Abstract
This paper is a continuation of RZhMat 1981, 7A438. Suppose R is a commutative ring generated by its group of units R* and there exist
such that
. Suppose also that ℑ is the Jacobson radical of R, and B(ℑ) is a subgroup of GL(n,R) consisting of the matrices a=(aij) such that aij∃ℑ for i>j. If a matrix a∃B(ℑ) is represented in the form a=udv, where u is upper unitriangular, d is diagonal, and v is lower unitriangular, then u,v∃〈D,aDa−1〉, where D=D(n,R) is the group of diagonal matrices. In particular, D is abnormal in B(ℑ)
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 of linear groups over a semilpcal ring,” Vestn. Leningr. Univ., No. 13, 16–24 (1976).
Z. I. Borevich, “A description of the subgroups of the full linear group that contain the group of diagonal matrices,” J. Sov. Math.,17, No. 2 (1981).
Z. I. Borevich and N. A. Vavilov, “Subgroups of the full linear group over a semilocal ring that contain the group of diagonal matrices,” Tr. Mat. Inst. Akad. Nauk SSSR,148, 43–57 (1978).
N. A. Vavilov, “Conjugacy of subgroups of the full linear group that contain the group of diagonal matrices,” Usp. Mat. Nauk,34, No. 5, 216–217 (1979).
N. A. Vavilov, “A Bruhat decomposition for subgroups containing the group of diagonal matrices,” J. Sov. Math.,24, No. 4 (1984).
N. A. Vavilov, “Subgroups of the full linear group over a semilocal ring that contain the group of diagonal matrices,” Vestn. Leningr. Univ., No. 1, 10–15 (1981).
N. A. Vavilov and E. V. Dybkova, “Subgroups of the full symplectic group that contain the group of diagonal matrices, J. Sov. Math.,24, No. 4 (1984).
J. McLaughlin, “Some groups generated by transvections,” Arch. Math.,18, No. 4, 364–368 (1967).
M. Newman, Integral Matrices, Academic Press, New York-London (1972).
G. M. Seitz, “Subgroups of finite groups of Lie type,” J. Algebra,61, No. 1, 16–27 (1979).
M. R. Stein, “Stability theorems for K1, K2, and related functors modeled on Chevalley groups,” Jpn. J. Math.4, No.1, 77–108 (1978).
N. A. Vavilov, “On subgroups of split orthogonal groups in even dimensions,” Bull. Acad. Polon. Sci., Ser. Sci.-Math.,29, Nos. 9–10, 383–387 (1981).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 50–61, 1982.
In conclusion, the author would like to thank Professor Z. I. Borevich for many useful discussions. The present paper (like [5, 12] and, in part [6, 7]) was mainly written in 1979–1980, while the author was at Wroclaw University. The author would like to thank his Wroclaw colleagues and, primarily, his adviser, Professor W. Narkiewicz, for creating ideal conditions under which to work.
Rights and permissions
About this article
Cite this article
Vavilov, N.A. A Bruhat decomposition for subgroups containing the group of diagonal matrices. II. J Math Sci 27, 2865–2874 (1984). https://doi.org/10.1007/BF01410740
Issue Date:
DOI: https://doi.org/10.1007/BF01410740