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

DOI:
https://doi.org/10.1090/S0002-9939-02-06514-0

Published electronically:
February 7, 2002

MathSciNet review:
1896401

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for a reductive algebraic group over an infinite field the group of rational points does not contain any noncentral finitely generated normal subgroups.

**[1]**S. Akbari and M. Mahdavi-Hezavehi,*Normal subgroups of**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*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**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**-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

Email:
gprasad@math.lsa.umich.edu

**Andrei S. Rapinchuk**

Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Email:
asr3x@weyl.math.virginia.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06514-0

Received by editor(s):
March 5, 2001

Published electronically:
February 7, 2002

Communicated by:
Rebecca Herb

Article copyright:
© Copyright 2002
American Mathematical Society