the subgroups of the complete linear group that contain the group of diagonal matrices. J Math Sci 17, 1718–1730 (1981). https://doi.org/10.1007/BF01091757
Issue Date:
DOI: https://doi.org/10.1007/BF01091757