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Subnormal subgroups of the groups of rational points of reductive algebraic groups

Authors: Gopal Prasad and Andrei S. Rapinchuk
Journal: Proc. Amer. Math. Soc. 130 (2002), 2219-2227
MSC (2000): Primary 20G15, 20G30, 22E46
Published electronically: February 7, 2002
MathSciNet review: 1896401
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Abstract: We prove that for a reductive algebraic group $G$ over an infinite field $K,$ the group of rational points $G(K)$ does not contain any noncentral finitely generated normal subgroups.

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  • [1] S. Akbari and M. Mahdavi-Hezavehi, Normal subgroups of $GL_n(D)$ are not finitely generated, Proc. AMS, 128(1999), 1627-1632. MR 2000j:16052
  • [2] S. Akbari, M. Mahdavi-Hezavehi and M.G. Mahmudi, Maximal subgroups of $GL_1(D),$ J. Algebra, 217(1999), 422-433. MR 2000d:16025
  • [3] Algebraic Number Theory, Proc. of an instructional conference organized by the London Mathematical Society, ed. by J.W.S. Cassels and A. Frölich, Academic Press, 1967. MR 35:6500
  • [4] A. Borel, Linear Algebraic Groups, GTM 126, Springer-Verlag, 1991. MR 92d:20001
  • [5] A. Borel and T.A. Springer, Rationality properties of linear algebraic groups II, Tohoku Math. J., 20(1968), 443-497. MR 39:5576
  • [6] C. Chevalley, On algebraic group varieties, J. Math. Soc. Japan, 6(1954), 303-324. MR 16:672g
  • [7] M.D. Fried and M. Jarden, Field Arithmetic, Springer-Verlag, Heidelberg, 1986. MR 89b:12010
  • [8] G. Harder, Eine Bemerkung zum schwachen Approximationssatz, Archive Math., 19(1968), 465-471. MR 39:2767
  • [9] S. Lang, Algebra, Addison-Wesley, 1965. MR 33:5416
  • [10] M. Mahdavi-Hezavehi, M.G. Mahmudi and S. Yasamin, Finitely generated subnormal subgroups of $GL_n(D)$ are central, J. Algebra, 225 (2000), 517-521. MR 2000j:20097
  • [11] R. Pink, The Mumford-Tate conjecture for Drinfeld-modules, Publ. Res. Inst. Math. Sci., 33(1997), 393-425. MR 98f:11062
  • [12] V.P. Platonov, Dieudonné's conjecture and the nonsurjectivity on $k$-points of coverings of algebraic groups, Soviet Math. Dokl., 216 (1974), 986-989. MR 50:7368
  • [13] V.P. Platonov and A.S. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Math. N 139, Academic Press, 1993. MR 95b:11039
  • [14] G. Prasad and A.S. Rapinchuk, Computation of the metaplectic kernel, Publ. Math. IHES, 84(1996), 90-187. MR 98i:22026
  • [15] A.S. Rapinchuk, Combinatorial theory of arithmetic groups, Preprint 20(420), 1990, Institute of Mathematics of the Academy of Sciences of BSSR (Minsk).
  • [16] A.S. Rapinchuk, Y. Segev and G.M. Seitz, Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable, preprint.
  • [17] J-P. Serre, Lie algebras and Lie groups, Lect. Notes Math., 1500, Springer-Verlag, 1992. MR 93h:17001

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Additional Information

Gopal Prasad
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Andrei S. Rapinchuk
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Received by editor(s): March 5, 2001
Published electronically: February 7, 2002
Communicated by: Rebecca Herb
Article copyright: © Copyright 2002 American Mathematical Society